COHOMOLOGY FOR QUANTUM GROUPS VIA THE GEOMETRY OF THE NULLCONE

C.P. BENDEL, D.K. NAKANO, B.J. PARSHALL, AND C. PILLEN

Abstract. Let ζ be a complex ℓth root of unity for an odd integer ℓ > 1. For any complex simple Lie algebra 𝔤, let u_{ζ} = u_{ζ}(𝔤) be the associated “small” quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U_{ζ} and as a quotient algebra of the De Concini–Kac quantum enveloping algebra U_{ζ}. It plays an important role in the representation theories of both U_{ζ} and U_{ζ} in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig’s Conjecture concerning the irreducible modules can only be formulated when p ≥ h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H^{•}(u_{ζ}, ℂ) of the small quantum group. When ℓ > h, this cohomology algebra has been calculated by Ginzburg and Kumar [GK]. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of 𝔤. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra u in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if M is a finite dimensional u_{ζ}-module, then H^{•}(u_{ζ}, M) is a finitely generated H^{•}(u_{ζ}, ℂ)-module, and we obtain new results on the theory of support varieties for u_{ζ}.

1. Introduction

1.1. Let 𝔤_{F} be a finite dimensional, restricted Lie algebra (as defined by Jacobson) over an algebraically closed field F of positive characteristic p, with restriction map x ↦ x^{[p]}, x ∈ 𝔤_{F}. The restricted enveloping algebra u := u(𝔤_{F}) of 𝔤_{F} is a finite dimensional cocommutative Hopf algebra. In general, the cohomology algebra A := H^{•}(u, F) is difficult to compute. However, Suslin-Friedlander-Bendel [SFB1, SFB2] proved that the (algebraic) scheme Spec A is homeomorphic to the closed subvariety N_{1}(𝔤_{F}) := {x ∈ 𝔤_{F} : x^{[p]} = 0}. We call N_{1}(𝔤_{F}) the restricted nullcone of 𝔤_{F}; it is a closed subvariety of the full nullcone N (𝔤_{F}) which consists of all [p]-nilpotent elements in 𝔤_{F}.

When 𝔤_{F} is the Lie algebra of a reductive algebraic group G over F, the above results can be considerably sharpened. For example, if p > h (the Coxeter number of G), then H^{2•}(u, F) ≅ F [N_{1}(𝔤_{F})], the coordinate algebra of N_{1}(𝔤_{F}) (cf. Friedlander-Parshall [FP2]

Research of the first author was supported in part by NSF grant DMS-0400558. Research of the second author was supported in part by NSF grant DMS-0400548. Research of the third author was supported in part by NSF grant DMS-0400996. 2000 Mathematics Subject Classification. Primary 20C, 20G; Secondary 20J06, 20G10.

and Andersen-Jantzen [AJ]). In fact, the condition p ≥ h implies that N_{1}(𝔤_{F}) = N (𝔤_{F}). However, when p ≤ h, there is no known calculation of H^{•}(u, F) (apart from some small rank cases). For all primes, N_{1}(𝔤_{F}) is an irreducible variety [NPV, (6.3.1) Cor.], [UGA2, Thm. 4.2] and is the closure of a G-orbit. These orbits have been determined in [CLNP], [UGA2, Thm. 4.2].

Now let 𝔤 = 𝔤_{ℂ} be a complex simple Lie algebra, and let U_{ζ} = U_{ζ}(𝔤) be the quantum enveloping algebra (Lusztig form) associated to 𝔤 at a primitive lth root of unity ζ ∈ ℂ. We regard U_{ζ} as an algebra over ℂ obtained by base change from the quantum enveloping algebra over the cyclotomic field ℚ(ζ). Here l > 1 is an odd integer, not divisible by 3 when 𝔤 has type G_{2}. The representation theory of U_{ζ}, when l = p, models or approximates the representation theory of G. For example, part of the “Lusztig program” to determine the characters of the irreducible G-modules amounts to showing, when l = p ≥ h, that the characters of the irreducible G-modules having high weights in the so-called Jantzen region coincide with the characters of the analogous irreducible modules for U_{ζ}. This result has been proved by Andersen-Jantzen-Soergel [AJS] for large p where an effective lower bound is unknown.

For all l, Lusztig [L] has also introduced an analog, denoted u_{ζ} := u_{ζ}(𝔤), of the restricted enveloping algebra u. Like u, u_{ζ} is a finite dimensional Hopf algebra, though it is in general not cocommutative. It plays a role in the representation theory of U_{ζ} much like that played by u in the representation theory of G. In particular, for l > h, Ginzburg-Kumar [GK] have calculated the cohomology algebra H^{•}(u_{ζ}, ℂ). By exact analogy with the cohomology of u, they prove that H^{2•}(u_{ζ}, ℂ) ≅ ℂ[N (𝔤)], the coordinate algebra of the nullcone N (𝔤) consisting of nilpotent elements in 𝔤. Recently, Arkhipov-Bezrukavnikov-Ginzburg [ABG, §1.4], taking [GK] as a method to pass from the representation theory of quantum groups to the geometry of the nullcone, provide a proof of Lusztig’s character formula for U_{ζ} when l > h. These important connections have made the small quantum group u_{ζ} an object of significant interest (cf. [ABBGM], [AG], [Be], [Lac1, Lac2]).

This paper presents new results on the cohomology of u_{ζ}. In particular, we compute the cohomology algebra H^{•}(u_{ζ}, ℂ) in the remaining cases when l ≤ h. Our results are explicitly described in Section 1.3 below. We prove in Sections 2–5 that H^{2•+1}(u_{ζ}, ℂ) = 0, while H^{2•}(u_{ζ}, ℂ) is isomorphic to, in most cases, the coordinate algebra of an explicitly described closed subvariety N (Φ_{0}) of N (𝔤) (constructed in a similar way to the variety N_{1}(𝔤_{F}) discussed above). In Section 2 we rigorously develop and present new results on the cohomology theory of parabolic subalgebras for quantum groups. The application of powerful tools from complex geometry represent at least one advantage that the quantum enveloping algebra situation has over that in positive characteristic. In Sections 3 and 5, we demonstrate how the Grauert-Riemenschneider theorem and the normality of the orbit closures which equal N (Φ_{0}) play a vital role in carrying out our cohomology calculations.

In Section 6, we prove that R := H^{•}(u_{ζ}, ℂ) is a finitely generated ℂ-algebra and if M is a finite dimensional u_{ζ}-module, then H^{•}(u_{ζ}, M) is finitely generated over R. Section 7 adapts some of our methods to the cohomology algebra H^{•}(u, F) in positive characteristic, making some ad hoc computations in [AJ] more transparent. In particular, we can identify key vanishing results, known over ℂ, and as yet unproved in positive characteristic, which would be sufficient to extend the calculations to positive characteristic p.

Finally, by building on results in Section 6, we define support varieties in the quantum setting in Section 8 and exhibit some new calculations on support varieties. The theory of support varieties for the restricted Lie algebra 𝔤_{F} attached to a reductive group G in positive characteristic provides evidence for beautiful connections between the representation theory of G and the structure and geometry of the restricted nullcone N_{1}(𝔤_{F}) ([NPV],[CLNP], [UGA1], [UGA2]). The results of this paper strongly reinforce this expectation (as do those in [ABG] mentioned above).

1.2. Notation. Let Φ be a finite, irreducible root system. Fix a set Π = {α_{1}, ⋯, α_{n}} of simple roots (labelled in the standard way [Bo, Appendix]), and let Φ^{+} (resp. Φ^{−}) be the corresponding set of positive (resp. negative) roots. Write Q = Q(Φ) := ℤΦ (the root lattice) and Q^{+} = Q^{+}(Φ) = ℕΦ^{+}. The set Φ spans a real vector space 𝔼 with positive definite inner product ⟨u, v⟩, u, v ∈ 𝔼, adjusted so that ⟨α, α⟩ = 2 if α ∈ Φ is a short root. Thus, if α ∈ Φ is a long root, then ⟨α, α⟩ = 4 (resp. 6) when Φ has type B_{n}, C_{n}, F_{4} (resp. G_{2}). For α ∈ Φ, put d_{α} := \frac{⟨α,α⟩}{2} ∈ \{1, 2, 3\}.

For α ∈ Φ, write α^{∨} = \frac{2}{⟨α,α⟩}α; thus, Φ^{∨} = \{α^{∨} : α ∈ Φ\} is the coroot system defined by Φ. For 1 ≤ i, j ≤ n, put c_{i,j} = ⟨α_{j}, α_{i}^{∨}⟩ ∈ ℕ. Then C := [c_{i,j}] is the Cartan matrix of Φ and DC is a symmetric matrix, setting D = diag[d_{α_{1}}, ⋯, d_{α_{n}}]. Let X_{+} ⊂ 𝔼 be the set of dominant weights, consisting of all ϖ ∈ 𝔼 satisfying ⟨ϖ, α_{i}^{∨}⟩ ∈ ℕ, i = 1, ⋯, n. Define the fundamental dominant weights ϖ_{1}, ⋯, ϖ_{n} by ⟨ϖ_{i}, α_{j}^{∨}⟩ = δ_{i,j}, so X_{+} = ℕϖ_{1} ⊕ ⋯ ⊕ ℕϖ_{n}. Put X = ℤϖ_{1} ⊕⋯⊕ℤϖ_{n} (the weight lattice). The group X is partially ordered: λ ≥ μ ⟺ λ − μ ∈ Q^{+}. For convenience, we will occasionally use the notation ϖ_{0} := 0. The Weyl group W is the finite group of orthogonal transformations of 𝔼 generated by reflections s_{α} : 𝔼 → 𝔼, u ↦ u − ⟨u, α^{∨}⟩α, α ∈ Φ. If S := {s_{α_{1}}, ⋯, s_{α_{n}}}, then (W, S) is a Coxeter system. Let ℓ : W → ℕ be the length function on W defined by the set S of simple reflections. If l is a positive integer, the affine Weyl group W_{l} = W ⋉ lQ is a group of affine transformations of 𝔼. If α ∈ Φ, m ∈ ℤ, let s_{α,m} : 𝔼 → 𝔼 be defined by s_{α,m}(u) = u − (⟨u, α^{∨}⟩ − ml)α. Then s_{α,m} ∈ W_{l}. If α_{0} ∈ Φ denotes the maximal short root, (W_{l}, S_{l}) is a Coxeter system, setting S_{l} := {s_{α_{1}}, ⋯, s_{α_{n}}, s_{α_{0},1}}. The extended affine Weyl group W̃_{l} is obtained by putting W̃_{l} = W ⋉lX. Although it need not be a Coxeter group, it contains W_{l} as a normal subgroup satisfying W̃_{l}/W_{l} ≅ X/Q. Denote the quotient map W̃_{l} → W ≅ W̃_{l}/lX by w ↦ \overline{w}. The Coxeter number of Φ is h = ⟨ρ, α_{0}^{∨}⟩ + 1 where ρ := ϖ_{1} + ⋯ + ϖ_{n} = \frac{1}{2} ∑_{α∈Φ^{+}} α is the Weyl weight. For w ∈ W_{l}, x ∈ 𝔼, we put w · x := w(x + ρ) − ρ. For J ⊆ Π, let Φ_{J} = Φ ∩ ℤJ be the root system of Φ generated by J.

Suppose that A and B are augmented algebras (over some common field) and that A is a subalgebra of B. Then A is called normal in B if BA_{+} = A_{+}B where A_{+} denotes the augmentation ideal of A. In this situation, put B//A := B/I, where I := BA_{+}.

1.3. Main results. Let G be the simple, simply connected algebraic group over ℂ with Lie algebra 𝔤 and root system Φ. Let 𝔱 be a fixed maximal toral subalgebra of 𝔤. Given α ∈ Φ, let 𝔤_{α} be the α-root space. Put 𝔟^{+} = 𝔱 ⊕ ⨁_{α∈Φ^{+}} 𝔤_{α} (the positive Borel subalgebra), and 𝔟 = 𝔱 ⊕ ⨁_{α∈Φ^{−}} 𝔤_{α} (the opposite Borel subalgebra).

For J ⊆ Π, let 𝔩_{J} = 𝔱 ⊕ ⨁_{α∈Φ_{J}} 𝔤_{α} be the Levi subalgebra containing 𝔱 and having root system Φ_{J}. Then 𝔭_{J} = 𝔩_{J} ⊕ 𝔲_{J} ⊇ 𝔟 is a parabolic subalgebra of 𝔤, where 𝔲_{J} = ⨁_{α∈Φ^{−}\Φ_{J}^{−}} 𝔤_{α} is the nilpotent radical of 𝔭_{J}. Let N (Φ_{J}) := G · 𝔲_{J}, where G acts on 𝔲_{J} ⊂ 𝔤 via the adjoint action.

Assumption 1.3.1. Throughout this paper let l > 1 be a fixed odd positive integer. If Φ has type G_{2}, then we assume that 3 does not divide l. In addition, we assume that l is not a bad prime for Φ. See Section 3.1 for the definition of a bad prime.

Let

Φ_{0} = Φ_{0,l} := {α ∈ Φ | ⟨ρ, α^{∨}⟩ ≡ 0 mod l}.

The restrictions on l guarantee (see Theorem 3.5.1) that there exists J ⊆ Π and w ∈ W such that

w(Φ_{0}) = Φ_{J}.

Set N (Φ_{0}) := N (Φ_{J}). If w′ ∈ W and J′ ⊆ Π also satisfy w′(Φ_{0}) = Φ_{J′}, the Johnston-Richardson Theorem [JR] guarantees that N (Φ_{0}) = N (Φ_{J′}), so that N (Φ_{0}) is well-defined.

Let ζ be a primitive lth root of unity, and let U_{ζ} := U_{ζ}(𝔤), u_{ζ} := u_{ζ}(𝔤) be the quantum enveloping algebras attached to the complex simple Lie algebra 𝔤 with root system Φ. These algebras were introduced in Section 1.1, and will be developed more fully in Section 2.2 below. The first major result of the paper is the computation of the cohomology algebra H^{•}(u_{ζ}, ℂ). This result extends the main theorem of Ginzburg-Kumar [GK, Main Thm.] which required that l > h.

Theorem 1.3.2. Let l be as in Assumption 1.3.1, and choose w ∈ W, J ⊆ Π such that w(Φ_{0}) = Φ_{J}.

(a) We have H^{2•+1}(u_{ζ}, ℂ) = 0, i. e., the cohomology of u_{ζ} vanishes in all odd degrees.

(b) The even degree cohomology algebra H^{2•}(u_{ζ}, ℂ) is described below.

(i) Suppose that l ∤ n + 1 when Φ is of type A_{n} and l ≠ 9 when Φ is of type E_{6}.

Then

H^{2•}(u_{ζ}, ℂ) ≅ ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[G ×_{P_{J}} 𝔲_{J}].

If we assume further that l ≠ 7, 9 when Φ is of type E_{8}, then

H^{2•}(u_{ζ}, ℂ) ≅ ℂ[N (Φ_{0})].

(ii) If Φ is of type A_{n} and l | n + 1 with n + 1 = l(m + 1), then

H^{2•}(u_{ζ}, ℂ) ≅ ind_{P_{J}}^{G} (⨁_{t=0}^{l−1} S^{\frac{2•−(m+1)t(l−t)}{2}} (𝔲_{J}^{∗}) ⊗ ϖ_{t(m+1)}) ,

where ϖ_{0} = 0. (iii) If Φ is of type E_{6} and l = 9, one can take J = {α_{4}}. Then

H^{2•}(u_{ζ}, ℂ) ≅ ind_{P_{J}}^{G} (S^{•}(𝔲_{J}^{∗}) ⊕ (S^{\frac{2•−12}{2}} (𝔲_{J}^{∗}) ⊗ ϖ_{1}) ⊕ (S^{\frac{2•−12}{2}} (𝔲_{J}^{∗}) ⊗ ϖ_{6})) .

This theorem will be proved in Section 5.5. In order to state our second main result, we need to make a very minor additional restriction on l.

Assumption 1.3.3. Let l > 1 be a fixed odd positive integer satifying Assumption 1.3.1. Moreover assume that when Φ is of type B_{n} or C_{n} that l > 3.

We also have the following finite generation result, which is proved in Section 6.

Theorem 1.3.4. Let l be as in Assumption 1.3.3.

(a) The algebra H^{•}(u_{ζ}, ℂ) is a finitely generated ℂ-algebra.

(b) For any finite dimensional u_{ζ}(𝔤)-module M, H^{•}(u_{ζ}, M) is finitely generated as a module for H^{•}(u_{ζ}, ℂ) (where the action is described in [PW, Rem. 5.3, Appendix]).

Theorem 1.3.4 implies that, given a finite dimensional u_{ζ}-module M, the support variety V_{𝔤}(M) can be defined as the maximal ideal spectrum Maxspec(R/J_{M}), letting J_{M} be the annihilator in R := H^{•}(u_{ζ}(𝔤), ℂ) for its natural action on H^{•}(u_{ζ}(𝔤), M ⊗ M^{∗}) (cf. [PW, §5]). A third major result gives a computation of the support varieties of Weyl modules.

Theorem 1.3.5. Let l be as in Assumption 1.3.3. Let λ ∈ X_{+} and let H_{ζ}^{0}(λ) be the induced (or Weyl) module for U_{ζ} of high weight λ. Suppose that (l, p) = 1 for any bad prime p of Φ. Then there exists a subset J ⊆ Π such that

V_{𝔤}(H_{ζ}^{0}(λ)) = G · 𝔲_{J}.

This result is proved in Section 8.2, where the subset J is explicitly identified in terms of the stabilizer set Φ_{λ,l} (cf. (3.1.1)). In particular, the theorem shows that the support variety of H_{ζ}^{0}(λ) is the closure of a Richardson orbit in N (𝔤). In Section 8.3, we show that even when l > h, if p | l where p is a bad prime for Φ, then V_{𝔤}(H_{ζ}^{0}(λ)) need not have the form G · 𝔲_{J}, contrary to the calculations given in [Ost].

1.4. Acknowledgements. The authors would like to acknowledge Bill Graham, Shrawan Kumar and Eric Sommers for several useful discussions at various stages of this project. We also thank Jesper Funch Thomsen for indicating how recent results of Christophersen can be used to compute G_{1}-cohomology for the group E_{6}, and two undergraduate students Alloys J. Samz and Sara Wood at the University of Wisconsin-Stout for their assistance with the MAGMA computations used in this paper. Finally, the first, second, and last authors acknowledge the hospitality and support of the Department of Mathematics at the University of Virginia in Spring 2006 and Spring 2007 during work on this project.

2. Quantum groups, actions, and cohomology

2.1. Listings. For this subsection let Φ be an arbitrary root system with associated Weyl group W. Let w_{0} ∈ W be the longest word and N = |Φ^{+}|. A reduced expression w_{0} = s_{β_{1}} ⋯ s_{β_{N}} (β_{i} ∈ Π) provides a listing

γ_{1} = β_{1}, γ_{2} = s_{β_{1}}(β_{2}), ⋯, γ_{N} = s_{β_{1}} ⋯ s_{β_{N−1}}(β_{N})

of Φ^{+}. Define a linear ordering on Φ^{+} by setting, for β, δ ∈ Φ^{+},

β ≺ δ ⟺ β = γ_{i}, δ = γ_{j} with i < j.

This ordering depends on the reduced expression chosen for w_{0}.

Now let J ⊆ Π, and put Φ_{J} = Φ ∩ ℤJ, the subroot system Φ_{J} generated by J. Set W_{J} = ⟨s_{α} : α ∈ J⟩, the Weyl group of Φ_{J}, let w_{0,J} ∈ W_{J} be the longest word in W_{J}, and let w_{J} := w_{0,J}w_{0}. Observe that w_{0} = w_{0,J}w_{J} satisfies ℓ(w_{0}) = ℓ(w_{0,J}) + ℓ(w_{J}). It will often be useful to fix a reduced expression for w_{0} by first choosing one for w_{0,J} and then a reduced expression for w_{J}. If |Φ_{J}^{+}| = M, then γ_{1} ≺ ⋯ ≺ γ_{M} lists the positive roots Φ_{J}^{+}, while γ_{M+1} ≺ ⋯ ≺ γ_{N} lists the remaining positive roots (those in Φ^{+}\Φ_{J}^{+}). As above, this ordering depends upon the choice of reduced expressions.

Since w_{0,J} is the longest word in W_{J}, for β ∈ J,

ℓ(w_{0,J}s_{β}) < ℓ(w_{0,J}).

Further, w_{J} is a distinguished right coset representative for W_{J} in W, so

(2.1.1)  ℓ(s_{β}w_{J}) > ℓ(w_{J})  and hence  ℓ(s_{β}w_{J}) = ℓ(w_{J}) + 1.

2.2. Quantum enveloping algebras. We return to our assumptions on Φ as in Section 1.2. Let A = ℚ[q, q^{−1}] be the algebra of Laurent polynomials in an indeterminate q. Then A has fraction field ℚ(q). We continue to restrict the integer l according to Assumption 1.3.1. Let ζ = \sqrt[l]{1} ∈ ℂ be a primitive lth root of unity and let k = ℚ(ζ) be the cyclotomic field generated by ζ over ℚ. We will regard k as an A-algebra by means of the homomorphism ℚ[q, q^{−1}] → k, q ↦ ζ.

The quantized enveloping algebra 𝕌_{q} = 𝕌_{q}(𝔤) of 𝔤 is the ℚ(q)-algebra with generators E_{α}, K_{α}^{±1}, F_{α}, α ∈ Π, and relations (R1)—(R6) listed in [Jan2, (4.3)] (whose notation we will generally follow, unless otherwise indicated). The Hopf algebra structure on 𝕌_{q}(𝔤) has comultiplication Δ, counit ϵ, and antipode S explicitly described in [Jan2, Ch. 4].

Let 𝕌_{q}^{+} denote the subalgebra of 𝕌_{q} generated by the E_{α} (α ∈ Π), 𝕌_{q}^{−} denote the subalgebra of 𝕌_{q} generated by the F_{α} (α ∈ Π), and 𝕌_{q}^{0} denote the subalgebra of 𝕌_{q} generated by the K_{α}^{±1} (α ∈ Π). Multiplication gives isomorphisms of vector spaces

𝕌_{q}^{+} ⊗ 𝕌_{q}^{0} ⊗ 𝕌_{q}^{−} →^{∼} 𝕌_{q} ←^{∼} 𝕌_{q}^{−} ⊗ 𝕌_{q}^{0} ⊗ 𝕌_{q}^{+}.

We will use analogous {+, −, 0}-notation upon specialization to ζ below, and the isomorphisms hold there as well.

The algebra 𝕌_{q} has two A-forms, U_{q}^{A} (due to Lusztig) and U_{q}^{A} (due to De Concini and Kac). After base change to k, they play roles analogous to the hyperalgebra of a reductive group and the universal enveloping algebra of its Lie algebra, respectively.

For a positive integer m and α ∈ Π, let E_{α}^{(m)} = \frac{E_{α}^{m}}{[m]_{α}^{!}} . Here [m]^{!} = [m][m − 1] ⋯ [1] with

[i] = \frac{q^{i}−q^{−i}}{q−q^{−1}} , while, in general, given f ∈ A and α ∈ Π, f_{α} is obtained from f by replacing

q by q_{α} := q^{d_{α}}. Defining F_{α}^{(m)} similarly, let U_{q}^{A} be the A-subalgebra of 𝕌_{q} generated by the E_{α}^{(m)}, F_{α}^{(m)}, K_{α}^{±1} for m ∈ ℕ, α ∈ Π.

Put

U_{ζ} = U_{ζ}(𝔤) := k ⊗_{A} U_{q}^{A}/⟨1 ⊗ K_{α}^{l} − 1 ⊗ 1, α ∈ Π⟩,

where ⟨⋯ ⟩ means “ideal generated by ⋯ ”. The Hopf algebra structure on 𝕌_{q} induces one on U_{q}^{A}, and then one on U_{ζ}. By abuse of notation, let E_{α}^{(m)}, etc., α ∈ Π, also denote the elements 1 ⊗ E_{α}^{(m)}, etc. in U_{ζ} (note that E_{α}^{(1)} = E_{α}). Then E_{α}^{l} = F_{α}^{l} = 0 and K_{α}^{l} = 1 in U_{ζ}. The elements E_{α}, K_{α}, F_{α}, α ∈ Π, generate a Hopf subalgebra, denoted u_{ζ} = u_{ζ}(𝔤), of U_{ζ}.

Next, define U_{q}^{A} to be the A-subalgebra of 𝕌_{q} generated by the E_{α}, F_{α}, K_{α}^{±1}. There is an inclusion of A-forms: U_{q}^{A} ⊆ U_{q}^{A}. Then set U_{k} := k ⊗_{A} U_{q}^{A}. Finally, put

U_{ζ} = U_{ζ}(𝔤) = U_{k}/⟨1 ⊗ K_{α}^{l} − 1 ⊗ 1, α ∈ Π⟩.

The (Hopf) algebra U_{ζ} has a central subalgebra Z such that u_{ζ} ≅ U_{ζ}//Z; see [DK] for more details. An explicit description of Z will be given in Section 2.7.

All the U_{ζ}-modules M considered in this paper will be assumed to be integrable and type 1. Usually, this will be assumed without mention, but occasionally we repeat it for emphasis. In other words, each E_{i}, F_{j} acts locally nilpotently on M. In addition, M = ⨁_{λ∈X} M_{λ}, where M_{λ} consists of all v ∈ M such that [ \begin{matrix}K_{α};m \\ n \end{matrix}] v = [ \begin{matrix}⟨λ,α⟩+m \\ n \end{matrix} ]_{q=ζ^{dα}} v. See [Jan2] for an explanation of the [ \begin{matrix}K_{α};m \\ n \end{matrix}].

2.3. Connections with algebraic groups. The Frobenius morphism is a surjective (Hopf) algebra homomorphism

Fr : U_{ζ} ↠ 𝕌(𝔤),

where 𝕌(𝔤) is the universal enveloping algebra of 𝔤. If I is the augmentation ideal of u_{ζ}, then Ker Fr = IU_{ζ} = U_{ζ}I. In other words, u_{ζ} is normal in U_{ζ} and U_{ζ}//u_{ζ} ≅ 𝕌(𝔤). By means of Fr, modules for the Lie algebra 𝔤 can be “pulled back” to give modules for U_{ζ}: given a 𝔤-module M, M^{[1]} = Fr^{∗}M denotes the U_{ζ} module obtained from M by making x ∈ U_{ζ} act as a linear transformation on M by Fr(x). If M is a locally finite 𝕌(𝔤)-module (e. g., finite dimensional), then M^{[1]} is an integrable, type 1 U_{ζ}-module. Conversely, a U_{ζ}-module N on which u_{ζ} acts trivially can be made into a 𝕌(𝔤)-module via the Frobenius map, so that N = M^{[1]} for a locally finite 𝕌(𝔤)-module M.

Let G be the complex simple, simply connected algebraic group having Lie algebra 𝔤. The category of locally finite 𝕌(𝔤)-modules is naturally isomorphic to the category of rational G-modules.

The set X_{+} of dominant weights indexes the irreducible modules for U_{ζ}. Given λ ∈ X_{+}, let L_{ζ}(λ) be the irreducible U_{ζ}-module of high weight λ. Also, let L(λ) be the irreducible rational G-module of high weight λ. We may identify L(λ) with the finite dimensional irreducible 𝕌(𝔤)-module of high weight λ.

The integral group algebra ℤX has basis denoted e(μ), μ ∈ X. Given a finite dimensional U_{ζ}-module M,

ch M = ∑_{μ∈X} dim M_{μ} e(μ) ∈ ℤX

denotes its (formal) character, where M_{μ} is the dimension of the μ-weight space for the action of U_{ζ}^{0} on M. Sometimes for emphasis we write ch_{ζ}M for ch M.

Similarly, if M is a finite dimensional rational G-module (resp. 𝕌(𝔤)-module), let ch M ∈ ℤX be its formal character with respect to the fixed maximal torus T (resp. Cartan subalgebra 𝔥 = Lie(T)).

2.4. Root vectors and PBW-basis. For each α ∈ Π, Lusztig has defined an automorphism T_{α} of 𝕌_{q}; here we will follow [Jan2, Ch. 8], where the reader can find more details. If s = s_{α} ∈ W is the simple reflection defined by α, we write T_{s} := T_{α}. Given any w ∈ W, let w = s_{β_{1}}s_{β_{2}} ⋯ s_{β_{n}} be a reduced expression. Then define T_{w} := T_{β_{1}} ⋯ T_{β_{n}} ∈ Aut(𝕌_{q}). The automorphism T_{w} is independent of the reduced expression of w. In other words, the automorphisms T_{α} extend to an action of the braid group of W on 𝕌_{q}.

Now let J ⊆ Π and fix a reduced expression w_{0} = s_{β_{1}} ⋯ s_{β_{N}} that begins with a reduced expression for w_{0,J} as in Section 2.1. If w_{0,J} = s_{β_{1}} ⋯ s_{β_{M}}, s_{β_{M+1}} ⋯ s_{β_{N}} is a reduced expression for w_{J} = w_{0,J}w_{0}. There is an ordering γ_{1} ≺ γ_{2} ≺ ⋯ ≺ γ_{N} of the positive roots, where γ_{i} = s_{β_{1}} ⋯ s_{β_{i−1}}(β_{i}). For γ = γ_{i} ∈ Φ^{+}, the “root vector” E_{γ} ∈ 𝕌_{q}^{+} is defined by

E_{γ} = E_{γ_{i}} := T_{s_{β_{1}}···s_{β_{i−1}}} (E_{β_{i}}) = T_{β_{1}} ⋯ T_{β_{i−1}}(E_{β_{i}}).

If γ is simple, the “new” E_{γ} agrees with the original generator E_{γ}. More generally, E_{γ} has weight γ. Similarly,

F_{γ} = F_{γ_{i}} := T_{s_{β_{1}}···s_{β_{i−1}}} (F_{β_{i}}) = T_{β_{1}} ⋯ T_{β_{i−1}}(F_{β_{i}}),

a root vector of weight −γ.

The subalgebra 𝕌_{q}^{+} (resp. 𝕌_{q}^{−}) has a PBW-like basis consisting of all monomials E_{γ_{1}}^{a_{1}} ⋯ E_{γ_{N}}^{a_{N}} (resp. F_{γ_{1}}^{a_{1}} ⋯ F_{γ_{N}}^{a_{N}}), a_{1}, ⋯, a_{n} ∈ ℕ. Using divided powers when necessary, one obtains PBWbases for the specialized quantum groups U_{ζ}^{+}, U_{ζ}^{−}, u_{ζ}^{+}, u_{ζ}^{−}, U_{ζ}^{+}, and U_{ζ}^{−}. The automorphisms T_{w} induce automorphisms on U_{q}^{A} and hence on U_{ζ}.

The monomial bases satisfy a key “commutativity” property originally observed by Levendorskiĭ and Soibelman. Here E_{γ_{i}}^{0} = 1 and F_{γ_{i}}^{0} = 1.

Lemma 2.4.1. ([DP, Thm. 9.3], [LS]) In 𝕌_{q}, we have for i < j

(a) E_{γ_{i}}E_{γ_{j}} = q^{⟨γ_{i},γ_{j}⟩}E_{γ_{j}}E_{γ_{i}} + (∗) where (∗) is an A-linear combination of monomials E_{γ_{1}}^{a_{1}} ⋯ E_{γ_{N}}^{a_{N}}, with a_{s} = 0 unless i < s < j;

(b) F_{γ_{i}}F_{γ_{j}} = q^{⟨γ_{i},γ_{j}⟩}F_{γ_{j}}F_{γ_{i}} + (∗) where (∗) is an A-linear combination of monomials F_{γ_{1}}^{a_{1}} ⋯ F_{γ_{N}}^{a_{N}}, with a_{s} = 0 unless i < s < j.

2.5. Levi and parabolic subalgebras. Given a subset J ⊆ Π, consider the Levi and parabolic Lie subalgebras 𝔩_{J} and 𝔭_{J} = 𝔩_{J} ⊕ 𝔲_{J} of 𝔤. We denote the respective universal enveloping algebras by 𝕌(𝔩_{J}) and 𝕌(𝔭_{J}). One can naturally define corresponding quantized enveloping algebras 𝕌_{q}(𝔩_{J}) and 𝕌_{q}(𝔭_{J}). As subalgebras of 𝕌_{q}, 𝕌_{q}(𝔩_{J}) is the subalgebra generated by {E_{α}, F_{α} : α ∈ J} ∪ {K_{α}^{±1} : α ∈ Π}, and 𝕌_{q}(𝔭_{J}) is the subalgebra generated by {E_{α} : α ∈ J} ∪ {F_{α}, K_{α}^{±1} : α ∈ Π}. For example, if J = ∅, then 𝔩_{J} = 𝔱, 𝔭_{J} = 𝔟, 𝕌_{q}(𝔩_{J}) = 𝕌_{q}^{0}, and 𝕌_{q}(𝔭_{J}) = 𝕌_{q}(𝔟) = 𝕌_{q}^{−} · 𝕌_{q}^{0}. Specializing gives the subalgebras U_{ζ}(𝔩_{J}), U_{ζ}(𝔭_{J}), u_{ζ}(𝔩_{J}), u_{ζ}(𝔭_{J}) of U_{ζ}, and U_{ζ}(𝔩_{J}) and U_{ζ}(𝔭_{J}) of U_{ζ}.

If we denote by 𝔭_{J}^{+} = 𝔩_{J} ⊕ 𝔲_{J}^{+} the opposite parabolic subalgebra (containing the positive Borel subalgebra 𝔟^{+}), then one can analogously consider 𝕌_{q}(𝔭_{J}^{+}).

2.6. The subalgebra 𝕌_{q}(𝔲_{J}). In this section, we define a subalgebra 𝕌_{q}(𝔲_{J}) which plays a role analogous to that of 𝕌(𝔲_{J}) ⊂ 𝕌(𝔭_{J}). As above, choose a reduced expression for w_{0} (beginning with one for w_{0,J}). Define 𝕌_{q}(𝔲_{J}) to be the subspace of 𝕌_{q}^{−} spanned by the F_{γ_{M+1}}^{a_{M+1}} ⋯ F_{γ_{N}}^{a_{N}}, a_{i} ∈ ℕ. By Lemma 2.4.1, 𝕌_{q}(𝔲_{J}) is a subalgebra of 𝕌_{q}^{−}. One can also verify directly that 𝕌_{q}(𝔲_{J}) is independent of the choice of reduced expression for w_{0}. However, this follows from a more general set-up which will be useful for other purposes as well.

Given any v ∈ W, we define as in [Jan2, 8.24] a subspace U^{−}[v] ⊂ 𝕌_{q}^{−} (resp. U^{+}[v] ⊂ 𝕌_{q}^{+}) as follows. Choose a reduced expression v = s_{η_{1}}s_{η_{2}} ⋯ s_{η_{t}}. For 1 ≤ i ≤ t, set f_{i} = T_{s_{η_{1}}s_{η_{2}}···s_{η_{i−1}}} (F_{η_{i}}) (resp. e_{i} = T_{s_{η_{1}}s_{η_{2}}···s_{η_{i−1}}} (E_{η_{i}})). The f_{i} (resp. e_{i}) are in some sense “root vectors” like those defined earlier. Then U^{−}[v] (resp. U^{+}[v]) is defined to be the span of all monomials of the form f_{1}^{a_{1}}f_{2}^{a_{2}} ⋯ f_{t}^{a_{t}} (resp. e_{1}^{a_{1}}e_{2}^{a_{2}} ⋯ e_{t}^{a_{t}}) for a_{i} ≥ 0. By [DKP, 2.2], U^{−}[v] (resp. U^{+}[v]) is an algebra and moreover is independent of the choice of reduced expression for v.

Since 𝕌_{q}(𝔲_{J}) = T_{w_{0,J}} (U^{−}[w_{J}]), 𝕌_{q}(𝔲_{J}) is a subalgebra of 𝕌_{q}(𝔭_{J}) ⊂ 𝕌_{q}, independent of the reduced expression for w_{J}. Further, since the automorphism T_{w_{0,J}} is also independent of the choice of reduced expression for w_{0,J}, the algebra 𝕌_{q}(𝔲_{J}) depends only on J, not on our above choices of reduced expressions. Following the procedure in Section 2.2, the algebra 𝕌_{q}(𝔲_{J}) can be specialized to give algebras U_{ζ}(𝔲_{J}), u_{ζ}(𝔲_{J}), and U_{ζ}(𝔲_{J}). For example, U_{ζ}(𝔲_{J}) is the subalgebra of U_{ζ}^{−} spanned by F_{γ_{M+1}}^{(a_{M+1})} ⋯ F_{γ_{N}}^{(a_{N})}, a_{i} ∈ ℕ.

Similarly one can define a subalgebra 𝕌_{q}(𝔲_{J}^{+}) ⊂ 𝕌_{q}(𝔭_{J}), and the corresponding specializations. Then 𝕌_{q}(𝔲_{J}^{+}) = T_{w_{0,J}} (U^{+}[w_{J}]). When J = ∅, 𝕌_{q}(𝔲_{J}) = 𝕌_{q}^{−} and 𝕌_{q}(𝔲_{J}^{+}) = 𝕌_{q}^{+}.

2.7. Adjoint Action. Given a Hopf algebra A, the adjoint action of A is defined by setting, for x, y ∈ A, Ad(x)(y) = ∑ x_{1}yS(x_{2}) where Δ(x) = ∑ x_{1} ⊗ x_{2} is the comultiplication and S is the antipode. We consider, in particular, the case A = 𝕌_{q}, where we record the formulas here on generators: (for m ∈ 𝕌_{q})

(2.7.1)  \begin{cases}Ad(E_{α})(m) & = E_{α}m − K_{α}mK_{α}^{−1}E_{α} \\ Ad(K_{α}^{±1})(m) & = K_{α}^{±1}mK_{α}^{∓1} \\ Ad(F_{α})(m) & = (F_{α}m − mF_{α})K_{α}. \end{cases}

By twisting the Hopf structure by the algebra involution ω of 𝕌_{q} given by ω(E_{α}) = F_{α}, ω(F_{α}) = E_{α}, and ω(K_{α}) = K_{α}^{−1}, we obtain another Hopf structure on 𝕌_{q} (cf. [Jan2, 3.8]). This procedure then gives an alternate adjoint action \overline{Ad}, which satisfies:

(2.7.2)  \begin{cases}\overline{Ad}(E_{α})(m) & = (E_{α}m − mE_{α})K_{α}^{−1} \\ \overline{Ad}(K_{α}^{±})(m) & = K_{α}^{±1}mK_{α}^{∓1} \\ \overline{Ad}(F_{α})(m) & = F_{α}m − K_{α}^{−1}mK_{α}F_{α}. \end{cases}

Proposition 2.7.1. The following stability results hold:

(a) The subalgebra 𝕌_{q}(𝔲_{J}^{+}) is stable under the Ad action of 𝕌_{q}(𝔭_{J}^{+}) on itself.

(b) The subalgebra 𝕌_{q}(𝔲_{J}) is stable under the \overline{Ad} action of 𝕌_{q}(𝔭_{J}) on itself.

Proof. We prove part (a). Part (b) follows from part (a) since, for x ∈ 𝕌_{q}(𝔭_{J}), \overline{Ad}(x) = ω ∘ Ad(ω(x)) ∘ ω. Since 𝕌_{q}(𝔭_{J}^{+}) is generated as an algebra by {E_{α}, K_{α}, K_{α}^{−1} : α ∈ Π} ∪ {F_{α} : α ∈ J}, it suffices to show that 𝕌_{q}(𝔲_{J}^{+}) is stable under the action of these elements. By (2.7.1), Ad(K_{α}) and Ad(K_{α}^{−1}) simply act by scalar multiplication on the weight spaces which span 𝕌_{q}(𝔲_{J}^{+}). Hence 𝕌_{q}(𝔲_{J}^{+}) is stable under Ad(K_{α}^{±1}). For α ∈ Φ^{+}\Φ_{J}^{+}, the stability under Ad(E_{α}) follows from the fact that 𝕌_{q}(𝔲_{J}^{+}) is an algebra. It remains to prove stability under Ad(F_{α}) and Ad(E_{α}) for α ∈ J. Fix α ∈ J and set β = −w_{0,J}(α) ∈ J.

By Section 2.6, 𝕌_{q}(𝔲_{J}^{+}) = T_{w_{0,J}} (U^{+}[w_{J}]). Given a reduced expression w_{J} = s_{η_{1}}s_{η_{2}} ⋯ s_{η_{t}}, U^{+}[w_{J}] is spanned by monomials of ordered root vectors:

E_{η_{1}}, T_{s_{η_{1}}} (E_{η_{2}}), …, T_{s_{η_{1}}···s_{η_{t−1}}} (E_{η_{t}}).

Since w_{J} = w_{0,J}w_{0}, w_{J}^{−1}(β) ∈ Π. Consider also U^{+}[w_{J}s_{w_{J}^{−1}(β)}]. Since w_{J}s_{w_{J}^{−1}(β)} = s_{β}w_{J}, by (2.1.1), ℓ(w_{J}s_{w_{J}^{−1}(β)}) = ℓ(w_{J})+1. Therefore, the ordered root vectors defining U^{+}[w_{J}s_{w_{J}^{−1}(β)}] are

E_{η_{1}}, T_{s_{η_{1}}} (E_{η_{2}}), …, T_{s_{η_{1}}···s_{η_{t−1}}} (E_{η_{t}}), T_{w_{J}} (E_{w_{J}^{−1}(β)}).

By [Jan2, Prop. 8.20] T_{w_{J}} (E_{w_{J}^{−1}(β)}) = E_{w_{J} (w_{J}^{−1}(β))} = E_{β}. Now U^{+}[w_{J}] is a subalgebra of U^{+}[w_{J}s_{w_{J}^{−1}(β)}] spanned by monomials not involving the last root vector E_{β}. Moreover, by Lemma 2.4.1, since E_{β} appears last in the ordering of root vectors in U^{+}[w_{J}s_{w_{J}^{−1}(β)}],

(2.7.3) if u ∈ U^{+}[w_{J}] ⊂ U^{+}[w_{J}s_{w_{J}^{−1}(β)}] is a monomial, then uE_{β} − q^{⟨wt(u),β⟩}E_{β}u ∈ U^{+}[w_{J}],

where wt(u) denotes the weight of u.

Now consider the subalgebra U^{+}[s_{β}w_{J}]. The ordered root vectors defining U^{+}[s_{β}w_{J}] are

E_{β}, T_{s_{β}}(E_{η_{1}}), T_{s_{β}}(T_{s_{η_{1}}} (E_{η_{2}})), …, T_{s_{β}}(T_{s_{η_{1}}···s_{η_{t−1}}} (E_{η_{t}})).

Then T_{s_{β}}(U^{+}[w_{J}]) is evidently a subalgebra of U^{+}[s_{β}w_{J}] spanned by monomials in all but the first root vector E_{β}. Moreover, by Lemma 2.4.1, since E_{β} occurs first in the root ordering, (2.7.4)

if u ∈ U^{+}[w_{J}] is a monomial, then E_{β}T_{s_{β}}(u) − q^{⟨wt(T_{s_{β}} (u)),β⟩}T_{s_{β}}(u)E_{β} ∈ T_{s_{β}}(U^{+}[w_{J}]).

We now show that Ad(F_{α}) preserves 𝕌_{q}(𝔲_{J}^{+}). Since α, β ∈ Π, by [Jan2, Prop. 8.20], F_{α} = T_{w_{0,J} s_{β}}(F_{β}). By [Jan2, 8.14(3)], F_{β} = −T_{s_{β}}(E_{β})K_{β}^{−1} = −T_{s_{β}}(E_{β}K_{β}). Combining these gives F_{α} = T_{w_{0,J} s_{β}}(−T_{s_{β}}(E_{β}K_{β})) = −T_{w_{0,J}} (E_{β}K_{β}).

Let T_{w_{0,J}} (u) for u ∈ U^{+}[w_{J}] be an arbitrary monomial element of 𝕌_{q}(𝔲_{J}^{+}). Then we have

Ad(F_{α})(T_{w_{0,J}} (u)) = (F_{α}T_{w_{0,J}} (u) − T_{w_{0,J}} (u)F_{α})K_{α}  by (2.7.1)
= (−T_{w_{0,J}} (E_{β}K_{β})T_{w_{0,J}} (u) + T_{w_{0,J}} (u)T_{w_{0,J}} (E_{β}K_{β}))K_{α} from above
= T_{w_{0,J}} ((uE_{β}K_{β} − E_{β}K_{β}u)K_{β}^{−1})  [Jan2, 8.18(3)]
= T_{w_{0,J}} (uE_{β} − E_{β}K_{β}uK_{β}^{−1})
= T_{w_{0,J}} (uE_{β} − q^{⟨wt(u),β⟩}E_{β}u)
∈ T_{w_{0,J}} (U^{+}[w_{J}]) = 𝕌_{q}(𝔲_{J}^{+})  by (2.7.3)

as claimed.

Lastly, we consider E_{α}. Again α, β ∈ Π, so E_{α} = T_{w_{0,J} s_{β}}(E_{β}). Let T_{w_{0,J}} (u) for u ∈ U^{+}[w_{J}] be an arbitrary monomial element of 𝕌_{q}(𝔲_{J}^{+}). Then we have

Ad(E_{α})(T_{w_{0,J}} (u)) = E_{α}T_{w_{0,J}} (u) − K_{α}T_{w_{0,J}} (u)K_{α}^{−1}E_{α}  by (2.7.1)
= T_{w_{0,J} s_{β}}(E_{β})T_{w_{0,J}} (u) − K_{α}T_{w_{0,J}} (u)K_{α}^{−1}T_{w_{0,J} s_{β}}(E_{β}) from above
= T_{w_{0,J} s_{β}}(E_{β}T_{s_{β}}(u) − K_{β}T_{s_{β}}(u)K_{β}^{−1}E_{β})  [Jan2, 8.18(3)]
= T_{w_{0,J} s_{β}}(E_{β}T_{s_{β}}(u) − q^{⟨wt(T_{s_{β}} (u)),β⟩}T_{s_{β}}(u)E_{β})
∈ T_{w_{0,J} s_{β}}(T_{s_{β}}(U^{+}[w_{J}])) = T_{w_{0,J}} (U^{+}[w_{J}]) = 𝕌_{q}(𝔲_{J}^{+}) by (2.7.4)

as claimed. _{□}

The definitions of Ad and \overline{Ad} now give the following.

Corollary 2.7.2. The algebra 𝕌_{q}(𝔲_{J}) is normal in 𝕌_{q}(𝔭_{J}) and the algebra 𝕌_{q}(𝔲_{J}^{+}) is normal in 𝕌_{q}(𝔭_{J}^{+}). Further, normality also holds for the specializations U_{ζ}(𝔲_{J}) ⊂ U_{ζ}(𝔭_{J}), u_{ζ}(𝔲_{J}) ⊂ u_{ζ}(𝔭_{J}), and U_{ζ}(𝔲_{J}) ⊂ U_{ζ}(𝔭_{J}) (as well as for the +-versions). Also, U_{ζ}(𝔩_{J}) ≅ U_{ζ}(𝔭_{J})//U_{ζ}(𝔲_{J}), u_{ζ}(𝔩_{J}) ≅ u_{ζ}(𝔭_{J})//u_{ζ}(𝔲_{J}), and U_{ζ}(𝔩_{J}) ≅ U_{ζ}(𝔭_{J})//U_{ζ}(𝔲_{J}).

Recall that we have an inclusion of A-forms U_{q}^{A} ⊆ U_{q}^{A}. While it is generally false that U_{q}^{A} is stable under the adjoint action of U_{q}^{A} on itself, this property does hold after base-change from A to a larger algebra B. More precisely, given l, let f_{l}(x) ∈ ℚ[x] denote the minimal polynomial for a primitive lth root of unity. Set B := ℤ[q, q^{−1}]_{(⟨f_{l}(q)⟩)}, i. e., the local ring determined by the maximal ideal ⟨f_{l}(q)⟩. Define U_{q}^{B} and U_{q}^{B} analogous to the way U_{q}^{A} and U_{q}^{A} were defined in Section 2.2. Then we still have an inclusion U_{q}^{B} ⊂ U_{q}^{B}.

Lemma 2.7.3. [ABG, Prop. 2.9.2(i)] The adjoint action of U_{q}^{B} stabilizes U_{q}^{B}. That is,

Ad(U_{q}^{B})(U_{q}^{B}) ⊆ U_{q}^{B}  and  \overline{Ad}(U_{q}^{B})(U_{q}^{B}) ⊆ U_{q}^{B}.

Hence, the adjoint action (either Ad or \overline{Ad}) of U_{ζ} defines an action on U_{ζ}.

Let Z ⊂ U_{ζ} be the central subalgebra such that U_{ζ}//Z ≅ u_{ζ} (cf. Section 2.2). In terms of root vectors, Z is the algebra generated by {E_{γ}^{l}, F_{γ}^{l} : γ ∈ Φ^{+}}. Given J, define a central subalgebra Z_{J} ⊂ U_{ζ}(𝔲_{J}) as Z_{J} := Z ∩ U_{ζ}(𝔲_{J}). Clearly Z_{J} is the subalgebra generated by {F_{γ}^{l} : γ ∈ Φ^{+}\Φ_{J}^{+}}. Further, we see that U_{ζ}(𝔲_{J})//Z_{J} ≅ u_{ζ}(𝔲_{J}). We now obtain the following generalization of [ABG, Prop. 2.9.2].

Corollary 2.7.4. Under the induced \overline{Ad}-action of U_{ζ} on U_{ζ} we have the following: (a) \overline{Ad}(U_{ζ}(𝔭_{J})) stabilizes U_{ζ}(𝔲_{J}), Z_{J}, and u_{ζ}(𝔲_{J}).

(b) The action of u_{ζ}(𝔭_{J}) on Z_{J} is trivial.

(c) \overline{Ad} induces an action of 𝕌(𝔭_{J}) on Z_{J}.

2.8. Finite dimensionality of cohomology groups. For any field k, given augmented k-algebras A ⊂ B with A normal in B (cf. Sections 1.2 and 2.7), there exists a Lyndon-Hochschild-Serre (LHS) spectral sequence (cf. [GK, §5.2, 5.3]). More precisely, we have the following result, which will be used below.

Lemma 2.8.1. Assume that B is flat as a right A-module. For any left B-module M, there is a natural action of B//A on H^{•}(A, M) which gives rise to a spectral sequence

E_{2}^{i,j} = H^{i}(B//A, H^{j}(A, M)) ⇒ H^{i+j}(B, M).

We return to the situation of the previous section. Under the \overline{Ad}-action of U_{ζ}(𝔭_{J}) on itself, U_{ζ}(𝔭_{J}) admits the structure of a Hopf module algebra (cf. [Mon, 4.1.1, 4.1.9]). The action of U_{ζ}(𝔭_{J}) on U_{ζ}(𝔭_{J}) can be extended to an action on the bar resolution which computes the cohomology H^{•}(U_{ζ}(𝔭_{J}), ℂ). Thus, there is a natural action of U_{ζ}(𝔭_{J}) on H^{•}(U_{ζ}(𝔭_{J}), ℂ) and further on H^{•}(U_{ζ}(𝔲_{J}), ℂ). In particular, U_{ζ}^{0} acts on these cohomology spaces.

Lemma 2.8.2. For each nonnegative integer n, H^{n}(U_{ζ}(𝔲_{J}), ℂ) is a finite dimensional.

Proof. If Z_{J} is the central subalgebra of U_{ζ}(𝔲_{J}) defined after Lemma 2.7.3, U_{ζ}(𝔲_{J})//Z_{J} ≅ u_{ζ}(𝔲_{J}). By Lemma 2.8.1, there is a spectral sequence

E_{2}^{i,j} = H^{i}(u_{ζ}(𝔲_{J}), H^{j}(Z_{J}, ℂ)) ⇒ H^{i+j}(U_{ζ}(𝔲_{J}), ℂ).

The subalgebra Z_{J} is central so it follows from [GK, Lemma 5.2.2] that the action of u_{ζ}(𝔲_{J}) on H^{•}(Z_{J}, ℂ) is trivial, and

E_{2}^{•,•} ≅ H^{•}(u_{ζ}(𝔲_{J}), ℂ) ⊗ H^{•}(Z_{J}, ℂ).

Since Z_{J} is the symmetric algebra based on the vector space 𝔲_{J}^{[1]}, we get that H^{•}(Z_{J}, ℂ) ≅ Λ^{•}(𝔲_{J}^{∗})^{[1]}. Moreover, u_{ζ}(𝔲_{J}) is finite dimensional, so H^{i}(u_{ζ}(𝔲_{J}), ℂ) is finite dimensional for any integer i. The result follows because the cohomology H^{n}(U_{ζ}(𝔲_{J}), ℂ) is a subquotient of ⊕_{i+j=n}E_{2}^{i,j} which is finite dimensional. _{□}

Also, the cohomology of U_{ζ}(𝔲_{J}) can be computed by means of the reduced bar resolution (cf. [Mac, Ch. X, §2]). Although it is not clear from this point of view that the cohomology is finite dimensional in any homological degree, it does have a natural U_{ζ}^{0}-action induced from the \overline{Ad}-action on U_{ζ}(𝔲_{J}). But then the above lemma establishes it is finite dimensional, and this fact implies each cohomology space is a weight module for U_{ζ}^{0}. We summarize this result as follows.

Corollary 2.8.3. The cohomology H^{•}(U_{ζ}(𝔲_{J}), ℂ) is a weight module for U_{ζ}^{0}. For any λ ∈ X, H^{•}(U_{ζ}(𝔲_{J}), ℂ)_{λ} is finite dimensional.

Proof. It remains only to establish the last statement. But if B̄_{•} is the reduced bar-resolution, then, given any weight μ, for large n, (B̄_{n})_{μ} = 0. _{□}

2.9. Spectral sequences and Euler characteristic. We study the cohomology of U_{ζ}(𝔲_{J}) and u_{ζ}(𝔲_{J}). To do this, we use a filtration on U_{ζ}(𝔲_{J}) (resp. u_{ζ}(𝔲_{J})) introduced for U_{ζ} in [DK].

Let A = U_{ζ}(𝔲_{J}). By Section 2.6, A has a basis of monomial elements

{F_{γ_{M+1}}^{a_{M+1}}F_{γ_{M+2}}^{a_{M+2}} ⋯ F_{γ_{N}}^{a_{N}} : a_{i} ∈ ℕ},

where N = |Φ^{+}| and M = |Φ_{J}^{+}|. For \overline{a} := (a_{M+1}, ⋯, a_{N}) ∈ ℕ^{N−M}, set

F\overline{a} := F_{γ_{M+1}}^{a_{M+1}}F_{γ_{M+2}}^{a_{M+2}} ⋯ F_{γ_{N}}^{a_{N}} .

Place a total (lexicographical) ordering ≺ on ℕ^{N−M} as follows. Put \overline{a} ≺ \overline{b} if and only if there exists M + 1 ≤ i ≤ N such that a_{i} < b_{i} and a_{j} = b_{j} for all j > i. With this ordering, one can define an ℕ^{N−M}-filtration on A. Given \overline{a} ∈ ℕ^{N−M}, let A\overline{a} be the subalgebra of A generated by

\{F\overline{b} : \overline{b} ⪯ \overline{a}\}.

By Lemma 2.4.1, this is a multiplicative filtration on A. That is, A\overline{a} · A\overline{b} ⊆ A_{\overline{a}+\overline{b}}.

Further, by Lemma 2.4.1 again, the associated graded algebra gr A is generated by {X_{α} : α ∈ Φ^{+}\Φ_{J}^{+}} subject to the relations:

X_{α} · X_{β} = ζ^{⟨α,β⟩}X_{β} · X_{α} if α ≺ β.

This filtration induces a filtration on u_{ζ}(𝔲_{J}) such that the algebra gr u_{ζ}(𝔲_{J}) is also generated by {X_{α} : α ∈ Φ^{+}\Φ_{J}^{+}} subject to the above relations, in addition to the condition:

X_{α}^{l} = 0 for α ∈ Φ^{+}\Φ_{J}^{+}.

Let Λ_{ζ,J}^{•} be the graded algebra with generators {x_{α} : α ∈ Φ^{+}\Φ_{J}^{+}} where deg(x_{α}) = 1 subject to the following relations:

x_{α} · x_{β} + ζ^{−⟨α,β⟩}x_{β} · x_{α} = 0 if α ≺ β;
x_{α}^{2} = 0 for α ∈ Φ^{+}\Φ_{J}^{+}.

There exists a graded (by degree) algebra isomorphism H^{•}(gr U_{ζ}(𝔲_{J}), ℂ) ≅ Λ_{ζ,J}^{•} (cf. [GK, Prop. 2.1]). This is also an isomorphism of U_{ζ}^{0}-modules, where Λ_{ζ,J}^{•} is regarded as a U_{ζ}^{0}- module by assigning x_{α} weight α.

Proposition 2.9.1. (a) In the character group ℤX, we have

∑_{n=0}^{∞}(−1)^{n} ch H^{n}(U_{ζ}(𝔲_{J}), ℂ) = ∑_{n=0}^{∞}(−1)^{n} ch Λ_{ζ,J}^{n}.

(b) If λ ∈ X is a weight of U_{ζ}^{0} in H^{n}(U_{ζ}(𝔲_{J}), ℂ), then λ is a weight of U_{ζ}^{0} in Λ_{ζ,J}^{n}.

Proof. Let A = U_{ζ}(𝔲_{J}). By Corollary 2.8.3 and [GK, Prop. 2.1] both H^{•}(A, ℂ) and H^{•}(gr A, ℂ) have weight space decompositions with finite dimensional weight spaces. Let A_{+} and gr A_{+} denote the augmentation ideals of A and gr A, respectively. Let C^{•}(A) and C^{•}(gr A) be the complexes obtained by taking duals of the respective reduced bar resolutions. More precisely, C^{n}(A) = Hom_{ℂ}((A_{+})^{⊗n}, ℂ) and C^{n}(gr A) = Hom_{ℂ}((gr A_{+})^{⊗n}, ℂ). Note that A_{+} and gr A_{+} are isomorphic as U_{ζ}^{0}-modules. The same holds for C^{n}(A) and C^{n}(gr A) and the differentials of both complexes are U_{ζ}^{0}-module maps. Thus, for a weight λ, H^{•}(A, ℂ)_{λ} and H^{•}(gr A, ℂ)_{λ} identify with the cohomologies of the complexes C^{•}(A)_{λ} and C^{•}(gr A)_{λ}, respectively.

By the Euler-Poincaré principle (cf. [GW, Lemma 7.3.11]),

χ(H^{•}(A, ℂ)_{λ}) := ∑_{n≥0}^{∞}(−1)^{n} dim H^{n}(A, ℂ)_{λ} = ∑_{n=0}^{∞}(−1)^{n} dim C^{n}(A)_{λ}

= ∑_{n=0}^{∞}(−1)^{n} dim C^{n}(gr A)_{λ} = ∑_{n=0}^{∞}(−1)^{n} dim H^{n}(gr A, ℂ)_{λ} := χ(H^{•}(gr A)_{λ}).

Part (a) follows now from the cohomology calculation for gr A given in [GK, Prop. 2.1].

Let A_{•} be the increasing filtration on A indexed by Λ := ℕ^{N−M}, viewed as a poset using the lexicographic ordering ≺ above. It induces a (downward) filtration on the complex C^{•}(A)_{λ} as follows. For γ, η ∈ Λ, set A_{+}_{γ} = A_{γ} ∩ A_{+}, and define

B_{[≺η]}^{n} = ∑ ; ∑ γ_{i}≺η A_{+}_{γ_{1}} ⊗ A_{+}_{γ_{2}} ⊗ ⋯ ⊗ A_{+}_{γ_{n}}, B_{[⪯η]}^{n} = ∑ ; ∑ γ_{i}⪯η A_{+}_{γ_{1}} ⊗ A_{+}_{γ_{2}} ⊗ ⋯ ⊗ A_{+}_{γ_{n}}.

Observe that B_{[≺η]}^{n} ⊆ B_{[⪯η]}^{n}, so setting, for λ ∈ X(T),

C^{n}(A)_{λ,[≺η]} = Hom_{ℂ}(A_{+}^{⊗n}/B_{[≺η]}^{n}, ℂ)_{λ}, C^{n}(A)_{λ,[⪯η]} = Hom_{ℂ}(A_{+}^{⊗n}/B_{[⪯η]}^{n}, ℂ)_{λ}

then C^{n}(A)_{λ,[⪯η]} ⊆ C^{n}(A)_{λ,[≺η]}. Moreover, if η, ζ ∈ Λ with ζ ≺ η then C^{n}(A)_{λ,[≺η]} ⊆ C^{n}(A)_{λ,[≺ζ]}.

The grading on gr A leads in a natural way to a grading of the complex C^{•}(gr A)_{λ}. For η ∈ Λ, C^{•}(gr A)_{λ,[η]} denotes the graded component corresponding to η, and we can identify C^{•}(A)_{λ,[≺η]}/C^{•}(A)_{λ,[⪯η]} with C^{•}(gr A)_{λ,[η]}. Also,

C^{•}(gr A)_{λ} ≅ ⨁_{η∈Λ} C^{•}(A)_{λ,[≺η]}/C^{•}(A)_{λ,[⪯η]}.

For a fixed weight λ, C^{n}(gr A)_{λ} ≠ 0, for finitely many n, and C^{•}(A)_{λ,[≺η]}/C^{•}(A)_{λ,[⪯η]} ≠ 0 for finitely many η. Let \overline{Λ} = \{η ∈ Λ : C_{λ,[≺η]}^{n}/C_{λ,[⪯η]}^{n} ≠ 0, for some n\}. Then \overline{Λ} is a finite totally ordered set in ℕ^{N−M} which induces a filtration (which can be indexed by ℕ) on C^{•}(gr A)_{λ}. Therefore, we have a spectral sequence,

E_{1}^{i,j} = (H^{i+j}(gr A, ℂ)_{λ})_{(i)} ⇒ H^{i+j}(A, ℂ)_{λ}.

This shows that H^{n}(U_{ζ}(𝔲_{J}), ℂ)_{λ} is a subquotient of (Λ_{ζ,J}^{n})_{λ}, and part (b) follows.

□

2.10. Induction functors. Let C (resp. C^{≤}) be the category of type 1, integrable representations of U_{ζ} (resp. U_{ζ}(𝔟)). The restriction functor res : C → C^{≤} has a right adjoint induction functor H_{ζ}^{0} = H^{0}(U_{ζ}/U_{ζ}(𝔟)) : C^{≤} → C defined by

H_{ζ}^{0}(M) = (M ⊗ k[U_{ζ}])^{U_{ζ}(𝔟)} ≅ F(Hom_{U_{ζ}(𝔟)}(U_{ζ}, M)).

In this expression k[U_{ζ}] denotes the coordinate algebra of U_{ζ}. Also, the functor F(−) assigns to any U_{ζ}-module the largest type 1, integrable submodule. We refer to [APW, (2.8), (2.10)] and [RH, (2.9)] for further discussion and explanation of notation.

For λ ∈ X_{+}, the induced module

∇_{ζ}(λ) := H^{0}(U_{ζ}/U_{ζ}(𝔟), λ)

has irreducible socle isomorphic to L_{ζ}(λ). In addition, there is an equality

ch ∇_{ζ}(λ) = ch L(λ) = ∑_{x∈W}(−1)^{ℓ(x)}e(w · λ)/ ∑_{x∈W}(−1)^{ℓ(x)}e(x · 0)

of formal characters, which is given by Weyl’s character formula.

We will also use the induction functors H^{0}(U_{ζ}/U_{ζ}(𝔭_{J}), −) (resp. H^{0}(U_{ζ}(𝔭_{J})/U_{ζ}(𝔟), −)) from the category of type 1, integrable U_{ζ}(𝔭_{J})-modules (resp. U_{ζ}(𝔟)-modules) to type 1, integrable U_{ζ}-modules (resp. U_{ζ}(𝔭_{J})-modules). Note that if λ is a one-dimensional U_{ζ}(𝔟)-module then H^{0}(U_{ζ}(𝔭_{J})/U_{ζ}(𝔟), λ) is trivial as a U_{ζ}(𝔲_{J})-module.

Let (X_{J})_{+} ⊆ X be the set of J-dominant weights, i. e., λ ∈ X belongs to (X_{J})_{+} if and only if ⟨λ, α^{∨}⟩ ∈ ℕ for all α ∈ J. The set (X_{J})_{+} indexes the irreducible (type 1, integrable) U_{ζ}(𝔩_{J})-modules. For λ ∈ (X_{J})_{+},

∇_{J,ζ}(λ) := H^{0}(U_{ζ}(𝔭_{J})/U_{ζ}(𝔟), λ)

has irreducible socle isomorphic to L_{J,ζ}(λ), the irreducible U_{ζ}(𝔩_{J})-module of high weight λ.

If λ ∈ (X_{J})_{+} satisfies ⟨λ + ρ, α^{∨}) = ℓ − 1 for all α ∈ J we call λ a J-Steinberg weight. Then ∇_{J,ζ}(λ) is a projective (and injective) irreducible U_{ζ}(𝔩_{J})-module (in the category of type 1, integrable modules). It remains irreducible, projective, and injective upon restriction to 𝔲_{ζ}(𝔩_{J}).

Finally, it will often be convenient to write ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} in place of H^{0}(U_{ζ}(𝔭_{J})/U_{ζ}(𝔟), −).

3. Computation of Φ_{0} and N (Φ_{0})

3.1. Recall that a prime p is called bad for the root system Φ provided that there exists a closed subsystem Φ′ of Φ such that Q/Q′ has p-torsion, where Q′ = ℤΦ′ is the root lattice of Φ′. If p is not bad, then p is called a good prime for Φ. Equivalently, p is good if and only if p does not appear as the coefficient of a simple root in the decomposition of α ∈ Φ as an integral linear combination of simple roots. The good primes for the various types of irreducible root systems are given as follows:

• Φ of type A_{n}, all p;

• Φ of type B_{n}, C_{n}, D_{n}, p ≥ 3;
• Φ of type E_{6}, E_{7}, F_{4}, G_{2}, p ≥ 5;
• Φ of type E_{8}, p ≥ 7.

Now let l > 1 be an integer. We will say that l is good for Φ provided that l is not divisible by a bad prime for Φ. Otherwise, l is bad for Φ. Additionally, a good integer l is said to be very good for Φ provided that if Φ has type A_{n}, then l and n + 1 are relatively prime. In cases when the root system is not irreducible, we will say that l is good (resp. very good) provided that it is good (resp. very good) for every irreducible component of Φ.

In this paper, the significance of good integers l comes about because of certain closed subsystems Φ_{λ,l} constructed from weights λ ∈ X. Precisely, define

Φ_{λ,l} = {α ∈ Φ | ⟨λ + ρ, α⟩ ≡ 0 mod l}.

In practice, when l is clear from context, we will write Φ_{λ} rather than Φ_{λ,l}. Obviously, Φ_{λ} is a closed subroot system of Φ. By our assumptions on l, each d_{α} is relatively prime to l (i. e., l is odd in types B_{n}, C_{n} and F_{4}, and, in type G_{2}, l is not divisible by 3) so we have

(3.1.1)  Φ_{λ} = {α ∈ Φ | ⟨λ + ρ, α^{∨}⟩ ≡ 0 mod l}.

Thus, because of our restrictions on l when working with U_{ζ}, etc., we can always take (3.1.1) as the definition of Φ_{λ}. It is useful to observe that, for any w ∈ W̃_{l},

\overline{w}(Φ_{λ}) = Φ_{w·λ}.

Lemma 3.1.1. Let l be an odd integer. Assume that l is good for Φ. For λ ∈ X, there exists a base Π′ for Φ such that Π′ ∩ Φ_{λ} is a base for Φ_{λ}. In particular, there exists a w ∈ W and a subset J ⊆ Π such that w(Φ_{λ}) = Φ_{J}.

Proof. This result follows immediately from [Bo, Prop. 24, IV, §7] since Q/Q_{λ} is l-torsion free, where Q_{λ} is the root lattice of Φ_{λ}. _{□}

The lemma may fail if l is not good; e. g., let Φ have type F_{4} with l = 3. Using [Bo, Appendix, Plate VIII] and taking λ = 0, we see that Φ_{0} has base

\{ϵ_{1} − ϵ_{2}, ϵ_{2} + ϵ_{4}\} ∪ \{−ϵ_{3}, \frac{1}{2}(ϵ_{1} + ϵ_{2} + ϵ_{3} − ϵ_{4})\},

and so has type A_{2} × A_{2}, and there is clearly no J ⊂ Π for which Φ_{J} has type A_{2} × A_{2}. It is sometimes useful to use the well-known algorithm of Dynkin, which states that the closed subsystems of Φ can be found by removing one or more nodes from the affine diagram of Φ, then repeating this process on a connected component of the resulting diagram. More generally (for type F_{4}), if 3 divides l, say l = 3l′, then the lemma fails for any λ = (l′ − 1)ρ.

A parabolic subgroup P_{J} = L_{J} ⋉U_{J} ⊇ B of the semisimple group G determines a Richardson orbit C_{J} in the nullcone N = N (𝔤) of 𝔤. Namely, P_{J} has a unique dense orbit in the Lie algebra 𝔲_{J} of U_{J}, and C_{J} is the G-orbit of any member of this P_{J}-orbit. If J, K are W -conjugate subsets of Π, then the Johnston-Richardson theorem implies that C_{J} = C_{K}. Hence, given λ ∈ X, if there exists w ∈ W and J ⊆ Π with w(Φ_{λ}) = Φ_{J}, then λ defines a Richardson class C_{λ} in N by setting C_{λ} = C_{J}. We recall that for any parabolic subgroup P_{J}, the corresponding Richardson class has Zariski closure \overline{C_{J}} = G · 𝔲_{J}. For λ ∈ X with w(Φ_{λ}) = Φ_{J} as above, we set N (Φ_{λ}) := G · 𝔲_{J} ⊆ N.

3.2. Given an integer l as in Assumption 1.3.1, of particular interest to us will be

Φ_{0} = Φ_{0,l} = {α ∈ Φ : ⟨ρ, α^{∨}⟩ ≡ 0 mod l}.

In the simply laced case, computing Φ_{0} entails computing the roots where l divides the height. For λ = 0, Lemma 3.1.1 in fact holds for this weaker condition on l, as will be discussed in the following sections, and culminating in Theorem 3.5.1.

3.3. Classical Lie algebras. In [CLNP, §3.1-3.7], explicit determinations were given of a J ⊂ Π so that w(Φ_{0}) = Φ_{J} when l = p is a good prime. The proofs work equally well when l satisfies Assumption 1.3.1. We use the root notation and ordering of Bourbaki [Bo].

Theorem 3.3.1. Let l be as in Assumption 1.3.1, 𝔤 be a classical simple Lie algebra with Φ of type A_{n} (resp. B_{n}), and h = n + 1 (resp. 2n) be the Coxeter number of Φ.

(a) If l ≥ h then N (Φ_{0}) = N (𝔤) and dim N (Φ_{0}) = |Φ|.

(b) Suppose that l < h where h − 1 = lm + s with m > 0 with and 0 ≤ s ≤ l − 1. Then N (Φ_{0}) = G · 𝔲_{J} where J ⊆ Π such that when

(i) Φ is of type A_{n},

Φ_{0} ≅ Φ_{J} ≅ \underbrace{A_{m} × ⋯ × A_{m}}_{s + 1 times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{l − s − 1 times};

where dim N (Φ_{0}) = n(n + 1) − m(lm + 2s − l + 2). (ii) Φ is of type B_{n},

Φ_{0} ≅ Φ_{J} ≅ \begin{cases}\underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s−1}{2} times} ×B_{\frac{m+1}{2}} & if s is even (m odd), \\ \underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s+1}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s−2}{2} times} ×B_{\frac{m}{2}} & if s is odd (m even). \end{cases}

Also,

dim N (Φ_{0}) = \begin{cases}2n^{2} − \frac{m(lm+2s−l+3)+1}{2} & if s is even (m odd), \\ 2n^{2} − \frac{m(lm+2s−l+3)}{2} & if s is odd (m even). \end{cases}

Theorem 3.3.2. Let l be as in Assumption 1.3.1, 𝔤 be a classical simple Lie algebra with Φ of type C_{n} (resp. D_{n}), and h = 2n (resp. 2n − 2) be the Coxeter number of Φ.

(a) If l ≥ h then N (Φ_{0}) = N (𝔤) and dim N (Φ_{0}) = |Φ|.

(b) Suppose that l < h where h + 1 = lm + s with m > 0 and 0 ≤ s ≤ l − 1. Then N (Φ_{0}) = G · 𝔲_{J} where J ⊆ Π such that when

(i) Φ is of type C_{n},

Φ_{0} ≅ Φ_{J} ≅ \begin{cases}\underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s−1}{2} times} ×C_{\frac{m−1}{2}} & if s is even, \\ \underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s−1}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s}{2} times} ×C_{\frac{m}{2}} & if s is odd. \end{cases}

Also,

dim N (Φ_{0}) = \begin{cases}2n^{2} − \frac{m(lm+2s−l−1)+1}{2} & if s is even (m odd), \\ 2n^{2} − \frac{m(lm+2s−l−1)}{2} & if s is odd (m even). \end{cases}

(iv) Φ is of type D_{n},

Φ_{0} ≅ Φ_{J} ≅ \begin{cases}\underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s−1}{2} times} ×D_{\frac{m+1}{2}} & if s is even and m ≥ 3, \\ \underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s−1}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s}{2} times} ×D_{\frac{m+2}{2}} & if s is odd, \\ \underbrace{A_{m} × ⋯ × A_{m}}_{\frac{s}{2} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{\frac{l−s−1}{2} times} & if s is even and m = 1. \end{cases}

Also,

dim N (Φ_{0}) = \begin{cases}2n^{2} − 2n − \frac{m(lm+2s−l+1)−1}{2} & if s is even (m odd), \\ 2n^{2} − 2n − \frac{m(lm+2s−l+1)}{2} & if s is odd (m even). \end{cases}

3.4. Exceptional Lie algebras. For the exceptional Lie algebras, the computation of Φ_{0} and N (Φ_{0}) can be carried out by hand. Note that dim N (Φ_{0}) = |Φ| − |Φ_{0}| and N (Φ_{0}) is the closure of a Richardson orbit. In most cases this information is enough to determine the orbit (cf. [CLNP, §4.2]). When this information is not sufficient, one can pin down the correct orbit by using the Weyl group (cf. [CLNP, §4.3]). In fact, for computational purposes in Section 4, for each value of l satisfying Assumption 1.3.1, we identify an explicit element w ∈ W and subset J ⊂ Π such that w(Φ_{0}) = Φ_{J}. Note that the choice of w and J is not unique in general. The computer package MAGMA [BC, BCP] was used to verify this. The tables providing the description of N (Φ_{0}), w, and J are presented in Section 9.1.

3.5. As a consequence of the aforementioned results, we have the following:

Theorem 3.5.1. Let l be as in Assumption 1.3.1. Then there exists w ∈ W and a subset J ⊆ Π such that w(Φ_{0,l}) = Φ_{J}.

3.6. Normality of orbit closures. In this subsection, we consider certain nilpotent orbit closures for a complex semisimple lie algebra 𝔤 = 𝔤_{ℂ} having root system Φ, etc. Let G be the complex semisimple, simply connected algebraic group of the same root type as 𝔤. There has been considerable work in determining the normal G-orbit closures in the nilpotent variety N = N (𝔤). Now fix an integer l satisfying Assumption 1.3.1. Let Φ_{0} = Φ_{0,l}. We will next verify that in almost all types N (Φ_{0}) is normal. In general the regular, subregular and minimal orbits have normal orbit closures. When Φ is of type A_{n} then all orbit closures are normal.

In the case when Φ is of type B_{n}, (resp. C_{n}, D_{n}), set N = 2n + 1 (resp. 2n, 2n). For X = A, (resp. B, C, D), let P_{X}(N) be the set of partitions of N parametrizing the set of nilpotent orbits for X_{N}. A precise description of P_{X}(N) is given in [CM, Thm. 5.1.2-5.1.4]. If λ ∈ P_{X}(N), let O_{λ} be the corresponding nilpotent X-orbit. In the case of type D, for very even partitions λ, there are two orbits corresponding to the partition λ. We will denote the two orbits by O_{λ}^{I} and O_{λ}^{II}.

Write N = ml + s where 0 ≤ s ≤ l − 1. One can use [UGA1, CLNP] to show that N (Φ_{0}) = \overline{O}_{λ_{X}} where λ = (l^{m}, s) and λ_{X} is the X-collapse of λ (see [CM, p. 99]). We will use the procedure given in [KP] to determine whether \overline{O}_{λ_{X}} is a normal variety. In order to check this, it suffices to look at all codimension two orbits inside of \overline{O}_{λ_{X}} and determine the singularity types of each one.

In order to determine the possible codimension two orbits, we start with the Hasse diagram in type A_{N} and use the collapsing procedure (which preserves the ordering in the Hasse diagram). For type A_{N}, one has the following configuration given in Figure 1. For those values of s for which the partition does not make sense, the orbit is not present. Put

\begin{matrix}λ & μ_{1} & μ_{2} & γ_{1} & γ_{2} & γ_{3} \\ (l^{m}, s) & (l^{m−1}, l − 1, s + 1) & (l^{m}, s − 1, 1) & (l^{m−1}, l − 2, s + 2) & (l^{m−1}, l − 1, s, 1) & (l^{m}, s − 2, 2) \end{matrix}

\begin{aligned}^{\overline{O}_{λ}} & ^{\overline{O}_{μ_{1}}} & ^{\overline{O}_{μ_{2}}} & _{\overline{O}_{γ_{1}}} & _{\overline{O}_{γ_{2}}} & _{\overline{O}_{γ_{3}}} \\ & Figure 1 \end{aligned}

If λ_{X} does not equal (μ_{1})_{X} or (μ_{2})_{X}, then the possible codimension two orbits in \overline{O}_{λ_{X}} are O_{(μ_{1})_{X}} and O_{(μ_{2})_{X}}. This occurs when X = B or D and s is odd or when X = C when s is even. In these cases, λ_{X} = (l^{m}, s) and

(μ_{1})_{X} = \begin{cases}(l^{m}, 2^{2}) & X = B, D, s = 1, l = 3, \\ (l^{m}, l − 2, 3) & X = B, D, s = 1, l ≥ 5, \\ (l^{m−1}, l − 2, s + 2) & X = B, D, 3 ≤ s ≤ l − 4, \\ (l^{m−1}, (l − 1)^{2}) & X = B, D, s = l − 2, \\ (l^{m−2}, (l − 1)^{2}, s + 2) & X = C \end{cases}

(μ_{2})_{X} = \begin{cases}(l^{m}, s − 2, 1^{2}) & X = B, D, 3 ≤ s ≤ l − 4, \\ (l^{m}, l − 4, 2) & X = B, D, s = l − 2, l ≥ 7, \\ (l^{m}, 1^{3}) & X = B, D, s = l − 2, l = 5, \\ (l^{m}, s − 1, 1) & X = C. \end{cases}

On the other hand, if X = B, D and s is even (s ≥ 2) then λ_{X} = (μ_{2})_{X} = (l^{m}, s − 1, 1). If s = 0, we can use the same type of analysis as above. Therefore, the possible codimension two orbits in \overline{O}_{λ_{X}} are O_{(μ_{1})_{X}} and O_{(γ_{3})_{X}}. Moreover,

(μ_{1})_{X} = \begin{cases}(l^{m−1}, l − 2, s + 2) & X = B, D, 0 ≤ s ≤ l − 4, \\ (l^{m−1}, (l − 2)^{2}, 1) & X = B, D, s = l − 3 \end{cases}

(γ_{3})_{X} = \begin{cases}(l^{m}, s − 3, 3) & X = B, D, 6 ≤ s, \\ (l^{m}, 2^{2}) & X = B, D, s = 4. \end{cases}

The last case occurs when X = C and s is odd. In this situation, λ_{C} = (μ_{1})_{X} = (l^{m−1}, l − 1, s + 1). The possible codimension two orbits are O_{(μ_{2})_{X}} and O_{(γ_{1})_{X}}. The collapse of these partitions is given by

(μ_{2})_{C} = \begin{cases}(l^{m−1}, l − 1, s − 1, 2) & 3 ≤ s, \\ (l^{m−1}, l − 1, 1^{2}) & s = 1 \end{cases}

(γ_{1})_{C} = \begin{cases}(l^{m−1}, l − 3, s + 3) & s ≤ l − 6, \\ (l^{m−1}, (l − 2)^{2}) & s = l − 4. \end{cases}

We will apply the procedure in [KP, §0.8] to verify the normality for N (Φ_{0}). According to [KP, Thm. 1], to show normality it suffices to look at the type of singularities in the classes of codimension two. In order to determine the types of singularities, one can strip off the common first rows and columns of λ_{X}, and the corresponding partitions give the codimension two classes. By the description above of these partitions, this process results in partitions with less than or equal to three parts. This shows that the singularities are not of “Type e” ([KP, Table 3.4]). Therefore, for the classical Lie algebras, N (Φ_{0}) is a normal variety.

In the exceptional types by using the tables in Section 9.1 and work of Sommers [So1, Thm. 1.1], it follows that N (Φ_{0}) is normal for E_{6}. For E_{7} the techniques used in [So2] can be used to verify the normality of N (Φ_{0}) in those cases [So3]. In the case of E_{8}, Sommers [So3] was able to verify that N (Φ_{0}) is normal in all cases except when l = 7, 9. We believe that normality will still hold for these last two cases, but a proof is unavailable at this time. For this reason we state our main results with care and precision. For F_{4}, one can can use [Br1, Thm. 1] to show that N (Φ_{0}) is normal, and for G_{2} the only orbits which realize N (Φ_{0}) are the regular and subregular orbits which have normal orbit closures.

We summarize the analysis in this subsection in the following theorem.

Theorem 3.6.1. Let l be as in Assumption 1.3.1 and J ⊆ Π so that N (Φ_{0}) = G · 𝔲_{J}. If Φ is of type E_{8}, assume that l ≠ 7, 9. Then N (Φ_{0}) is a normal variety.

3.7. Resolution of singularities. We maintain the notation of the previous subsection. Let J ⊆ Π and P_{J} be the associated parabolic subgroup. The moment map G×_{P_{J}} 𝔲_{J} → G·𝔲_{J} is a resolution of singularities if C_{G}(x) ⊆ P_{J} where x is a Richardson element contained in 𝔲_{J}. When the moment map is a resolution of singularities and G · x has a normal orbit closure, then

ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[G ×_{P_{J}} 𝔲_{J}] ≅ ℂ[G · 𝔲_{J}]

(e.g. [So1, p. 290] [KP, §0.7]).

Theorem 3.7.1. Let l be as in Assumption 1.3.1 and J ⊆ Π so that N (Φ_{0}) = G · 𝔲_{J}. If Φ is of type E_{8}, assume that l ≠ 7, 9. Then

ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[G · 𝔲_{J}].

Proof. In Theorem 3.6.1, we showed that N (Φ_{0}) is a normal variety under the hypothesis of the theorem. So it suffices to prove that C_{G}(x) ⊆ P_{J} where x ∈ 𝔲_{J}. First observe that C_{G}(x)^{0} ⊆ P_{J} [Car, Cor. 5.2.2] where C_{G}(x)^{0} is the identity component of C_{G}(x). For type A_{n}, all orbits are Richardson. The centralizer is generated by the connected component and toral elements, so the centralizer is contained in P_{J}. So in this case we have verified the theorem. Second, if the Richarson orbit corresponding to x is even, then P_{J} coincides with the “Bala-Carter” parabolic and in this case we have C_{G}(x) ⊆ P_{J} (e.g. [Jan3, p. 93 Rem.]). For the exceptional groups, we can use the description of N (Φ_{0}) given in Section 9.1 and the tables in [Car, p. 401-407] to verify that either the orbit is even or the centralizer is connected.

We are left to verify our claim for the remaining classical groups of types B_{n}, C_{n} and D_{n}. Let N = 2n+1 (resp. 2n, 2n) for X = B_{n} (resp. C_{n}, D_{n}). Define integers m, s by N = lm+s where 0 ≤ s ≤ l − 1. Set λ = (l^{m}, s) and recall that N (Φ_{0}) = \overline{O}_{λ_{X}} where λ_{X} is the X-collapse of λ.

For type B_{n} we have

λ_{B} = \begin{cases}(l^{m}, s) & if s is odd or s = 0, \\ (l^{m}, s − 1, 1) & if s is even and s ≠ 0. \end{cases}

In either case each of the nonzero parts are odd so the associated weighted Dynkin diagram has even entries [CM, §5.3]. Therefore, the orbit O_{λ_{B}} is even, and the centralizer is contained in P_{J}. A similar argument works to show that for type D_{n} the orbit O_{λ_{D}} is even.

We are left to analyze type C_{n}. In this case

λ_{C} = \begin{cases}(l^{m}, s) & if m is even (s even), \\ (l^{m−1}, l − 1, s + 1) & if m is odd (s odd). \end{cases}

In the first case when m is even (s even), let b be the number of distinct even nonzero parts. So b = 1 and by [CM, p. 92] the component group A(O_{λ_{C}}) ≅ (ℤ/2ℤ)^{b−1}. Therefore, the centralizer is connected in this case.

Now suppose that m is odd (s odd). In this case the component group is isomorphic to ℤ/2ℤ so we need to appeal to a different line of reasoning. In [H, Cor. 7.7], Hesselink provides a necessary and sufficient criterion for having C_{G}(x) ⊆ P_{J}. Here λ_{C} satisfies condition (i) of [H, Cor. 7.7] where ϵ = 1 and λ_{C} ∈ Pai(2n, m − 1) (see [H, §6.1] for the definitions). _{□}

4. Combinatorics with the Steinberg module

One of the key steps in computing the cohomology H^{•}(u_{ζ}(𝔤), ℂ) for l > h entails showing that the u_{ζ}(𝔱) invariants on H^{•}(U_{ζ}(𝔲), ℂ) is one-dimensional by looking at weights of Λ_{ζ,∅}^{•}. This is far from being true for l ≤ h. For small l, a more intricate analyis is needed, where we consider the multiplicity of a certain “Steinberg module” in H^{•}(U_{ζ}(𝔲_{J}), ℂ). This computation will then be used in Sections 5 and 6 to make the desired cohomology computations.

4.1. Steinberg weights. If l satisfies Assumption 1.3.1, by Theorem 3.5.1, we can choose w ∈ W such that w(Φ_{0}) = Φ_{w·0} = Φ_{J}.

Lemma 4.1.1. For all α ∈ J, ⟨w · 0, α^{∨}⟩ = l − 1.

Proof. Since w · 0 = w(ρ) − ρ, the claim is equivalent to showing that l = ⟨w(ρ), α^{∨}⟩ = ⟨ρ, w^{−1}(α)^{∨}⟩ for all α ∈ J. But w^{−1}(J) is the unique set Π_{0} of simple roots for Φ_{0} contained in Φ_{0}^{+} = Φ^{+} ∩ Φ_{0}. So, the lemma asserts that ⟨ρ, β^{∨}⟩ = l for all β ∈ Π_{0}. Therefore, although w ∈ W is not uniquely determined, if the lemma holds for one choice of w, it holds for all choices. Now, in the exceptional cases, for each l, an element w ∈ W and a subset J ⊆ Π satisfying w(Φ_{0}) = Φ_{J} are identified in Section 9.1. In these cases, the lemma can be checked directly by using the tables in Section 9.2.

Now assume that Φ has classical type A_{n}, B_{n}, C_{n}, or D_{n}. We can also assume that Π_{0} ≠ ∅. Then Φ_{0} consists of all roots α such that the coroot α^{∨} has height ht(α^{∨}) = ⟨ρ, α^{∨}⟩ divisible by ℓ. In particular, ℓ < h. If β^{∨} is any coroot of height m > 0, then, for any positive integer i, 3 ≤ i < m, it is easy to see (in each possible case), that we can write β^{∨} = δ^{∨} + γ^{∨} for δ, γ ∈ Φ satisfying ht(δ^{∨}) = i. Then β = aδ + bγ, where a, b are positive rational numbers. If β ∈ Φ_{0}^{+} with ht(β^{∨}) = tℓ, t > 1, we can thus write β^{∨} = δ^{∨} + γ^{∨} with δ, γ ∈ Φ_{0}^{+}. If β ∈ Π_{0}, it follows that ht(β^{∨}) = ℓ, i. e., ⟨ρ, β^{∨}⟩ = ℓ, as required. _{□}

Set M := (ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗}. By the lemma, we see that w · 0 is a J-Steinberg weight (see Section 2.10). The module M is therefore isomorphic to a “Steinberg” type module on U_{ζ}(𝔩_{J}) that remains irreducible if viewed as a u_{ζ}(𝔩_{J})-module. The highest weight of M is −w_{0,J}(w·0) and the lowest weight of M is −w · 0.

4.2. Weights of Λ_{ζ,J}^{•}. By Section 2.7, the \overline{Ad}-action induces an action of U_{ζ}(𝔭_{J}) (and hence also of u_{ζ}(𝔭_{J})) on U_{ζ}(𝔲_{J}). This defines an action of U_{ζ}(𝔭_{J}) on the cohomology H^{•}(U_{ζ}(𝔲_{J}), ℂ). See also Section 2.8. In Theorem 4.3.1 below, we determine

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗}, H^{•}(U_{ζ}(𝔲_{J}), ℂ)).

By Lemma 2.9.1(b), it follows that, as a U_{ζ}^{0}-module, H^{•}(U_{ζ}(𝔲_{J}), ℂ) is a subquotient of Λ_{ζ,J}^{•}. The key ingredient to proving Theorem 4.3.1 is the following computation about weights in Λ_{ζ,J}^{•}.

Proposition 4.2.1. Let l be as in Assumption 1.3.1. Choose w ∈ W such that w(Φ_{0}) = Φ_{J}. Let γ be a J-dominant weight of Λ_{ζ,J}^{i} and write γ = −w_{0,J}(w · 0) + lν for some ν ∈ X(T).

(a) Suppose that l ∤ n + 1 when Φ is of type A_{n} and l ≠ 9 when Φ is of type E_{6}. Then γ = −w_{0,J}(w · 0) (i. e., ν = 0) and i = ℓ(w).

(b) If Φ is of type A_{n} with n + 1 = l(m + 1) and w is as defined in (4.8.1), then γ is one of the following, for 0 ≤ t ≤ l − 1:

γ = −w_{0,J}(w · 0) + lϖ_{t(m+1)}  with  i = ℓ(w) + (m + 1)t(l − t).

We set ϖ_{0} = 0.

(c) If Φ is of type E_{6} and l = 9 (assuming that w and J are as in Section 9.1), then γ is one of the following:

γ = −w_{0,J}(w · 0) with i = ℓ(w) = 8,
γ = −w_{0,J}(w · 0) + ℓϖ_{1} with i = 20,
γ = −w_{0,J}(w · 0) + ℓϖ_{6} with i = 20.

One should observe that the weight ν in the statement of the proposition must necessarily be J-dominant by Lemma 4.1.1. The proposition will be proved below. See Section 4.4.

4.3. Multiplicity of the Steinberg module. Assuming that Proposition 4.2.1 holds, we can now determine how often the “Steinberg module” M (introduced in Section 4.1) appears in H^{•}(u_{ζ}(𝔲_{J}), ℂ). When l ≥ h, w = 1, J = ∅, and M = ℂ.

Theorem 4.3.1. Let l be as in Assumption 1.3.1. Choose w ∈ W such that w(Φ_{0}) = Φ_{J}.

(a) Suppose that l ∤ n + 1 when Φ is of type A_{n} and l ≠ 9 when Φ is of type E_{6}. Then as U_{ζ}(𝔩_{J})-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{i}(U_{ζ}(𝔲_{J}), ℂ)) = \begin{cases}ℂ & if i = ℓ(w) \\ 0 & otherwise. \end{cases}

(b) If Φ is of type A_{n} with n + 1 = l(m + 1) and w is as defined in (4.8.1), then as U_{ζ}(𝔩_{J})-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{i}(U_{ζ}(𝔲_{J}), ℂ))

= \begin{cases}ℂ & if i = ℓ(w) \\ lϖ_{t(m+1)} ⊕ lϖ_{(l−t)(m+1)} & if i = ℓ(w) + (m + 1)t(l − t) \\ & for 1 ≤ t ≤ (l − 1)/2 \\ 0 & otherwise. \end{cases}

(c) If Φ is of type E_{6} and l = 9 (assuming that w and J are as in Section 9.1), then as U_{ζ}(𝔩_{J})-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{i}(U_{ζ}(𝔲_{J}), ℂ)) = \begin{cases}lϖ_{1} ⊕ lϖ_{6} & if i = 20 \\ ℂ & if i = ℓ(w) = 8 \\ 0 & otherwise. \end{cases}

Proof. Since Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗}, H^{•}(U_{ζ}(𝔲_{J}), ℂ)) is a module for U_{ζ}(𝔩_{J}) on which u_{ζ}(𝔩_{J}) acts trivially, it is also a (finite dimensional—hence completely reducible) module for the universal enveloping algebra 𝕌(𝔩_{J}) (see Section 2.3). Thus, if Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w ·

0)^{∗}, H^{•}(U_{ζ}(𝔲_{J}), ℂ)) ≠ 0, any U_{ζ}(𝔩_{J})-composition factor will be of the form L_{J}(ν)^{[1]} for a J-dominant weight ν. In other words, there must be a nonzero U_{ζ}(𝔩_{J})-homomorphism

(4.3.1)  (ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗} ⊗ L_{J}(ν)^{[1]} → H^{•}(U_{ζ}(𝔲_{J}), ℂ).

Hence, the weight −w_{0,J}(w · 0) + lν must appear in H^{•}(U_{ζ}(𝔲_{J}), ℂ), and so also in Λ_{ζ,j}^{•} by Lemma 2.9.1(b). The theorem now follows from Proposition 4.2.1 if each weight listed therein does indeed give rise to a non-trivial homomorphism as in (4.3.1).

By Lemma 2.9.1(a),

(4.3.2)  ∑_{n=0}^{∞}(−1)^{n} ch H^{n}(U_{ζ}(𝔲_{J}), ℂ) = ∑_{n=0}^{dim(𝔲_{J})}(−1)^{n} ch Λ_{ζ,J}^{n}.

Since H^{•}(𝕌(𝔲_{J}), ℂ) can be computed from Λ^{•}(𝔲_{J}^{∗}) considered as a complex (with appropriately defined differential), we similarly have

(4.3.3)  ∑_{n=0}^{dim(𝔲_{J})}(−1)^{n} ch H^{n}(𝕌(𝔲_{J}), ℂ) = ∑_{n=0}^{dim(𝔲_{J})}(−1)^{n} ch Λ^{n}(𝔲_{J}^{∗}).

Clearly, ch_{ζ} Λ_{ζ,J}^{n} = ch Λ^{n}(𝔲_{J}^{∗}). Hence (4.3.2) and (4.3.3) give

(4.3.4)  ∑_{n=0}^{∞}(−1)^{n} ch H^{n}(U_{ζ}(𝔲_{J}), ℂ) = ∑_{n=0}^{dim(𝔲_{J})}(−1)^{n} ch H^{n}(𝕌(𝔲_{J}), ℂ).

Let ^{J}W = {x ∈ W | x(Φ^{−}) ∩ Φ^{+} ⊂ Φ^{+}\Φ_{J}^{+}} be the set of distinguished right W_{J}-coset representatives in W. By [W, Thm. 2.5.1.3],

(4.3.5)  ∑_{n=0}^{dim(𝔲_{J})}(−1)^{n} ch H^{n}(𝕌(𝔲_{J}), ℂ) = ∑_{x∈^{J} W}(−1)^{ℓ(x)}ch L_{J}(−w_{0,J}(x · 0)).

Also, (cf. [HK, Prop. 3.4.5])

ch_{ζ} ind_{U_{ζ}(𝔟_{𝔩}_{J})}^{U_{ζ}(𝔩_{J})}(−w_{0,J}(x · 0)) = ch L_{J}(−w_{0,J}(x · 0)).

Combining this with (4.3.4) and (4.3.5) gives

(4.3.6)  ∑_{n=0}^{∞}(−1)^{n} ch H^{n}(U_{ζ}(𝔲_{J}), ℂ) = ∑_{x∈^{J} W}(−1)^{ℓ(x)} ch ind_{U_{ζ}(𝔟_{𝔩}_{J})}^{U_{ζ}(𝔩_{J})}(−w_{0,J}(x · 0)).

The weights γ in Proposition 4.2.1 are all “Steinberg weights” whose induced module is injective over U_{ζ}(𝔩_{J}). Hence, they do not appear as a composition factor in any other induced module. For part (a), since only one such weight occurs, it gives rise to a composition factor on the right-hand side of (4.3.6) which cannot be canceled out by any other factor. Hence it appears as well on the left-hand side which completes the proof. For parts (b) and (c), while multiple “Steinberg weights” appear, these weights are all distinct and hence give rise to distinct composition factors on the right-hand side of (4.3.6) which cannot cancel each other out. Hence they all appear on the left-hand side as well. _{□}

Remark 4.3.2. In those cases where the cohomology is two-dimensional, the weights are not strongly linked (cf. [Jan1, II.6.4]). Hence the isomorphisms in the theorem also hold as U_{ζ}(𝔭_{J})-modules.

4.4. Proof of Proposition 4.2.1. The remainder of Section 4 is devoted to proving Proposition 4.2.1. Note first of all that the weight −w_{0,J}(w · 0) does appear in Λ_{ζ,J}^{•} in degree ℓ(w) (cf. [GW, 7.3], [FP1, Prop. 2.2]). So the goal is to show that (in most cases) a weight ν satisfying the hypothesis must in fact be zero. In Sections 4.5–4.6, the classical root systems will be considered. For these, we will mainly work with the ϵ-basis that represents Φ [Bo, p. 250] and ⟨−, −⟩ will always denote the ordinary Euclidean inner product. In Section 4.5, we first show that ⟨ν, α^{∨}⟩ = 0 for all α ∈ J. To do this, for each of the classical root systems X ∈ {A_{n}, B_{n}, C_{n}, D_{n}} we show the existence of an element δ_{X} ∈ X(T) with the following properties:

(4.4.1)  max_{λ}⟨λ, δ_{X}⟩ := max{⟨λ, δ_{X}⟩ | λ a weight of Λ_{ζ,J}^{•}} = ⟨−w_{0,J}(w · 0), δ_{X}⟩ and
(4.4.2)  ⟨ϖ_{j}, δ_{X}⟩ > 0 for all fundamental weights ϖ_{j} corresponding to α_{j} ∈ J.

When Φ is of type A_{n} or C_{n}, one can choose δ_{X} = ∑_{α∈J} α^{∨} and the maximum value in (4.4.1) turns out to be simply (l − 1)|J|.

Now assume that a δ_{X} satisfying properties (4.4.1) and (4.4.2) exists and that −w_{0,J}(w · 0) + lν is a J-dominant weight of Λ_{ζ,J}^{i} for some J-dominant weight ν. Then

⟨−w_{0,J}(w · 0), δ_{X}⟩ + l⟨ν, δ_{X}⟩ = ⟨−w_{0,J}(w · 0) + lν, δ_{X}⟩ ≤ max_{λ}⟨λ, δ_{X}⟩ = ⟨−w_{0,J}(w · 0), δ_{X}⟩.

This forces ν to vanish on J. To show that ν = 0, it remains to show that ⟨ν, α^{∨}⟩ = 0 for α ∈ Π\J. In Section 4.6, this is shown for types B_{n}, C_{n}, D_{n}. Sections 4.7 and 4.8 are devoted to dealing with type A_{n} and showing how the “extra” weights arise in Proposition 4.2.1.

In Section 4.9, similar ideas along with direct computations will be used to deal with the exceptional root systems.

4.5. The weight δ_{X}. In this section, we construct a weight δ_{X} which satisfies properties (4.4.1) and (4.4.2). We will first consider the case m ≥ 2 in Theorems 3.3.1 and 3.3.2.

For Φ of type A_{n} and

Φ_{0} ≅ Φ_{J} ≅ \underbrace{A_{m} × ⋯ × A_{m}}_{r_{1} times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{r_{2} times};

let

δ_{A} = (\underbrace{1, 0, …, 0 − 1, …, 1, 0, …, 0, −1}_{r_{1} times}, \underbrace{1, 0, …, 0, −1, …, 1, 0, …, 0, −1}_{r_{2} times})

in the orthonormal basis describing Φ in n + 1-dimensional Euclidean space 𝔼 [Bo, p. 250]. Note that the first r_{1}-groupings of (1, 0, …, 0, −1) have m + 1-components, while the last r_{2}- groupings of (1, 0, …, 0, −1) have m-components. In this case, δ_{A} = ∑_{α∈J} α^{∨}, and evidently ⟨ϖ_{j}, δ_{A}⟩ > 0 for all ϖ_{j} corresponding to simple roots in J (property (4.4.2)).

To show property (4.4.1), let λ be a weight of Λ_{ζ,J}^{i}. Then λ is a sum of distinct positive roots not in Φ_{J}^{+}. If β is a positive root then ⟨β, δ_{A}⟩ = 0, ±1, ±2 (i. e. β = ϵ_{i} − ϵ_{j} with i < j). Set

A[t] = {β ∈ Φ^{+}\Φ_{J}^{+} | ⟨β, δ_{A}⟩ = t}.

A quick count shows that

|A[2]| = \frac{(r_{1} + r_{2} − 1)(r_{1} + r_{2})}{2}
|A[1]| = (m − 1)(r_{1} + r_{2} − 1)r_{1} + (m − 2)(r_{1} + r_{2} − 1)r_{2}.

Therefore,

max_{λ}⟨λ, δ_{A}⟩ = 2|A[2]| + |A[1]|
= (r_{1} + r_{2} − 1)(mr_{1} + (m − 1)r_{2})
= (r_{1} + r_{2} − 1)|J|.

According to Theorem 3.3.1, we can set r_{1} = s + 1 and r_{2} = l − s − 1. Consequently,

max_{λ}⟨λ, δ_{A}⟩ = (l − 1)|J| = ⟨−w_{0,J}(w · 0), δ_{A}⟩

because A_{n} is simply laced.

For the other classical Lie algebras, let Φ be of type X_{n} where X = B, C, D with

Φ_{0} ≅ Φ_{J} ≅ \underbrace{A_{m} × ⋯ × A_{m}}_{r_{1} times} × \underbrace{A_{m−1} × … A_{m−1}}_{r_{2} times} ×X_{q};

with

δ_{X} = (\underbrace{1, 0, …, 0 − 1, …, 1, 0, …, 0, −1}_{r_{1} times}, \underbrace{1, 0, …, 0, −1, …, 1, 0, …, 0, −1}_{r_{2} times}, 1, 0, …, 0).

Notice that in all cases −w_{0,J}δ_{X} = δ_{X}. By using [GW, p. 102], one can verify that ⟨ϖ_{j}, δ_{X}⟩ > 0 for all ϖ_{j} that correspond to simple roots in J. If β is a positive root in X_{n} then ⟨β, δ_{X}⟩ = 0, ±1, ±2, so set X[t] = {β ∈ Φ^{+}\Φ_{J}^{+} | ⟨β, δ_{X}⟩ = t}. By using our computations for type A_{n} with consideration of other positive roots in X_{n}, it follows that

|X[2]| = \begin{cases}(r_{1} + r_{2})^{2} & X_{n} = B_{n}; \\ (r_{1} + r_{2})(r_{1} + r_{2} + 1) & X_{n} = C_{n}; \\ (r_{1} + r_{2})^{2} & X_{n} = D_{n} \end{cases}

and

|X[1]| = \begin{cases}2(r_{1} + r_{2})(r_{1}(m − 1) + r_{2}(m − 2)) + (2q − 1)(r_{1} + r_{2}) & X_{n} = B_{n}; \\ 2(r_{1} + r_{2})(r_{1}(m − 1) + r_{2}(m − 2)) + 2(q − 1)(r_{1} + r_{2}) & X_{n} = C_{n}; \\ 2(r_{1} + r_{2})(r_{1}(m − 1) + r_{2}(m − 2)) + 2(q − 1)(r_{1} + r_{2}) & X_{n} = D_{n}. \end{cases}

Hence,

max_{λ}⟨λ, δ_{X}⟩ = 2|X[2]| + |X[1]|

= \begin{cases}2(r_{1} + r_{2})(r_{1}m + r_{2}(m − 1)) + (2q − 1)(r_{1} + r_{2}) & X_{n} = B_{n}; \\ 2(r_{1} + r_{2})(r_{1}m + r_{2}(m − 1)) + 2q(r_{1} + r_{2}) & X_{n} = C_{n}; \\ 2(r_{1} + r_{2})(r_{1}m + r_{2}(m − 1)) + 2(q − 1)(r_{1} + r_{2}) & X_{n} = D_{n}. \end{cases}

On the other hand, by using the expression of ρ for the classical Lie algebras in terms of the ϵ-basis (see [GW, p. 107]), one sees that

⟨−w_{0,J}(w · 0), δ_{X}⟩ = \begin{cases}(l − 1)(r_{1}m + r_{2}(m − 1)) + (l − 1)(q − \frac{1}{2}) & X_{n} = B_{n}; \\ (l − 1)(r_{1}m + r_{2}(m − 1)) + (l − 1)(q) & X_{n} = C_{n}; \\ (l − 1)(r_{1}m + r_{2}(m − 1)) + (l − 1)(q − 1) & X_{n} = D_{n}. \end{cases}

In Theorems 3.3.1 and 3.3.2, r_{1} + r_{2} = \frac{l−1}{2} for all root systems Φ of types B_{n}, C_{n}, or D_{n}, and so max_{λ}⟨λ, δ_{X}⟩ = ⟨−w_{0,J}(w · 0), δ_{X}⟩ as desired.

Next we consider the case m = 1. Again let Φ be of type X_{n} where X = A, B, C, D with

Φ_{0} ≅ Φ_{J} ≅ \begin{cases}\underbrace{A_{1} × ⋯ × A_{1}}_{r times} & for X_{n} = A_{n}, C_{n}, or D_{n}. \\ \underbrace{A_{1} × ⋯ × A_{1}}_{r times} ×A_{1} & for X_{n} = B_{n}. \end{cases}

and

δ_{X} = \begin{cases}(\underbrace{1, −1, …, 1, −1}_{r times}, \underbrace{0, …, 0}_{z times}) & for X_{n} = A_{n}, C_{n}, or D_{n}; \\ (\underbrace{1, −1, …, 1, −1}_{r times}, \underbrace{0, …, 0}_{z times}, 1) & for X_{n} = B_{n}. \end{cases}

As above, one can verify that ⟨ϖ_{j}, δ_{X}⟩ > 0 for all ϖ_{j} that correspond to simple roots in J. Also, we conclude that

|X[2]| = \begin{cases}\frac{r(r−1)}{2} & X_{n} = A_{n}; \\ r^{2} & X_{n} = B_{n}, C_{n}; \\ r(r − 1) & X_{n} = D_{n}. \end{cases}

and

|X[1]| = \begin{cases}rz & X_{n} = A_{n}; \\ 2rz & X_{n} = C_{n}, D_{n}; \\ 2rz + r + z & X_{n} = B_{n}. \end{cases}

Hence,

max_{λ}⟨λ, δ_{X}⟩ = 2|X[2]| + |X[1]| = \begin{cases}r(r + z − 1) & X_{n} = A_{n}; \\ (r + \frac{1}{2})(2r + 2z) & X_{n} = B_{n}; \\ r(2r + 2z) & X_{n} = C_{n}; \\ r(2(r + z − 1)) & X_{n} = D_{n}. \end{cases}

On the other hand, by using the expression of ρ for the classical Lie algebras in terms of the ϵ-basis (see [GW, p. 107]) one sees that

⟨−w_{0,J}(w · 0), δ_{X}⟩ = \begin{cases}r(l − 1) & X_{n} = A_{n}, C_{n}, D_{n}; \\ (r + \frac{1}{2})(l − 1) & X_{n} = B_{n}. \end{cases}

By Theorems 3.3.1 and 3.3.2, r is the number of copies of A_{1} and z = n + 1 − 2r (resp. n − 2r − 1, n − 2r) in type A_{n} (resp. B_{n}, C_{n} or D_{n}). One obtains that

r + z = \begin{cases}l & X_{n} = A_{n}; \\ \frac{l−1}{2} & X_{n} = B_{n}, C_{n}; \\ \frac{l+1}{2} & X_{n} = D_{n}. \end{cases}

Again, we see that max_{λ}⟨λ, δ_{X}⟩ = ⟨−w_{0,J}(w · 0), δ_{X}⟩.

4.6. Types B_{n}, C_{n}, D_{n}. In this subsection Φ is always of type B_{n}, C_{n} or D_{n}. Under this assumption we show that the only weight ν satisfying the hypothesis of Proposition 4.2.1 is the zero weight. Note that our restriction on the root systems and l being odd implies that gcd(l, (X : Q)) = 1. For any such weight ν we observe that lν ∈ Q because −w_{0,J}(w · 0) + lν is a weight of Λ_{ζ,J}^{i}. It follows that ν ∈ Q.

Our results in Section 4.5 show that both −w_{0,J}(w · 0) and −w_{0,J}(w · 0) + lν consist of a sum of all the roots in X[1] ∪ X[2] with some additional terms involving roots in X[0]. Therefore, lν is a sum of distinct roots in X[0] ∪ −X[0].

We express δ_{X} and ν in the ϵ-basis as δ_{X} = ∑_{i=1}^{n} δ_{X,i}ϵ_{i} and ν = ∑_{i=1}^{n} ν_{i}ϵ_{i}, respectively. Notice that ν ∈ Q implies that ν ∈ ℤϵ_{1} ⊕ ⋯ ⊕ ℤϵ_{n}. Define the following sets

(4.6.1)  S_{(a,b)} = {(i, j) : δ_{X,i} = a, δ_{X,j} = b, and i < j} and S_{(a)} = {i : δ_{X,i} = a}.

The case when Φ is of type B_{n} will be discussed in detail. The verification that ν = 0 for the cases when Φ is of type C_{n} and D_{n} are left to the reader.

Any positive root of the form ϵ_{i} − ϵ_{j} in B[0] satisfies (i, j) ∈ S_{(0,0)} ∪ S_{(1,1)} ∪ S_{(−1,−1)}, any positive root of the form ϵ_{i} + ϵ_{j} in B[0] satisfies (i, j) ∈ S_{(0,0)} ∪ S_{(1,−1)} ∪ S_{(−1,1)} and any positive root of the form ϵ_{i} in B[0] satisfies i ∈ S_{(0)}. Using (4.6.1), we can express

lν = ∑_{(i,j)∈S_{(1,1)}} m_{i,j}(ϵ_{i} − ϵ_{j}) + ∑_{(i,j)∈S_{(−1,−1)}} m̃_{i,j}(ϵ_{i} − ϵ_{j})

+ ∑_{(i,j)∈S_{(1,−1)}} n_{i,j}(ϵ_{i} + ϵ_{j}) + ∑_{(i,j)∈S_{(−1,1)}} ñ_{i,j}(ϵ_{i} + ϵ_{j})

+ ∑_{(i,j)∈S_{(0,0)}} p_{i,j}(ϵ_{i} − ϵ_{j}) + ∑_{(i,j)∈S_{(0,0)}} p̃_{i,j}(ϵ_{i} + ϵ_{j}) + ∑_{i∈S_{(0)}} q_{i}ϵ_{i}

with m_{i,j}, m̃_{i,j}, n_{i,j}, ñ_{i,j}, q_{i}, p_{i,j}, p̃_{i,j} = 0, ±1.

The above expression shows that if δ_{B,i} = 1 then lν_{i} = ∑_{j}(m_{i,j} + n_{i,j} + m_{j,i} + ñ_{j,i}). For fixed i, it follows from Theorem 3.3.1 that the number of pairs of the form (i, j) together with those of form (j, i) in S_{(1,1)} are less than (l + 1)/2. A similar counting argument shows that the number of pairs of the form (i, j) in S_{(1,−1)} together with the number of pairs of the form (j, i) in S_{(−1,1)} are also less than (l − 1)/2. Therefore, | ∑_{j}(m_{i,j} + n_{i,j} + m_{j,i} + ñ_{j,i})| ≤ (l − 1) and δ_{B,i} = 1 implies ν_{i} = 0. Similarly, one can argue that δ_{B,i} = −1 implies ν_{i} = 0.

Any simple root ϵ_{i} − ϵ_{i+1} ∈ Π\J satisfies either δ_{B,i} = −1 and δ_{B,i+1} = 1 or δ_{B,i} = 0 and δ_{B,i+1} = 0. It follows from above that, for any α ∈ Π\J,

⟨lν, α^{∨}⟩ = ⟨ ∑_{i∈S_{(0)}} q_{i}ϵ_{i} + ∑_{(i,j)∈S_{(0,0)}} p_{i,j}(ϵ_{i} − ϵ_{j}) + ∑_{(i,j)∈S_{(0,0)}} p̃_{i,j}(ϵ_{i} + ϵ_{j}), α^{∨}⟩.

For m > 1, there are no roots in Π\J with δ_{B,i} = 0 and δ_{B,i+1} = 0 and the inner product on the right-hand side vanishes. Since ν vanishes on J, one concludes ν = 0, as desired.

Assume m = 1 and let i be such that δ_{B,i} = 0, then lν_{i} = q_{i} + ∑_{j}(p_{i,j} + p̃_{i,j} + p_{j,i} + p̃_{j,i}). Theorem 3.3.1 implies that for fixed i there are less than (l − 1)/2 pairs of the form (i, j) or (j, i) in S_{(0,0)}. It follows that l|ν_{i}| = |q_{i} + ∑_{j}(p_{i,j} + p̃_{i,j} + p_{j,i} + p̃_{j,i})| < l. This forces ν = 0.

4.7. Type A_{n}. In this (and the next) subsection Φ is always of type A_{n} with simple roots α_{1}, …, α_{n}. We will show that a weight ν satisfying the hypothesis of Proposition 4.2.1 equals the zero weight unless l divides n + 1. For μ ∈ Q, μ_{i} always denotes the coefficient of μ in its expansion in terms of the ϵ-basis (i. e., μ = ∑_{i=1}^{n+1} μ_{i}ϵ_{i}).

First consider the case when m > 1 in Theorem 3.3.1. Let

Φ_{0} ≅ Φ_{J} ≅ \underbrace{A_{m} × ⋯ × A_{m}}_{s + 1 times} × \underbrace{A_{m−1} × ⋯ × A_{m−1}}_{l − s − 1 times}

where n = lm + s and 0 ≤ s ≤ l − 1.

We start out by choosing a particular w ∈ W with w(Φ_{0}) = Φ_{J}. Partition the set \{\frac{n}{2}, \frac{n}{2} − 1, ..., −\frac{n}{2} + 1, −\frac{n}{2}\}, i. e., the set of coordinates of ρ in the ϵ-basis, into its congruence classes modulo l and order each congruence class in decreasing order. Then we order the congruence classes according to the size of their largest element from highest to lowest. The resulting array is the coordinate vector of a W -conjugate of ρ. We denote this conjugate by wρ and w ∈ W denotes the unique permutation that sends ρ to wρ. If we identify the Weyl group with the symmetric group in n + 1 letters, then w can be described as follows.

(4.7.1)  For 1 ≤ i ≤ n + 1 define s_{i}, t_{i} via i − 1 = s_{i}l + t_{i} with 0 ≤ t_{i} ≤ l − 1.

(4.7.2)  Then w(ϵ_{i}) = ϵ_{w(i)} where w(i) = \begin{cases}t_{i}(m + 1) + s_{i} + 1 & if 0 ≤ t_{i} ≤ s \\ t_{i}m + s_{i} + s + 2 & if s + 1 ≤ t_{i} ≤ l − 1. \end{cases}

(4.7.3)  Moreover, w_{0,J}w(i) = \begin{cases}(t_{i} + 1)(m + 1) − s_{i} & if 0 ≤ t_{i} ≤ s \\ (t_{i} + 1)m − s_{i} + s + 1 & if s + 1 ≤ t_{i} ≤ l − 1. \end{cases}

For any u ∈ W define Q(u) := {α ∈ Φ^{+} | uα ∈ Φ^{−}}. Using (4.7.1) and (4.7.2) we find that [GW, 7.3]

(4.7.4)  Q(w) = {ϵ_{i} − ϵ_{j} | s_{i} < s_{j}, t_{i} > t_{j}} and w · 0 = ∑_{{γ∈Q(w)}} −w(γ).

Define

f(i) = \begin{cases}w_{0,J}w(lm + i) = 1 + (i − 1)(m + 1) & if 1 ≤ i ≤ s + 1 \\ w_{0,J}w(l(m − 1) + i) = 1 + (i − 1)m + (s + 1) & if s + 2 ≤ i ≤ l, \end{cases}

and

(4.7.5)  g(i) = \begin{cases}w_{0,J}w(i) = i(m + 1) & if 1 ≤ i ≤ s + 1 \\ w_{0,J}w(i) = im + (s + 1) & if s + 2 ≤ i ≤ l. \end{cases}

Next set δ^{i} = ϵ_{f(i)} − ϵ_{g(i)} for 1 ≤ i ≤ l. Then δ_{A} = ∑_{i=1}^{l} δ^{i} = ∑_{i=1}^{l} ϵ_{f(i)} − ϵ_{g(i)}.

The α_{g(i)} = ϵ_{g(i)} − ϵ_{f(i+1)} are precisely the simple roots contained in Π but not in J.

Using the notation introduced in (4.6.1), we partition A[0] into the following subsets:

R_{(0,0)}^{+} = {ϵ_{i} − ϵ_{j} ∈ A[0] | (i, j) ∈ S_{(0,0)}},
R_{(1,1)}^{+} = {ϵ_{i} − ϵ_{j} ∈ A[0] | (i, j) ∈ S_{(1,1)}} = {ϵ_{f(i)} − ϵ_{f(j)} | 1 ≤ i < j ≤ l},
R_{(−1,−1)}^{+} = {ϵ_{i} − ϵ_{j} ∈ A[0] | (i, j) ∈ S_{(−1,−1)}} = {ϵ_{g(i)} − ϵ_{g(j)} | 1 ≤ i < j ≤ l}.

In addition we set R_{(a,a)} = R_{(a,a)}^{+} ∪ −R_{(a,a)}^{+}, a ∈ {−1, 0, 1}. Notice that both sets R_{(1,1)} and R_{(−1,−1)} form root systems of type A_{l−1} with simple roots

(4.7.6)  β_{i} := ϵ_{f(i)} − ϵ_{f(i+1)} and τ_{i} := ϵ_{g(i)} − ϵ_{g(i+1)},

respectively.

Next define

S^{+} := {ϵ_{f(i)} − ϵ_{f(j)} | 1 ≤ i ≤ s + 1, s + 2 ≤ j ≤ l} and S := S^{+} ∪ −S^{+}.

Then it follows from (4.7.3) through (4.7.5) that

(4.7.7)  R_{(−1,−1)}^{+} ∩ w_{0,J}w(Q(w)) = ∅ and R_{(1,1)}^{+} ∩ w_{0,J}w(Q(w)) = S^{+}.

One concludes that the weight −w_{0,J}(w ·0) is the sum of all roots in A[1]∪A[2] together with certain roots in R_{(0,0)}^{+} and the roots in S^{+}. The elements of S^{+} can also be characterized as those roots in R_{(1,1)}^{+} that contain β_{s+1}. It is important to note that no roots of R_{(−1,−1)}^{+} contribute to −w_{0,J}(w · 0).

Next assume that λ is a weight of Λ_{ζ,J}^{•} such that ⟨λ + w_{0,J}(w · 0), α⟩ = 0 for all α ∈ J. Set ν = λ + w_{0,J}(w · 0). Using the β-basis of R_{(1,1)} and the γ-basis of R_{(−1,−1)}, we express ν in the form

(4.7.8)  ν = ∑_{i=1}^{l−1} k_{i}β_{i} + ∑_{i=1}^{l−1} l_{i}τ_{i} + ∑_{η∈R_{(0,0)}^{+}} m_{η}η.

Since ν is the zero weight when restricted to J, one observes that ⟨ν, δ^{i}⟩ = 0 for 1 ≤ i ≤ l. Since ⟨ν, δ^{1}⟩ = 0, k_{1} − l_{1} = 0 and, inductively, it follows from ⟨ν, δ^{i}⟩ = 0 that k_{i} = l_{i} for 1 ≤ i ≤ l − 1. Moreover, it follows from (4.7.7) that all k_{i} and l_{i} are nonnegative. One concludes that ν is a sum of distinct roots in R_{(0,0)} together with distinct roots in R_{(−1,−1)}^{+} and in R_{(1,1)}^{+}\S^{+}.

Finally, assume that λ_{1} and λ_{2} are two weights of Λ_{ζ,J}^{•} such that ⟨λ_{i} + w_{0,J}(w · 0), α⟩ = 0 for all α ∈ J. For example λ_{1} could be of the form −w_{0,J}(u · 0) for some u ≠ w with u(Φ_{0}) = Φ_{J} and λ_{2} could be equal to −w_{0,J}(u · 0) + lν, where ν is the zero weight when restricted to J. It follows from our above arguments that λ_{2} − λ_{1} is a sum of distinct roots in R_{(−1,−1)} ∪R_{(1,1)}\S ∪R_{(0,0)}. The elements in R_{(1,1)}\S form a root system of type A_{s} ×A_{l−s−2}, spanned by the roots {β_{1}, …, β_{s}}∪{β_{s+2}, …, β_{l−1}}, as defined in (4.7.6). We can decompose λ_{2} − λ_{1} = γ_{1} + γ_{2} + γ_{3} where the support of γ_{1} lies entirely in the type A_{s} component of R_{(1,1)}\S, the support of γ_{2} lies entirely in the type A_{l−s−2} component of R_{(1,1)}\S, and the support of γ_{3} lies entirely in R_{(−1,−1)} ∪ R_{(0,0)}. Next observe that α_{g(i)} = β_{i} − δ^{i}. It follows that ⟨λ_{2} − λ_{1}, α_{g(i)}⟩ = ⟨λ_{2} − λ_{1}, β_{i}⟩ = ⟨γ_{1} + γ_{2}, β_{i}⟩. The inner product of λ_{2} − λ_{1} with the roots in Π\J is then given by the following:

(4.7.9)  ⟨λ_{2} − λ_{1}, α_{g(i)}⟩ = \begin{cases}⟨γ_{1}, β_{i}⟩ & if 1 ≤ i ≤ s, \\ ⟨γ_{1} + γ_{2}, β_{i}⟩ & if i = s + 1, \\ ⟨γ_{2}, β_{i}⟩ & if s + 2 ≤ i ≤ l − 1. \end{cases}

Since γ_{1} is a sum of distinct roots of a root system of type A_{s} a direct computation shows that |⟨γ_{1}, β_{i}⟩| ≤ s + 1. Similarly, |⟨γ_{2}, β_{i}⟩| ≤ l − s − 1. Now λ_{2} − λ_{1} ∈ lX implies that either λ_{1} = λ_{2} or l = s + 1. Hence, Proposition 4.2.1 holds for type A_{n} as long as l does not divide n + 1.

Suppose now that m = 1. Then

Φ_{0} ≅ Φ_{J} ≅ \underbrace{A_{1} × ⋯ × A_{1}}_{s + 1 times} .

Essentially the same argument as above works here as well. We highlight some of the differences and leave the details to the interested reader. The definition of w holds as above with m = 1. Define f(i), g(i), δ^{i}, δ_{A}, and α_{g(i)} just as above with m = 1. Note that for i ≥ s + 2, f(i) = g(i) and δ^{i} = 0. In the definitions of R_{(1,1)}^{+} and R_{(−1,−1)}^{+}, we have 1 ≤ i < j ≤ s + 1, and the root systems R_{(1,1)} and R_{(−1,−1)} are of type A_{s}, while S^{+} = ∅. For s + 2 ≤ i ≤ l − 1, the β_{i} = τ_{i} form a basis for the root system R_{(0,0)} of type A_{l−s−2}. The equivalent of (4.7.7) is now

R_{(a,a)}^{+} ∩ w_{0,J}w(Q(w)) = ∅ where a ∈ {1, 0, −1}.

In (4.7.8), the index i should run from i = 1 to i = s. Next we consider λ_{2} − λ_{1}, as defined above. We decompose λ_{2} − λ_{1} = γ_{1} + γ_{2} + γ_{3} with the support of γ_{1} in R_{(1,1)}, the support of γ_{2} in R_{(0,0)} and the support of γ_{3} in R_{(−1,−1)}. Then equation (4.7.9) remains valid. Hence one obtains the same conclusion.

4.8. Type A_{n} with l dividing n + 1. We continue to identify the Weyl group with the symmetric group in n + 1 letters. Assume throughout this subsection that l divides n + 1. Then l = s + 1 and the definition of w given in (4.7.1) and (4.7.2) can be simplified to

(4.8.1)  w(ϵ_{i}) = ϵ_{w(i)} where w(i) = t_{i}(m + 1) + s_{i} + 1.

We now follow the arguments used in [AJ, 6.2]. Recall that n + 1 = (m + 1)l. We define the element σ ∈ W as follows:

σ = (1, 2, ..., l)(l + 1, l + 2, ..., 2l) ⋯ (ml + 1, ml + 2, ..., (m + 1)l).

Direct computation shows that w(σ^{t} · 0) = −lϖ_{g(t)} for 1 ≤ t ≤ l − 1. Setting ϖ_{g(0)} = 0 yields

(4.8.2)  w · 0 = wσ^{t} · 0 + lϖ_{g(t)} for 0 ≤ t ≤ l − 1.

We find that

Q(wσ^{t}) = {ϵ_{i} − ϵ_{j} | s_{i} < s_{j}, σ^{t}(t_{i} + 1) > σ^{t}(t_{j} + 1)}
∪ {ϵ_{i} − ϵ_{j} | s_{i} = s_{j}, t_{i} < t_{j} and σ^{t}(t_{i} + 1) > σ^{t}(t_{j} + 1)}.

The cardinality of the first set in the above union is equal to the cardinality of Q(w) (see (4.7.4)) while the second set can be identified with Q(σ^{t}). Using this decomposition of Q(wσ^{t}), [GW, 7.3], and [AJ, 6.2(3)], we conclude that

(4.8.3)  ℓ(wσ^{t}) = ℓ(w) + ℓ(σ^{t}) = ℓ(w) + (m + 1)t(l − t).

Next, assume that lν −w_{0,J}(w·0) is a weight of Λ_{ζ,J}^{•} such that ⟨lν, α⟩ = 0 for all α ∈ J. The discussion in Section 4.7 shows that lν is a sum of distinct roots in R_{(−1,−1)}^{+}∪R_{(0,0)}∪R_{(1,1)}^{+}. We can decompose ν = γ_{1}+γ_{2} where the support of γ_{1} lies entirely in R_{(1,1)}^{+} and the support of γ_{2} lies entirely in R_{(−1,−1)}^{+} ∪ R_{(0,0)}. As before, the inner product ⟨lν, α_{g(i)}⟩ = ⟨lν, β_{i}⟩ = ⟨lγ_{1}, β_{i}⟩ is completely determined by the contribution coming from R_{(1,1)}^{+}, the positive roots of a type A_{l−1} root system. By [AJ, 2.2, 6.1] one concludes that either γ_{1} = 0 or lγ_{1} = lxκ_{i}, where κ_{i} denotes the fundamental weight corresponding to the simple root β_{i} of the root system of type A_{l−1} and x is in the corresponding Weyl group. But γ_{1} is a sum of positive roots. Hence lxκ_{i} must also be a sum of positive roots. But this can only happen if lxκ_{i} = lκ_{i}. Therefore, we must have lν = lϖ_{g(i)}. Finally, by (4.8.2), for each 1 ≤ i ≤ l − 1, we have

−w_{0,J}(w · 0) + lϖ_{g(i)} = −w_{0,J}(wσ^{i} · 0),

and the latter weight is a weight of Λ_{ζ,J}^{•} (cf. [GW, 7.3],[FP1, Prop. 2.2]). Further, by (4.8.3), this lies in degree ℓ(wσ^{i}) = ℓ(w) + (m + 1)t(l − i). Since g(i) = i(m + 1), the result follows.

4.9. Exceptional Lie algebras: In this subsection, we assume that the root system Φ is of exceptional type. We show that if ν satisfies the hypothesis of Proposition 4.2.1, then ν = 0 except in the case when Φ is of type E_{6} and l = 9. Note that in the excluded case l is divisible by (X : ℤΦ) = 3. Our goal is to show that if −w_{0,J}(w · 0) + lν with ν being J-dominant is a weight of Λ_{ζ,J}^{•}, then ν = 0. For the exceptional Lie algebras, an explicit choice of w and J is listed in Section 9.1. One can then explicitly compute the value of −w_{0,J}(w · 0). This was again done with the aid of MAGMA [BC, BCP] and the results are given in tables in Section 9.2. The weights in the tables are listed with respect to the basis {ϖ_{1}, …, ϖ_{n}} of fundamental dominant weights.

Since the dimension of Λ_{ζ,J}^{•} is finite, with the aid of a computer, one could in principle compute all possible weights of Λ_{ζ,J}^{•} and compare them to −w_{0,J}(w · 0) modulo l. For types F_{4} and G_{2}, this can readily be done and one finds that ν = 0. For type E_{n}, the size of Λ_{ζ,J}^{•} is sufficiently large as to make the computations somewhat impractical on a typical desktop computer. As such, we present an alternative approach which makes use of some of the ideas from the preceding sections on classical root systems to show directly that ν = 0 or reduce the computations to a more manageable number.

In what follows, let δ^{∨} = ∑_{α∈J} α^{∨}. This is a slight abuse of notation since δ^{∨} may not equal (∑_{α∈J} α)^{∨} but should not lead to any confusion. Recall that ⟨−w_{0,J}(w·0), δ^{∨}⟩ = (l−1)|J|. As in Section 4.2.1, set max_{λ}⟨λ, δ^{∨}⟩ := max{⟨λ, δ^{∨}⟩ | λ a weight of Λ_{ζ,J}^{i}}. As with the classical cases, the key to showing that ν = 0 is that max_{λ}⟨λ, δ^{∨}⟩ is in general “close” to (l − 1)|J|.

To make this more precise, set

E[t] := {β ∈ Φ^{+}\Φ_{J}^{+} | ⟨β, δ^{∨}⟩ = t}.

Then we can decompose Φ^{+}\Φ_{J}^{+} as a disjoint union:

Φ^{+}\Φ_{J}^{+} = E[< 0] ∪ E[0] ∪ E[> 0]

where

E[> 0] = ∪_{t>0}E[t] and E[< 0] = ∪_{t<0}E[t].

That is, we separate the positive roots into those which give a positive, zero, or negative inner product with δ^{∨}. Since weights in Λ_{ζ,J}^{•} are composed of sums of distinct positive roots from Φ^{+}\Φ_{J}^{+}, clearly,

max_{λ}⟨λ, δ^{∨}⟩ = ⟨ ∑_{β∈E[>0]} β, δ^{∨}⟩.

For convenience, for the remainder of this section, set λ := ∑_{β∈E[>0]} β. One might think of λ as a conical representative of those weights having maximum inner product with δ^{∨}. Any other such element would be of the form λ + z where z is a sum of distinct roots which lie in E[0]. With the aid of MAGMA the weight λ can be readily computed for a given l, w, and J. This is given in the tables in Section 9.2.

Let x = −w_{0,J}(w · 0) + lν be a J-dominant weight of Λ_{ζ,J}^{•}. To show that ν = 0, we show that ⟨ν, α^{∨}⟩ = 0 for all α ∈ Π. We separate this into two cases: α ∈ J and α ∈ Π\J. The first case follows if it can be shown that

|⟨−w_{0,J}(w · 0), δ^{∨}⟩ − ⟨λ, δ^{∨}⟩| < l.

This can be done by direct calculation (see tables in Section 9.2) and holds in all but one case (type E_{8} with l = 7, which is dealt with separately).

We now outline the basic process for showing the second case. Once the first case is known, that is, ⟨ν, α^{∨}⟩ = 0 for α ∈ J, we have that ⟨x, δ^{∨}⟩ = ⟨−w_{0,J}(w · 0) + lν, δ^{∨}⟩ = ⟨−w_{0,J}(w · 0), δ^{∨}⟩ = (l − 1)|J|. To show that ⟨ν, α^{∨}⟩ = 0 for α ∈ Π\J, we express x in the form x = λ−a+b+z where a, b, and z consist of sums of distinct roots lying in E[> 0], E[< 0], and E[0], respectively. While z can consist of arbitrarily many elements from E[0], a and b are constrained to consist of a small number of elements from E[> 0] or E[< 0] depending upon how close ⟨λ, δ^{∨}⟩ is to (l − 1)|J|. In each case, this can be explicitly described.

Given α ∈ Π\J, ⟨−w_{0,J}(w · 0), α^{∨}⟩ and ⟨λ, α^{∨}⟩ can be explicitly computed. Also, one can find bounds A and B (integers) such that for an arbitirary linear combination z as above, one has A ≤ ⟨z, α^{∨}⟩ ≤ B. Using the expression of x as λ + a − b + z, and considering the possibilities for a and b, one can then obtain bounds A′ ≤ ⟨x, α^{∨}⟩ ≤ B′. In those cases needed, such bounds are listed in Section 9.2. On the other hand,

⟨x, α^{∨}⟩ = ⟨−w_{0,J}(w · 0) + lν, δ^{∨}⟩ = ⟨−w_{0,J}(w · 0), α^{∨}⟩ + l⟨να^{∨}⟩.

Comparing this to the bounds obtained previously, one often finds that ⟨ν, α^{∨}⟩ must be zero. When the bounds allow for a nonzero ν, MAGMA is used to verify that no solutions exist (except in type E_{6} when l = 9) by checking all possibilities for a, b, and z. When needed for efficiency, the known bounds can be used to limit the possible choices for z. The basic details for each case are given below. l = 11: Here λ = −w_{0,J}(w · 0). Hence x = λ + z (i. e., a and b must be empty), and z would need to equal lν. However, for α ∈ Π\J, one finds that −4 ≤ ⟨z, α^{∨}⟩ ≤ 6. Hence, z cannot equal 11ν unless ν = 0.

l = 9: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. Also E[> 0] = E[1] and E[< 0] = E[−1]. So there are three ways we can express x in the form λ − a + b + z:

(1) x = λ − x_{1} − x_{2} + z where x_{1}, x_{2} ∈ E[1] (x_{1} ≠ x_{2}),
(2) x = λ − x_{1} + y_{1} + z where x_{1} ∈ E[1] and y_{1} ∈ E[−1],
(3) x = λ + y_{1} + y_{2} + z where y_{1}, y_{2} ∈ E[−1] (y_{1} ≠ y_{2}).

Using MAGMA, we compute all possibilities. In case (1), we find precisely one pair of elements in E[1] that works with z being an empty sum. That is, −w_{0,J}(w · 0) is a sum of eight distinct roots in E[1]. In case (2), no sums over E[0] work. In case (3), we find however two cases where λ plus two elements of E[−1] and eight elements of E[0] equals −w_{0,J}(w · 0) + 9ν for a J-dominant weight ν. In one case ν = ϖ_{1} and in the other ν = ϖ_{6}. The reader should be aware that these weights give rise to the exceptions stated in Theorem 1.3.2(b)(iii).

l = 7: Here ⟨−w_{0,J}(w·0), δ^{∨}⟩ = ⟨λ, δ^{∨}⟩. Hence x must be of the form x = λ+z. For α ∈ Π\J, one finds the bounds listed in Section 9.2. For each α, in order to have x = −w_{0,J}(w · 0) + 7ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 5: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by one. So there are two ways we can express x in the form λ − a + b + z:

(1) x = λ − x_{1} + z where x_{1} ∈ E[1],
(2) x = λ + y_{1} + z where y_{1} ∈ E[−1].

Using MAGMA, we compute all possibilities and find that only one such x works - in case (1) with z being the sum of a pair of elements from E[0]. Again, x = −w_{0,J}(w · 0), i. e., ν = 0.

Type E_{7}:

l = 17: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, for α ∈ Π\J, one finds that −9 ≤ ⟨z, α^{∨}⟩ ≤ 10. Hence, z cannot equal 17ν unless ν = 0.

l = 15: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. So there are three ways we can express x in the form λ − a + b + z just as in the type E_{6}, l = 9 case. For α ∈ Π\J, one finds the bounds listed in Section 9.2. For each α, in order to have x = −w_{0,J}(w · 0) + 15ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 13: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, for α ∈ Π\J, one finds that −6 ≤ ⟨z, α^{∨}⟩ ≤ 8. Hence, z cannot equal 13ν unless ν = 0.

l = 11: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, for α ∈ Π\J, one finds that −4 ≤ ⟨z, α^{∨}⟩ ≤ 6. Hence, z cannot equal 11ν unless ν = 0.

l = 9: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. Here, for all β ∈ E[> 0], ⟨β, δ^{∨}⟩ ∈ {1, 2, 3}, and for all β ∈ E[< 0], ⟨β, δ^{∨}⟩ ∈ {−1, −2, −3}. So there are five ways we can express x in the form λ − a + b + z. We leave the details to the interested reader.

For α ∈ Π\J, one finds the bounds listed in Section 9.2. For each α, in order to have x = −w_{0,J}(w · 0) + 9ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 7: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by three. Here, for all β ∈ E[> 0], ⟨β, δ^{∨}⟩ ∈ {1, 2, 3}, and for all β ∈ E[< 0], ⟨β, δ^{∨}⟩ ∈ {−1, −2, −3}. So there are ten ways we can express x in the form λ − a + b + z. In this case, for α ∈ Π\J, the bounds on ⟨x, α^{∨}⟩ allow for the possibility that ⟨ν, α^{∨}⟩ ≠ 0. Using MAGMA, we compute all possibilities. We find precisely one case that works (of form x = λ − a where a is a sum of three distinct roots in E[1] and b and z are empty) with x = −w_{0,J}(w · 0), i. e., ν = 0.

l = 5: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by three. As in the l = 7 case, there are ten ways we can express x in the form λ − a + b + z. Here Π\J = {α_{4}} and one finds that −10 ≤ ⟨x, α_{4}^{∨}⟩ ≤ −6. Since ⟨−w_{0,J}(w · 0), α_{4}^{∨}⟩ = −9, in order to have x = −w_{0,J}(w · 0) + 5ν, we must have ⟨ν, α_{4}^{∨}⟩ = 0.

Type E_{8}:

l = 29: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, for α ∈ Π\J, one finds that −16 ≤ ⟨z, α^{∨}⟩ ≤ 18. Hence, z cannot equal 29ν unless ν = 0.

l = 27: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. There are three ways we can express x in the form λ − a + b + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 27ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 25: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by four. There are five ways we can express x in the form λ − a + b + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 25ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 23: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ = ⟨λ, δ^{∨}⟩. Hence x = λ + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 23ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 21: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by four. Here, for all β ∈ E[> 0], ⟨β, δ^{∨}⟩ ∈ {1, 2}, and for all β ∈ E[< 0], ⟨β, δ^{∨}⟩ ∈ {−1, −2}. So there are fourteen ways we can express x in the form λ − a + b + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 21ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 19: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ = ⟨λ, δ^{∨}⟩. Hence x = λ + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 19ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 17: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ = ⟨λ, δ^{∨}⟩. Hence x = λ + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 17ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 15: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by eight. Here, for all β ∈ E[> 0], ⟨β, δ^{∨}⟩ ∈ {1, 2, 3}, and for all β ∈ E[< 0], ⟨β, δ^{∨}⟩ ∈ {−1, −2, −3}. As a result, there are numerous ways we can express x in the form λ − a + b + z. Further, for α ∈ Π\J, the bounds on ⟨x, α^{∨}⟩ allow for the possibility that ⟨ν, α^{∨}⟩ ≠ 0. By analyzing the constraints placed on a, b, and z (to afford a nonzero ν), the resulting possiblities are all computed with MAGMA, and one finds that in fact no nonzero solution exists.

l = 13: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by three. There are ten ways we can express x in the form λ − a + b + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 13ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 11: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. There are five ways we can express x in the form λ − a + b + z. For α ∈ Π\J one finds the bounds listed in Section 9.2. In order to have x = −w_{0,J}(w · 0) + 11ν, we must have ⟨ν, α^{∨}⟩ = 0.

l = 9: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by three. There are ten ways we can express x in the form λ − a + b + z. For α ∈ Π\J, the bounds on ⟨x, α^{∨}⟩ allow for the possibility that ν ≠ 0. To reduce the number of possibilities that need to be checked, observe that there is a sizable difference between ⟨λ, α_{1}^{∨}⟩ = 18 and ⟨−w_{0,J}(w · 0), α_{1}^{∨}⟩ = 8. Since ⟨ν, α_{1}^{∨}⟩ = 0, we must have ⟨−a + b + z, α_{1}^{∨}⟩ = −10. We find that ⟨−a + b, α_{1}^{∨}⟩ ≥ −3 and ⟨z, α_{1}^{∨}⟩ ≥ −8. This reduces the possibilities to a number manageable for MAGMA to compute all the possible cases. The only x that works is precisely −w_{0,J}(w · 0), i. e., ν = 0.

l = 7: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by 8 which is larger than 7. Here we may not immediately conclude that ⟨ν, α^{∨}⟩ = 0 for all α ∈ J. However, since ⟨ν, α^{∨}⟩ ≥ 0 for all α ∈ J, this must be true for all but possibly one α for which one could have ⟨ν, α^{∨}⟩ = 1.

Suppose the latter case holds, then we would have ⟨x, δ^{∨}⟩ = 49 whereas ⟨λ, δ^{∨}⟩ = 50. So there would be only two ways in which x could occur (x = λ − x_{1} + z where x_{1} ∈ E[1] or x = λ+y_{1} +z where y_{1} ∈ E[−1]). Here Π\J = {α_{4}} and one finds that −21 ≤ ⟨x, α_{4}^{∨}⟩ ≤ −19. On the other hand, ⟨−w_{0,J}(w · 0), α_{4}^{∨}⟩ = −17. Since ⟨ν, α_{4}^{∨}⟩ is an integer, these bounds show that we cannot have x = −w_{0,J}(w·0)+7ν for any ν, contradicting our assumption. Therefore ⟨ν, α^{∨}⟩ = 0 for all α ∈ J.

Now, our standard argument can be used to show that ⟨ν, α_{4}^{∨}⟩ = 0. Since ⟨−w_{0,J}(w·0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by 8 there are numerous ways we can express x in the form λ − a + b + z. One finds that −21 ≤ ⟨x, α^{∨}⟩ ≤ −12. Since ⟨−w_{0,J}(w · 0), α_{4}^{∨}⟩ = −17, In order to have x = −w_{0,J}(w · 0) + 7ν, we must have ⟨ν, α_{4}^{∨}⟩ = 0.

Type F_{4}:

l = 11: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, for α ∈ Π\J, one finds that −4 ≤ ⟨z, α^{∨}⟩ ≤ 5. Hence, z cannot equal 11ν unless ν = 0.

l = 9: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by two. There are five ways we can express x in the form λ − a + b + z. For α ∈ Π\J, the bounds on ⟨x, α^{∨}⟩ allow for the possibility that ⟨ν, α^{∨}⟩ ≠ 0. Using MAGMA, we compute all possibilities and see that in fact the only solution is when ν = 0.

l = 7: In this case, ⟨−w_{0,J}(w · 0), δ^{∨}⟩ and ⟨λ, δ^{∨}⟩ differ by one. There are two ways we can express x in the form λ − a + b + z. For α ∈ Π\J, the bounds on ⟨x, α^{∨}⟩ allow for the possibility that ⟨ν, α^{∨}⟩ ≠ 0. Using MAGMA, we compute all possibilities and see that in fact the only solution is when ν = 0. l = 5: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. Here Π\J = {α_{2}}, and one finds that 0 ≤ ⟨z, α_{2}^{∨}⟩ ≤ 1. Hence, z cannot equal 5ν unless ν = 0.

Type G_{2}:

l = 5: Here λ = −w_{0,J}(w · 0). Hence x = λ + z, and z would need to equal lν. However, Π\J = {α_{2}}, and the only choice for z (other than empty) is z = (0, 1) with ⟨z, α_{2}^{∨}⟩ = 1. Hence, z cannot equal 5ν unless ν = 0.

5. Cohomology of H^{•}(u_{ζ}(𝔤), ℂ)

The identification of H^{•}(u_{ζ}(𝔤), ℂ) for small l will proceed in steps. Our computations are motivated by the determination of the support variety V_{𝔤}(F) which may be identified with the restricted nullcone N_{1}(𝔤). By [CLNP], the restricted nullcone N_{1}(𝔤) may be identfied with G · 𝔲_{J} for some subset J ⊂ Π. When p ≥ h, we simply have J = ∅ and 𝔲_{J} = 𝔲. To understand the cohomology of u_{ζ}(𝔤), we consider the parabolic subgroup P_{J} associated to this subset J ⊂ Π with w(Φ_{0}) = Φ_{w·0} = Φ_{J}. In Section 5.1, the cohomology of u_{ζ}(𝔤) will be related to that of u_{ζ}(𝔭_{J}), and in Section 5.2, the latter cohomology will be related to the cohomology of u_{ζ}(𝔲_{J}). Section 5.3 presents a key computation for H^{•}(u_{ζ}(𝔲_{J}), ℂ) which will then be used in Section 5.5 to compute H^{•}(u_{ζ}(𝔤), ℂ)

Throughout this section, fix J ⊆ Π. While the goal of this paper is to make cohomological computations in the case that l < h, we note that the arguments are also valid for l ≥ h. In that case, we would have Φ_{0} = ∅, J = ∅, and w = Id. Then P_{J} = B, U_{J} = U, L_{J} = T, 𝔭_{J} = 𝔟, 𝔲_{J} = 𝔲, 𝔩_{J} = 𝔱, and the module M = (ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗} = ℂ. As such, our results encompass the known results for large l in [GK].

5.1. Let w ∈ W with w(Φ_{0}) = Φ_{J}. By Lemma 4.1.1, w · 0 is dominant on J. There exists a spectral sequence (cf. [Jan1, I 4.5]):

(5.1.1)  E_{2}^{i,j} = R^{i}ind_{U_{ζ}(𝔭_{J})}^{U_{ζ}(𝔤)}R^{j}ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0 ⇒ R^{i+j}ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔤)}w · 0.

But, w ·0 is dominant on J so R^{j}ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w ·0 = 0 for j > 0. This spectral sequence collapses and yields by [A, Cor. 3.8]

(5.1.2)  R^{i}ind_{U_{ζ}(𝔭_{J})}^{U_{ζ}(𝔤)}(ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0) = R^{i}ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔤)}w · 0 = \begin{cases}ℂ & if i = ℓ(w) \\ 0 & if i ≠ ℓ(w). \end{cases}

The following spectral sequence provides a connection between the cohomology of u_{ζ}(𝔭_{J}) and that of u_{ζ}(𝔤).

Theorem 5.1.1. Let w ∈ W such that w(Φ_{0}) = Φ_{w·0} = Φ_{J} where J ⊆ Π. There exists a first quadrant spectral sequence

E_{2}^{i,j} = R^{i} ind_{P_{J}}^{G} H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0) ⇒ H^{i+j−ℓ(w)}(u_{ζ}(𝔤), ℂ).

Proof. We follow the construction in [Jan1, I 6.12]. Consider the functors

F_{1}(−) = Hom_{u_{ζ}(𝔤)}(ℂ, ind_{U_{ζ}(𝔭_{J})}^{U_{ζ}(𝔤)}(−)) and F_{2}(−) = ind_{P_{J}}^{G} Hom_{u_{ζ}(𝔭_{J})}(ℂ, −)

with F_{1}, F_{2} : U_{ζ}(𝔭_{J})-mod → U_{ζ}(𝔤)-mod (or G-mod). The reader should observe that we are implicitly using the Frobenius map (cf. Section 2.3) and the following identification of functors:

ind_{P_{J}}^{G} (−) ≅ ind_{U_{ζ}(𝔭_{J})//u_{ζ}(𝔭_{J})}^{U_{ζ}(𝔤)//u_{ζ}(𝔤)}(−).

The functors F_{1} and F_{2} are naturally isomorphic and there exist two spectral sequences:

E_{2}^{i,j} = H^{i}(u_{ζ}(𝔤), R^{j}ind_{U_{ζ}(𝔭_{J})}^{U_{ζ}(𝔤)}(ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)) ⇒ (R^{i+j}F_{1})(ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0);

(5.1.3)  E_{2}^{i,j} = R^{i}ind_{P_{J}}^{G} H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0) ⇒ (R^{i+j}F_{2})(ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)

necessarily converging to the same abutment.

By (5.1.2), the first spectral sequence collapses and we can identify

(R^{•+ℓ(w)}F_{1})(ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0) ≅ H^{•}(u_{ζ}(𝔤), R^{ℓ(w)}ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔤)}w · 0) ≅ H^{•}(u_{ζ}(𝔤), ℂ).

Combining this with the second spectral sequence (5.1.3) proves the theorem. _{□}

5.2. We can reidentify the E_{2}-terms of the spectral sequence in Theorem 5.1.1 by using the spectral sequence in Lemma 2.8.1 for u_{ζ}(𝔲_{J}) ⊴ u_{ζ}(𝔭_{J}). Note that u_{ζ}(𝔭_{J})//u_{ζ}(𝔲_{J}) ≅ u_{ζ}(𝔩_{J}). We have

E_{2}^{i,j} = H^{i}(u_{ζ}(𝔩_{J}), H^{j}(u_{ζ}(𝔲_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)) ⇒ H^{i+j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0).

All of the cohomology groups involved admit an action of U_{ζ}(𝔭_{J}) induced from the \overline{Ad}-action. Further, the action of the subalgebra u_{ζ}(𝔭_{J}) is trivial. Hence, there is an action of 𝕌(𝔭_{J}) (or equivalently P_{J}). Moreover, the spectral sequence preserves this action.

Since ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w·0 is trivial as a U_{ζ}(𝔲_{J})-module, the left-hand side of the spectral sequence may be reidentified as follows:

(5.2.1) E_{2}^{i,j} = H^{i}(u_{ζ}(𝔩_{J}), H^{j}(u_{ζ}(𝔲_{J}), ℂ) ⊗ ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0) ⇒ H^{i+j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0).

Proposition 5.2.1. Let w ∈ W and J ⊂ Π be such that w(Φ_{0}) = Φ_{J}. Then for all j ≥ 0 there is an isomorphism of 𝕌(𝔭_{J})-modules

H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0) ≅ Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{j}(u_{ζ}(𝔲_{J}), ℂ)).

Proof. Since ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0 is projective as a u_{ζ}(𝔩_{J})-module, in the spectral sequence (5.2.1),

E_{2}^{i,j} = 0 for all i > 0. Thus the spectral sequence collapses giving for all j ≥ 0

H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0) ≃ Hom_{u_{ζ}(𝔩_{J})}(ℂ, H^{j}(u_{ζ}(𝔲_{J}), ℂ) ⊗ ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)
≃ Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{j}(u_{ζ}(𝔲_{J}), ℂ)).
□

5.3. The following theorem now gives an identification as U_{ζ}^{0}-modules (or equivalently 𝕌(𝔥)-modules where 𝔥 ⊂ 𝔤 is the Cartan subalgebra) of the Hom-group appearing in Proposition 5.2.1.

Theorem 5.3.1. Let l be as in Assumption 1.3.1 and w ∈ W such that w(Φ_{0}) = Φ_{J}.

(a) Suppose that l ∤ n + 1 when Φ is of type A_{n} and l ≠ 9 when Φ is of type E_{6}. Then as U_{ζ}^{0}-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{s}(u_{ζ}(𝔲_{J}), ℂ)) ≅ S^{\frac{s−ℓ(w)}{2}} (𝔲_{J}^{∗})^{[1]}.

(b) If Φ is of type A_{n} with n + 1 = l(m + 1) and w ∈ W is as defined in (4.8.1), then as U_{ζ}^{0}-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{s}(u_{ζ}(𝔲_{J}), ℂ)) ≅ ⨁_{t=0}^{l−1} S^{\frac{s−ℓ(w)−(m+1)t(l−t)}{2}} (𝔲_{J}^{∗})^{[1]} ⊗ lϖ_{t(m+1)},

where ϖ_{0} = 0.

(c) If Φ is of type E_{6} and l = 9 (assuming that w and J are as in Section 9.1), then as U_{ζ}^{0}-modules

Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗}, H^{s}(u_{ζ}(𝔲_{J}), ℂ)) ≅ S^{\frac{s−ℓ(w)}{2}} (𝔲_{J}^{∗})^{[1]} ⊕ (S^{\frac{s−20}{2}} (𝔲_{J}^{∗})^{[1]} ⊗ lϖ_{1})
⊕ (S^{\frac{s−20}{2}} (𝔲_{J}^{∗})^{[1]} ⊗ lϖ_{6}) .

Proof. For convenience set M = (ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0)^{∗} and G(−) = Hom_{u_{ζ}(𝔩_{J})}(M, −). Since the module M is injective (equivalently projective) as a u_{ζ}(𝔩_{J})-module, the functor G(−) = Hom_{u_{ζ}(𝔩_{J})}(M, −) is an exact functor.

The argument will proceed by induction on successive quotients of U_{ζ}(𝔲_{J}) (cf. [GK, 2.4]). Let N = |Φ^{+}\Φ_{J}^{+}|. Note that N previously was used to denote |Φ^{+}|, but this should not cause any confusion here. As in Section 2, choose any fixed ordering of root vectors f_{1}, f_{2}, …, f_{N} in U_{ζ}(𝔲_{J}) corresponding to the positive roots in Φ^{+}\Φ_{J}^{+}. For purposes of this argument, the precise ordering is irrelevent. Recall that each f_{i}^{l} is central in U_{ζ}(𝔲_{J}). For 0 ≤ i ≤ N, set A_{i} = U_{ζ}(𝔲_{J})//⟨f_{1}^{l}, f_{2}^{l}, …, f_{i}^{l}⟩ where ⟨… ⟩ denotes “the subalgebra generated by … ” (with A_{0} = U_{ζ}(𝔲_{J})). Note that A_{N} = u_{ζ}(𝔲_{J}). For 1 ≤ i ≤ N, set B_{i} = ⟨f_{i}^{l}⟩ ⊂ A_{i−1} be the subalgebra generated by f_{i}^{l}. Note that each B_{i} is a polynomial algebra in one variable. Note also that B_{i} is normal (in fact central) in A_{i−1}, and A_{i−1}//B_{i} ≅ A_{i}. For 0 ≤ i ≤ N, let V_{i} be an i-dimensional vector space with basis {x_{1}, x_{2}, …, x_{i}}. Further consider V_{i} as a U_{ζ}^{0}-module by letting x_{i} have weight γ_{i}. That is, each x_{i} is “dual” to the element f_{i}. In particular, V_{N} ≅ 𝔲_{J}^{∗}. Here V_{0} = {0}.

Consider first part (a). We prove inductively for 0 ≤ i ≤ N that as U_{ζ}^{0}-modules we have

G( H^{s}(A_{i}, ℂ))) ≅ \begin{cases}S^{r}(V_{i})^{[1]} & if s = 2r + ℓ(w) \\ 0 & else. \end{cases}

The case i = N is precisely the statement of the theorem. For i = 0, this is precisely Theorem 4.3.1 where by convention we take S^{0}(V_{0}) = ℂ.

Assume now that the claim is true for i − 1, and we will show that it is true for i. We will make use of the LHS spectral sequence of Lemma 2.8.1 for B_{i}⊴A_{i−1}. Since B_{i} is a polynomial algebra in one variable, its cohomology is an exterior algebra in one variable. Precisely, for each i, we have as a U_{ζ}^{0}-module:

H^{b}(B_{i}, ℂ) = \begin{cases}ℂ & if b = 0 \\ ℂ_{i}^{[1]} & if b = 1 \\ 0 & else, \end{cases}

where ℂ_{i} is a one-dimensional vector space with basis element x_{i} (i. e., of weight γ_{i}). The spectral sequence is

E_{2}^{a,b} = H^{a}(A_{i}, H^{b}(B_{i}, ℂ)) ⇒ H^{a+b}(A_{i−1}, ℂ).

The algebra A_{i} acts on B_{i} via the \overline{Ad}-action. By Corollary 2.7.4(b), this action is trivial. Therefore this spectral sequence can be rewritten as

E_{2}^{a,b} = H^{a}(A_{i}, ℂ) ⊗ H^{b}(B_{i}, ℂ) ⇒ H^{a+b}(A_{i−1}, ℂ).

Alternately, this easily follows from the above description of H^{b}(B_{i}, ℂ).

Again using the \overline{Ad}-action, u_{ζ}(𝔩_{J}) acts on all terms of the spectral sequence and this action is preserved by the differentials. Since the functor G(−) is exact, we can apply it to the above spectral sequence and obtain a new spectral sequence. For convenience, we abusively use the same name:

E_{2}^{a,b} = G( H^{a}(A_{i}, ℂ) ⊗ H^{b}(B_{i}, ℂ)) ⇒ G( H^{a+b}(A_{i−1}, ℂ)).

The above description of H^{b}(B_{i}, ℂ) shows that u_{ζ}(𝔩_{J}) acts trivially on it. And hence, this spectral sequence may be rewritten as

E_{2}^{a,b} = G( H^{a}(A_{i}, ℂ)) ⊗ H^{b}(B_{i}, ℂ) ⇒ G( H^{a+b}(A_{i−1}, ℂ)).

Observe that E_{2}^{a,b} = 0 for b ≥ 2. That is, the spectral sequence consists of at most two nonzero rows. So only the first differential d_{2} : E_{2}^{a,1} → E_{2}^{a+2,0} could potentially be nonzero. The first row E_{2}^{a,0} = H^{a}(A_{i}, ℂ) is precisely what we are trying to identify inductively. Note also that for all a, the second row E_{2}^{a,1} ≅ E_{2}^{a,0} ⊗ ℂ_{i}^{[1]}. In particular, E_{2}^{a,1} ≠ 0 if and only if E_{2}^{a,0} ≠ 0.

By the inductive hypothesis, we know that the abutment

G( H^{a+b}(A_{i−1}, ℂ)) ≅ \begin{cases}S^{r}(V_{i−1})^{[1]} & if a + b = 2r + ℓ(w) \\ 0 & else. \end{cases}

In particular, G( H^{a+b}(A_{i−1}, ℂ)) = 0 for a + b < ℓ(w).

Let A ≥ 0 be least value of a such that E_{2}^{a,0} ≠ 0. Hence, A is necessarily the least value of a such that E_{2}^{a,1} ≠ 0. In particular, E_{2}^{A−2,1} = 0 and hence E_{∞}^{A,0} ≅ E_{2}^{A,0}/d_{2}(E_{2}^{A−2,1}) = E_{2}^{A,0}. By the inductive hypothesis, we conclude that A = ℓ(w). So for all a < ℓ(w) we have

G( H^{a}(A_{i}, ℂ)) = 0.

Next we claim that E_{2}^{ℓ(w)+a,0} = 0 = E_{2}^{ℓ(w)+a,1} for all odd a > 0. This can be seen inductively on a. For example, since E_{2}^{ℓ(w)−1,1} = 0,

E_{2}^{ℓ(w)+1,0} = E_{2}^{ℓ(w)+1,0}/d_{2}(E_{2}^{ℓ(w)−1,1}) ≅ E_{∞}^{ℓ(w)+1,0} ⊂ G( H^{ℓ(w)+1}(A_{i−1}, ℂ)) = 0.

Inductively, for odd a > 0, we similarly have E_{2}^{ℓ(w)+a−2,1} = 0 and so

E_{2}^{ℓ(w)+a,0} = E_{2}^{ℓ(w)+a,0}/d_{2}(E_{2}^{ℓ(w)+a−2,1}) ≅ E_{∞}^{ℓ(w)+a,0} ⊂ G( H^{ℓ(w)+a}(A_{i−1}, ℂ)) = 0.

In other words, for all odd a > 0, we have (as claimed)

G( H^{ℓ(w)+a}(A_{i}, ℂ)) = 0.

Summarizing: our spectral sequence has E_{2}^{a,b} = 0 for a < ℓ(w) and E_{2}^{ℓ(w)+a,b} = 0 for all odd a > 0. That is, the columns are initially zero and then begin to alternate nonzero (potentially) and zero thereafter.

Further, for all even a ≥ 0, we then have

ker{d_{2} : E_{2}^{ℓ(w)+a,1} → E_{2}^{ℓ(w)+a+2,0}} ⊂ E_{∞}^{ℓ(w)+a,1} ⊂ G( H^{ℓ(w)+a+1}(A_{i−1}, ℂ)) = 0.

Therefore d_{2} : E_{2}^{ℓ(w)+a,1} → E_{2}^{ℓ(w)+a+2,0} is injective. Hence, we have for all even a ≥ 0,

E_{2}^{ℓ(w)+a,0}/E_{2}^{ℓ(w)+a−2,1} ≅ E_{2}^{ℓ(w)+a,0}/d_{2}(E_{2}^{ℓ(w)+a−2,1}) ≅ E_{∞}^{ℓ(w)+a,0}
≅ G( H^{ℓ(w)+a}(A_{i−1}, ℂ)) = S^{a/2}(V_{i−1})^{[1]}.

So we have a short exact sequence of U_{ζ}^{0}-modules:

0 → E_{2}^{ℓ(w)+a−2,1} → E_{2}^{ℓ(w)+a,0} → S^{a/2}(V_{i−1})^{[1]} → 0.

But identifying these E_{2}-terms gives

0 → G( H^{ℓ(w)+a−2}(A_{i}, ℂ)) ⊗ ℂ_{i}^{[1]} → G( H^{ℓ(w)+a}(A_{i}, ℂ)) → S^{a/2}(V_{i−1})^{[1]} → 0.

Inducting now on even a ≥ 0, we may assume that

G( H^{ℓ(w)+a−2}(A_{i}, ℂ)) = S^{(a−2)/2}(V_{i})^{[1]}.

The short exact sequence becomes

0 → S^{(a−2)/2}(V_{i})^{[1]} ⊗ ℂ_{i}^{[1]} → G( H^{ℓ(w)+a}(A_{i}, ℂ)) → S^{a/2}(V_{i−1})^{[1]} → 0

or setting a = 2r,

0 → S^{r−1}(V_{i})^{[1]} ⊗ ℂ_{i}^{[1]} → G( H^{ℓ(w)+2r}(A_{i}, ℂ)) → S^{r}(V_{i−1})^{[1]} → 0.

Hence as a U_{ζ}^{0}-module,

G( H^{ℓ(w)+2r}(A_{i}, ℂ)) ≅ (S^{r−1}(V_{i})^{[1]} ⊗ ℂ_{i}^{[1]}) ⊕ S^{r}(V_{i−1})^{[1]}.

The left hand factor consists of r-fold symmetric powers in the x_{j} which contain at least one x_{i}, while the righthand factor consists of r-fold symmetric powers in x_{j} with 1 ≤ j ≤ i − 1. Hence

G( H^{ℓ(w)+2r}(A_{i}, ℂ)) ≅ S^{r}(V_{i})^{[1]}

which along with the above conclusions verifies the inductive claim and hence part (a) of the theorem.

For parts (b) and (c) a similar argument can be used whose details are left to the interested reader. Of crucial importance here is the degrees in which the “extra” classes arise in parts (b) and (c) of Theorem 4.3.1. For example, consider part (b). We have G( H^{i}(U_{ζ}(𝔲_{J}), ℂ)) = 0 unless i = ℓ(w) + (m + 1)t(l − t) for 0 ≤ t ≤ l − 1. Observe that t(l − t) (and, hence, (m + 1)t(l − t)) is necessarily even. Thus the extra cohomology classes appear in degrees having the same parity as ℓ(w). As such, in the above argument, we will have a similar phenomenon happening in the spectral sequence: E_{2}^{a,b} = 0 for a < ℓ(w) or a = ℓ(w) + a′ with a′ > 0 being odd, and the analogous argument will give the claim. Similary in part (c), we see that the extra cohomology classes lie in a degree with the same parity as ℓ(w).

□

5.4. As mentioned previously, the Hom-groups in Theorem 5.3.1 admit an action of 𝕌(𝔭_{J}) induced from the \overline{Ad}-action. On the other hand 𝕌(𝔭_{J}) acts naturally on 𝔲_{J} by the adjoint action or on 𝔲_{J}^{∗} by the coadjoint action. This can be further extended to an action on S^{•}(𝔲_{J}^{∗}). With this action, the isomorphisms in Theorem 5.3.1 also hold as 𝕌(𝔭_{J})-modules (or equivalently P_{J}-modules).

Lemma 5.4.1. The isomorphisms of Theorem 5.3.1 also hold as 𝕌(𝔭_{J})-modules where the actions are as described above.

Proof. Consider first the generic case - part (a). We use notation as in the proof of Theorem 5.3.1. Let Z_{J} = ⟨f_{1}^{l}, …, f_{N}^{l}⟩ ⊂ U_{ζ}(𝔲_{J}) be the central subalgebra such that U_{ζ}(𝔲_{J})//Z_{J} ≅ u_{ζ}(𝔲_{J}) (cf. also Section 2.7). Consider the spectral sequence

E_{2}^{a,b} = H^{a}(u_{ζ}(𝔲_{J}), H^{b}(Z_{J}, ℂ)) ⇒ H^{a+b}(U_{ζ}(𝔲_{J}), ℂ)

given by Lemma 2.8.1. This is a spectral sequence of U_{ζ}(𝔭_{J})-modules. Since u_{ζ}(𝔭_{J}) acts trivially on H^{b}(Z_{J}, ℂ) (cf. Corollary 2.7.4(b)), this may be rewritten as

E_{2}^{a,b} = H^{a}(u_{ζ}(𝔲_{J}), ℂ) ⊗ H^{b}(Z_{J}, ℂ)) ⇒ H^{a+b}(U_{ζ}(𝔲_{J}), ℂ).

Further, applying the functor G(−) we get a new spectral sequence (using the same name)

E_{2}^{a,b} = G( H^{a}(u_{ζ}(𝔲_{J}), ℂ)) ⊗ H^{b}(Z_{J}, ℂ) ⇒ G( H^{a+b}(U_{ζ}(𝔲_{J}), ℂ)),

whose differentials still preserve the action of U_{ζ}(𝔭_{J}). Since u_{ζ}(𝔲_{J}) acts trivially on both H^{•}(u_{ζ}(𝔲_{J}), ℂ) and H^{•}(U_{ζ}(𝔲_{J}), ℂ), and u_{ζ}(𝔩_{J}) ≅ u_{ζ}(𝔭_{J})//u_{ζ}(𝔲_{J}), we have

G( H^{•}(u_{ζ}(𝔲_{J}), ℂ)) = Hom_{u_{ζ}(𝔩_{J})}(M, H^{•}(u_{ζ}(𝔲_{J}), ℂ)) ≅ Hom_{u_{ζ}(𝔭_{J})}(M, H^{•}(u_{ζ}(𝔲_{J}), ℂ))

and

G( H^{•}(U_{ζ}(𝔲_{J}), ℂ)) = Hom_{u_{ζ}(𝔩_{J})}(M, H^{•}(U_{ζ}(𝔲_{J}), ℂ)) ≅ Hom_{u_{ζ}(𝔭_{J})}(M, H^{•}(U_{ζ}(𝔲_{J}), ℂ)).

Therefore, u_{ζ}(𝔭_{J}) acts trivially on this new spectral sequence, and so this is a spectral sequence of 𝕌(𝔭_{J})-modules.

By Theorem 4.3.1, the abutment is nonzero only when a + b = ℓ(w) in which case it is the trivial module ℂ. And by Theorem 5.3.1, as 𝕌(𝔥)-modules, we know that G( H^{a}(u_{ζ}(𝔲_{J}), ℂ)) ≅

S^{\frac{a−ℓ(w)}{2}} (𝔲_{J}^{∗}). Also, the algebra Z_{J} is a polynomial algebra and so its cohomology as an algebra is an exterior algebra. Moreover, by [ABG, Cor. 2.9.6], as 𝕌(𝔟)-modules, H^{•}(Z_{J}, ℂ) ≅ Λ^{•}(𝔲_{J}^{∗}) (i. e., the ordinary exterior algebra on 𝔲_{J}^{∗}), where the action of 𝕌(𝔟) on Λ^{•}(𝔲_{J}^{∗}) is given by the coadjoint action. Since H^{•}(Z_{J}, ℂ) is a 𝕌(𝔭_{J})-module and the coadjoint action on Λ^{•}(𝔲_{J}^{∗}) can be extended to 𝔭_{J}, by applying ind_{𝕌(𝔟)}^{𝕌(𝔭_{J})}(−), we see that H^{•}(Z_{J}, ℂ) ≅ Λ^{•}(𝔲_{J}^{∗}) as 𝔭_{J}-modules.

As in Theorem 5.3.1, E_{2}^{a,b} = 0 for a < ℓ(w) and E_{2}^{ℓ(w)+a,b} = 0 for odd a. Consider the term

E_{2}^{ℓ(w)+2,0} = G( H^{ℓ(w)+2}(u_{ζ}(𝔲_{J}), ℂ)) ≅ S^{1}(𝔲_{J}^{∗}) = 𝔲_{J}^{∗},

where the latter identifications are as 𝕌(𝔥)-modules. We will see this also holds as 𝕌(𝔭_{J})- modules. Since E_{∞}^{ℓ(w),1} = 0 and E_{∞}^{ℓ(w)+2,0} = 0, the differential d_{2} : E_{2}^{ℓ(w),1} → E_{2}^{ℓ(w)+2,0} is an isomorphism of 𝕌(𝔭_{J})-modules.

Since E_{2}^{ℓ(w),1} = H^{1}(Z_{J}, ℂ) ≅ 𝔲_{J}^{∗} as a 𝕌(𝔭_{J})-module and E_{2}^{ℓ(w)+2,0} = G( H^{ℓ(w)+2}(u_{ζ}(𝔲_{J}), ℂ)) ≅ 𝔲_{J}^{∗} as a 𝕌(𝔥)-module, this latter identification must also hold as a 𝕌(𝔭_{J})-module.

As already noted, the 𝕌(𝔭_{J})-structure of the terms

E_{2}^{ℓ(w),b} = H^{b}(Z_{J}, ℂ) ≅ Λ^{b}(𝔲_{J}^{∗})

is determined by the coadjoint action. Further, we can now say that the 𝕌(𝔭_{J})-structure of the terms

E_{2}^{ℓ(w)+2,b} = G( H^{ℓ(w)+2}(u_{ζ}(𝔲_{J}), ℂ)) ⊗ H^{b}(Z_{J}, ℂ) ≅ 𝔲_{J}^{∗} ⊗ Λ^{b}(𝔲_{J}^{∗})

is determined by the coadjoint action. Therefore, since E_{∞}^{ℓ(w)+4,0} = 0, the 𝕌(𝔭_{J})-structure of

E_{2}^{ℓ(w)+4,0} = G( H^{ℓ(w)+4}(u_{ζ}(𝔲_{J}), ℂ)) ≅ S^{2}(𝔲_{J}^{∗})

(with isomorphism as a 𝕌(𝔥)-module), which is determined by the structure on E_{2}^{ℓ(w),b} and E_{2}^{ℓ(w)+2,b}, must also be determined by the coadjoint action. Inductively, we conclude that indeed the 𝕌(𝔭_{J})-action on

G( H^{a}(u_{ζ}(𝔲_{J}), ℂ)) ≅ S^{\frac{a−ℓ(w)}{2}} (𝔲_{J}^{∗})

is given by the coadjoint action.

For parts (b) and (c), a similar argument may be used. The key here (as in Theorem 5.3.1) is that the additional classes in G( H^{i}(𝔲_{J}, ℂ)) lie in degrees which have the same parity as ℓ(w). Further, the additional classes have distinct (nonzero) weights which are not strongly linked as noted in Remark 4.3.2. _{□}

5.5. Proof of Theorem 1.3.2. Let us recall the following vanishing result due to Broer [Br2, Thm. 2.2] and extended to a more general setting by Sommers [So1, Prop. 4.2]. Let J be an arbitrary subset of Π. Then

(5.5.1)  R^{i}ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ⊗ λ = 0

for i > 0 and λ ∈ X(P_{J})_{+} where X(P_{J})_{+} is the set of dominant weights inside the character group X(P_{J}). This vanishing result is proved using the Grauert-Riemenschneider theorem. We can now prove the first of our main theorems which provides a precise description of the cohomology of quantum groups at roots of unity in the case that ζ is a primitive lth root of unity.

Proof. (i) According to Proposition 5.2.1, Theorem 5.3.1(a), and Lemma 5.4.1, we have, as 𝕌(𝔭_{J})-modules,

H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})} w · 0) ≅ S^{\frac{j−ℓ(w)}{2}} (𝔲_{J}^{∗}).

By substituting this identification into the spectral sequence given in Theorem 5.1.1, we have

E_{2}^{i,j} = R^{i} ind_{P_{J}}^{G} S^{\frac{j−ℓ(w)}{2}} (𝔲_{J}^{∗}) ⇒ H^{i+j−ℓ(w)}(u_{ζ}(𝔤), ℂ).

We can now apply (5.5.1) to conclude that E_{2}^{i,j} = 0 for i > 0, thus the spectral sequence collapses to yield

H^{s}(u_{ζ}(𝔤), ℂ) = ind_{P_{J}}^{G} S^{\frac{s}{2}} (𝔲_{J}^{∗}).

This gives part (a) for those types listed in (b)(i). According to Theorem 3.7.1, ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[G · 𝔲_{J}] ≅ ℂ[N (Φ_{0})], which gives part (b)(i).

(ii) In this case we have the following spectral sequence (using Proposition 5.2.1, Theorem 5.3.1(b), and Lemma 5.4.1),

E_{2}^{i,j} = ⨁_{t=0}^{l−1} R^{i} ind_{P_{J}}^{G} S^{\frac{j−ℓ(w)−(m+1)t(l−t)}{2}} (𝔲_{J}^{∗}) ⊗ ϖ_{t(m−1)} ⇒ H^{i+j−ℓ(w)}(u_{ζ}(𝔤), ℂ).

One can again apply (5.5.1) because we are tensoring the symmetric algebra by weights in X(P_{J})_{+}, thus the spectral sequence collapses and yields part (ii) of (a) and (b). One can argue similarly for part (iii) using Theorem 5.3.1(c). In that case ℓ(w) = 8. _{□}

6. Finite Generation.

This section provides provide a proof of Theorem 1.3.4 in Section 1.3.

6.1. For part (a), if l ∤ n + 1 when Φ is of type A_{n}, l ≠ 9 when Φ is of type E_{6}, and l ≠ 7, 9 when Φ is of type E_{8}, then H^{•}(u_{ζ}(𝔤), ℂ) is by Theorem 1.3.2 the coordinate algebra of the affine variety N (Φ_{0}), so the cohomology algebra is a finitely generated ℂ-algebra. If l | n + 1 when Φ is of type A_{n} or l = 9 when Φ is of type E_{6}, we have H^{•}(u_{ζ}(𝔤), ℂ) ≅ ⨁ ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗})⊗B_{t} being a finite sum where each B_{t} is finite dimensional. Thus, the algebra R := H^{•}(u_{ζ}(𝔤), ℂ) is finitely generated as a module over the subring ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[N (Φ_{0})] so, by the Hilbert Basis Theorem, R is finitely generated. Lastly, if l = 7, 9 when Φ is of type E_{8}, then H^{•}(u_{ζ}(𝔤), ℂ) ≅ ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ≅ ℂ[G ×_{P_{J}} 𝔲_{J}] (cf. Section 3.7). Let O denote the corresponding nilpotent orbit (cf. Section 9.1). That is, G · 𝔲_{J} = \overline{O}. Since the moment map G ×_{P_{J}} 𝔲_{J} → G · 𝔲_{J} is a resolution of singularities (that portion of the proof of Theorem 3.7.1 still holds), ℂ[G ×_{P_{J}} 𝔲_{J}] ≅ ℂ[O] (cf. [KP] or [So1]). We have ℂ[\overline{O}] ⊆ ℂ[O] and, by [Jan3, Prop. 8.3], ℂ[O] is the integral closure of ℂ[\overline{O}] in its field of fractions. Since ℂ[\overline{O}] = ℂ[G · 𝔲_{J}] is finitely generated, ℂ[O] ≅ ℂ[G ×_{P_{J}} 𝔲_{J}] ≅ H^{•}(u_{ζ}(𝔤), ℂ) is also finitely generated by [Mat, p. 261-63] and [Kun, Cor. 2.4].

Remark 6.1.1. Part (a) of Theorem 1.3.4 holds if l is as in Assumption 1.3.1.

6.2. Part (b) will be proved in Section 6.3. We first make some preliminary observations. In Section 2.9, a filtration was introduced on U_{ζ}(𝔲_{J}) which can be restricted to u_{ζ}(𝔲_{J}). Since u_{ζ}(𝔲_{J}) is finite dimensional, the induced filtration on the cobar complex computing the cohomology H^{•}(u_{ζ}(𝔲_{J}), ℂ) is finite. As in the proof of part (b) of Proposition 2.9.1, there is a spectral sequence as follows.

Lemma 6.2.1. There is a spectral sequence

E_{1}^{i,j} = H^{i+j}(gr u_{ζ}(𝔲_{J}), ℂ)_{(i)} ⇒ H^{i+j}(u_{ζ}(𝔲_{J}), ℂ)

of graded algebras and U_{ζ}^{0}-modules.

Note that since the filtration on u_{ζ}(𝔲_{J}) is finite, this spectral sequence has only finitely many columns, and hence eventually stops.

Globally, we have E_{1}^{•,•} ≅ H^{•}(gr u_{ζ}(𝔲_{J}), ℂ). By [GK, Prop. 2.3.1], there exists a graded algebra isomorphism

(6.2.1)  H^{•}(gr u_{ζ}(𝔲_{J}), ℂ) ≅ S^{•}(𝔲_{J}^{∗}) ⊗ Λ_{ζ,J}^{•}.

This is also an isomorphism of U_{ζ}^{0}-modules with u_{ζ}^{0} acting trivially on the symmetric algebra. Further, under the isomorphism (6.2.1),

(6.2.2)  H^{n}(gr u_{ζ}(𝔲_{J}), ℂ) ≅ ⨁_{2a+b=n} S^{a}(𝔲_{J}^{∗})^{[1]} ⊗ Λ_{ζ,J}^{b}.

By the isomorphism (6.2.1), E_{1}^{•,•} is finitely generated over a subalgebra which is isomorphic (as algebras and U_{ζ}^{0}-modules) to S^{•}(𝔲_{J}^{∗})^{[1]}. For notational convenience, we abusively consider S^{•}(𝔲_{J}^{∗})^{[1]} as a subalgebra of E_{1}^{•,•}. Under mild conditions on l, this subalgebra consists of universal cycles.

Proposition 6.2.2. Let l satisfy Assumption 1.3.3 and J ⊆ Π. In the spectral sequence of Lemma 6.2.1, d_{1}(S^{•}(𝔲_{J}^{∗})^{[1]}) = 0.

Proof. From (6.2.2), the submodule S^{1}(𝔲_{J}^{∗})^{[1]} is identified with a submodule of H^{2}(gr u_{ζ}(𝔲_{J}), ℂ). As such, the image of S^{1}(𝔲_{J}^{∗})^{[1]} under d_{1} must lie in

H^{3}(gr u_{ζ}(𝔲_{J}), ℂ) ≅ (S^{1}(𝔲_{J}^{∗})^{[1]} ⊗ Λ_{ζ,J}^{1}) ⊕ Λ_{ζ,J}^{3}.

A U_{ζ}^{0}-homogeneous element x_{σ} of S^{1}(𝔲_{J}^{∗})^{[1]} has weight lσ for some σ ∈ Φ^{+} − Φ_{J}^{+}. Hence, by weight considerations, if the image of x_{σ} is not zero, it cannot lie in S^{1}(𝔲_{J}^{∗})^{[1]} ⊗ Λ_{ζ,J}^{1}. On the other hand, for x_{σ} to have nonzero image in Λ_{ζ,k}^{3}, we must have lσ = γ_{1} + γ_{2} + γ_{3} for three distinct (positive) roots γ_{i} ∈ Φ^{+} − Φ_{J}^{+}. Under the given conditions on l, this is not possible. To see this, we argue by the type of Φ.

For any weight η which lies in the positive root lattice, we can write η = ∑_{β∈Π} n_{η,β}β for unique n_{η,β} ∈ ℤ_{≥0}. Set γ := γ_{1} + γ_{2} + γ_{3} for γ_{i} as above. In order to have γ = lσ, l must divide n_{γ,β} for each β ∈ Π.

In type A_{n}, for each γ_{i} and β ∈ Π, we have n_{γ_{i},β} ≤ 1. Hence n_{γ,β} ≤ 3, and the claim immediately follows for l > 3. For l = 3, we could only have 3σ = γ if γ_{1} = γ_{2} = γ_{3} which contradicts our assumption.

For types B_{n}, C_{n}, and D_{n}, n_{γ,β} ≤ 6. Hence the claim follows if l ≥ 7. Suppose l = 5 and n_{γ,β} = 5 for some β ∈ Π. Without a loss of generality we may assume that n_{γ_{1},β} = 2, n_{γ_{2},β} = 2, and n_{γ_{3},β} = 1. Observe that there is necessarily some β′ ∈ Π such that n_{γ_{1},β′} = 1, n_{γ_{2},β′} = 1, and n_{γ_{3},β′} ∈ {0, 1}. Thus n_{γ,β′} ∈ {2, 3} and so cannot be divisible by 5. In types B_{n} and C_{n} when l = 3, the condition can be satisfied (e.g., in type B_{2}, α_{1} + (α_{1} + α_{2}) + (α_{1} + 2α_{2}) = 3(α_{1} + α_{2})).

On the other hand, in type D_{n}, the condition is still not possible when l = 3. To see this, suppose on the contrary that 3σ = γ_{1} + γ_{2} + γ_{3} = γ. If n_{γ_{i},β} ≤ 1 for each γ_{i} and all β ∈ Π, then we are done as in type A_{n}. Suppose now that n_{γ_{1},β} = 2 for some β ∈ Π. Recall that if η is a positive root with n_{η,β} = 2, then (in the standard Bourbaki ordering)

η = α_{i} + ⋯ α_{j} + 2α_{j+1} + ⋯ + 2α_{n−2} + α_{n−1} + α_{n}

for some j ≥ i. Hence we have n_{γ_{1},α_{n−2}} = 2, n_{γ_{1},α_{n−1}} = 1 and n_{γ_{1},α_{n}} = 1. In order to have n_{γ,α_{n−1}} = 3 = n_{γ,α_{n}}, we must also have n_{γ_{2},α_{n−1}} = n_{γ_{2},α_{n}} = n_{γ_{3},α_{n−1}} = n_{γ_{3},α_{n}} = 1. To have n_{γ,α_{n−2}} divisible by 3, there two cases to consider: either n_{γ_{2},α_{n−2}} = 1 and n_{γ_{3},α_{n−2}} = 0 or n_{γ_{2},α_{n−2}} = 2 = n_{γ_{3},α_{n−2}}. However, there are no such roots γ_{3} satisifying n_{γ_{3},α_{n−2}} = 0, n_{γ_{3},α_{n−1}} = 1, and n_{γ_{3},α_{n}} = 1. So the first case is not possible.

Suppose the latter case holds. Then for each i, n_{γ_{i},α_{n−3}} ∈ {1, 2}. If these numbers are not all 2 or all 1, then we are done. If each n_{γ_{i},α_{n−3}} = 2, inductively computing n_{γ_{i},β} for β = α_{m} with m < n − 3 either we are done or we come to the case that n_{γ_{i},β} = 1 for each i. Continuing from that case, since the γ_{i} are distinct, there is some β′ = α_{m′} with m′ < m such that n_{γ_{1},β′} = 1, n_{γ_{2},β′} ∈ {0, 1}, and n_{γ_{3},β′} = 0. Hence n_{γ,β′} is not divisible by 3, and we are done.

For the exceptional types, one can check “by hand”, using for example MAGMA, that the root condition lσ = γ_{1} + γ_{2} + γ_{3} cannot hold for l > 3.

Hence, under our assumptions on l, we must have d_{1}(S^{1}(𝔲_{J}^{∗})^{[1]}) = 0. Since the differentials in the spectral sequence are derivations with respect to the cup product, it follows that d_{1}(S^{•}(𝔲_{J}^{∗})^{[1]}) = 0. _{□}

6.3. We now prove part (b) of Theorem 1.3.4. Let M be a finite dimensional u_{ζ}(𝔤)-module. Without a loss of generality, we may assume that M is an irreducible u_{ζ}(𝔤)-module because of the following proposition which is easily proved by using induction on the composition length of the module and the long exact sequence in cohomology.

Proposition 6.3.1. Let R := H^{•}(u_{ζ}(𝔤), ℂ) and M be a finite dimensional u_{ζ}(𝔤)-module. Suppose that H^{•}(u_{ζ}(𝔤), S) is finitely generated over R for all irreducible u_{ζ}(𝔤)-modules S. Then H^{•}(u_{ζ}(𝔤), M) is finitely generated over R.

Let S be an irreducible u_{ζ}(𝔤)-module. Since S lifts to a U_{ζ}(𝔤)-module, there exists a spectral sequence of R = ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗})-modules (obtained in a manner analogous to that of Theorem 5.1.1):

E_{2}^{i,j} = R^{i}ind_{P_{J}}^{G} H^{j}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0 ⊗ S) ⇒ H^{i+j−ℓ(w)}(u_{ζ}(𝔤), S).

Using this spectral sequence, it suffices to show that D := H^{•}(u_{ζ}(𝔭_{J}), ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0 ⊗ S)

is finitely generated over S^{•}(𝔲_{J}^{∗})^{[1]} because then E_{2}^{i,•} is finitely generated over R for each i. Further, this spectral sequence stops (i.e., E_{r} = E_{∞} for r sufficiently large) because the higher right derived functors R^{i}ind_{P_{J}}^{G} − vanish when i > dim G/P_{J}. Thus E_{∞} is finitely generated over R, and H^{•}(u_{ζ}(𝔤), S) is finitely generated over R.

To show that D is finitely generated over S^{•}(𝔲_{J}^{∗})^{[1]}, observe that one can use the LHS spectral sequence (Lemma 2.8.1) to show that

D ≅ Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗}, H^{•}(u_{ζ}(𝔲_{J}), S)).

By the same principles as used in the proposition above, it suffices to show finite generation when S is an irreducible u_{ζ}(𝔭_{J})-module. Note that irreducible u_{ζ}(𝔭_{J})-modules are obtained by inflating irreducible u_{ζ}(𝔩_{J})-modules, so u_{ζ}(𝔲_{J}) acts trivially on S. Now we have

D ≅ Hom_{u_{ζ}(𝔩_{J})}((ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗} ⊗ S^{∗}, H^{•}(u_{ζ}(𝔲_{J}), ℂ)).

But, (ind_{U_{ζ}(𝔟)}^{U_{ζ}(𝔭_{J})}w · 0)^{∗} ⊗ S^{∗} is a projective u_{ζ}(𝔩_{J})-module. Thus we need to show that

D_{P} := Hom_{u_{ζ}(𝔩_{J})}(P, H^{•}(u_{ζ}(𝔲_{J}), ℂ)) is finitely generated over S^{•}(𝔲_{J}^{∗})^{[1]} where P is an arbitrary projective indecomposable u_{ζ}(𝔩_{J})-module. Observe that H^{•}(u_{ζ}(𝔲_{J}), ℂ) ≅ ⊕_{P} D_{P} where the sum is taken over all projective indecomposable u_{ζ}(𝔩_{J})-modules. We are now reduced to proving that H^{•}(u_{ζ}(𝔲_{J}), ℂ) is a finitely generated S^{•}(𝔲_{J}^{∗})^{[1]}-module.

Consider the spectral sequence of Lemma 6.2.1 and the first differential d_{1}. By Proposition 6.2.2, d_{1}(S^{•}(𝔲_{J}^{∗})^{[1]}) = 0, so by the argument given in [Ba], it follows that H^{•}(u_{ζ}(𝔲_{J}), ℂ) is finitely generated over S^{•}(𝔲_{J}^{∗})^{[1]}.

7. Frobenius kernels

7.1. Let G be a reductive connected algebraic group scheme defined over 𝔽_{p} and F be an algebraically closed field of characteristic p > 0. Let Fr : G → G be the Frobenius map and G_{r} = ker Fr^{r} where Fr^{r} is the composition of Fr with itself r-times. Friedlander and Parshall [FP2] computed H^{•}(G_{1}, F) for p ≥ 3(h − 1) by showing that H^{2•}(G_{1}, F) can be identified with the coordinate algebra of N and that the odd degree cohomology vanishes. Andersen and Jantzen [AJ] provided a different proof and showed this also holds when p > h. In the same paper, they provide some ad hoc calculations of the G_{1}-cohomology for small primes. In this section we will demonstrate how these calculations fit into our general set up.

Given a rational G-module M, let M^{(1)} denote the module obtained by first applying the Frobenius morphism to G and then acting as given. Note that the Frobenius kernel G_{1} acts trivially on such a module. Conversely, given a rational G-module N on which G_{1} acts trivially, there exists a rational G-module M such that N = M^{(1)}. We write M = N^{(−1)} (cf. [Jan1, II 3.16]).

7.2. For J ⊂ Π, we formulate two assumptions. The first assumption involves Grauert-Riemenschneider vanishing:

(A1) R^{i}ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) = 0 for i > 0.

(A1)′ R^{i}ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}) ⊗ λ = 0 for i > 0 and λ ∈ X(P_{J})_{+}.

The second assumption on J is a condition on the normality of the closure of the Richardson orbit defined by J:

(A2) The variety G · 𝔲_{J} is normal.

These two assumptions hold in the case when the field is ℂ and were used to compute the cohomology for quantum groups. When F is an algebraically closed field of characterstic p > 0, much less is known about the validity of (A1) and (A2). In this situation, the nullcone N is still normal and Kumar, Lauritzen, and Thomsen [KLT] have shown that the closure of the subregular orbit is normal. Thomsen [Th, Prop. 7] has also shown that closures of Richardson orbits whose corresponding set J consists of a set of pairwise orthogonal short simple roots are normal. Donkin [D] proved that all orbit closures in type A_{n} are normal. Christophersen [C] has recently determined the nilpotent orbits for type E_{6} which have normal orbit closure when p ≥ 5.

7.3. One can now use the assumptions (A1) and (A2) to formulate when the cohomology algebra for G_{1} identifies with the coordinate algebra on the restricted nullcone. Note that the proof here is simpler than in the quantum group setting due to the fact that the exterior algebra Λ^{•}(𝔲_{J}^{∗}) has a natural structure as a P_{J}-module whereas the quantized exterior algebra Λ_{ζ,J}^{•} does not admit a natural structure as a U_{ζ}(𝔭_{J})-module.

Theorem 7.3.1. Let G be a reductive algebraic group and assume that p ≥ 3 and p is a very good prime. Let w ∈ W such that w(Φ_{0}) = Φ_{J}.

(a) If J ⊆ Π satisfies (A1), then (i) H^{2•}(G_{1}, F) ≅ ind_{P_{J}}^{G} S^{•}(𝔲_{J}^{∗}); (ii) H^{2•+1}(G_{1}, F) = 0.

(b) If J ⊆ Π satisfies (A1) and (A2), then (i) H^{2•}(G_{1}, F) ≅ F [N (Φ_{0})] (ii) H^{2•+1}(G_{1}, F) = 0.

Proof. Let w be in W such that w(Φ_{0}) = Φ_{w·0} = Φ_{J} where J ⊆ Π. One can use an argument similar to that given in Section 5 with the functors

Hom_{G_{1}}(ℂ, ind_{P_{J}}^{G} (−)) and ind_{P_{J} /(P_{J})_{1}}^{G/G_{1}}(Hom_{(P_{J})_{1}}(ℂ, −))

(from P_{J}-mod to G/G_{1}-mod) to construct a first quadrant spectral sequence

(7.3.1)  E_{2}^{i,j} = [R^{i} ind_{P_{J}}^{G} (H^{j}((P_{J})_{1}, ind_{B}^{P_{J}} w · 0)^{(−1)})]^{(1)} ⇒ H^{i+j−ℓ(w)}(G_{1}, F).

Next one can apply the LHS spectral sequence and the fact that ind_{B}^{P_{J}} w · 0 is an injective (L_{J})_{1}-module to conclude that

(7.3.2)  Hom_{(L_{J})_{1}}(F, ind_{B}^{P_{J}} w · 0 ⊗ H^{j}((U_{J})_{1}, F)) = H^{j}((P_{J})_{1}, ind_{B}^{P_{J}} w · 0).

In order to compute H^{•}((U_{J})_{1}, F), there is a first quadrant spectral sequence of P_{J}-modules [Jan1, I 9.16]:

(7.3.3)  E_{0}^{i,j} = S^{i}(𝔲_{J}^{∗})^{(1)} ⊗ Λ^{j}(𝔲_{J}^{∗}) ⇒ H^{2i+j}((U_{J})_{1}, F).

Since ind_{B}^{P_{J}} w·0 is an injective (L_{J})_{1}-module, we can compose the spectral sequence in (7.3.3) with Hom_{(L_{J})_{1}}(F, ind_{B}^{P_{J}} w · 0 ⊗ −) and use (7.3.2) to construct a spectral sequence:

(7.3.4) E_{0}^{i,j} = S^{i}(𝔲_{J}^{∗})^{(1)} ⊗ Hom_{(L_{J})_{1}}(F, ind_{B}^{P_{J}} w · 0 ⊗ Λ^{j}(𝔲^{∗})) ⇒ H^{2i+j}((P_{J})_{1}, ind_{B}^{P_{J}} w · 0).

Now one can use the same analysis as in Section 4 with the Steinberg module to conclude that (analogous to Theorem 4.3.1)

Hom_{(L_{J})_{1}}(F, ind_{B}^{P_{J}} w · 0 ⊗ Λ^{j}(𝔲_{J}^{∗})) ≅ \begin{cases}F & if j = ℓ(w) \\ 0 & else. \end{cases}

Therefore, the spectral sequence (7.3.4) collapses and yields:

H^{i}((P_{J})_{1}, ind_{B}^{P} w · 0) ≅ S^{\frac{i−ℓ(w)}{2}} (𝔲_{J}^{∗})^{(1)}.

By using this isomorphism and assumption (A1), the spectral sequence (7.3.1) collapses and we obtain part (a) of the theorem. For part (b), we simply use (A2) and Theorem 3.7.1. _{□}

Once again we can consider the case not covered in the preceding theorem which happens only when Φ is of type A_{n}. This result encompasses Andersen and Jantzen’s computation when p = n + 1 [AJ, §6.A] (corresponding to m = 0). The proof follows the line of reasoning given in Section 5.5 and Theorem 7.3.1. Details are left to the reader.

Theorem 7.3.2. Let G = GL_{n}(F). Assume that p ≥ 3 and p | n + 1 with n + 1 = l(m + 1). Let w ∈ W such that w(Φ_{0}) = Φ_{J}. If J ⊆ Π satisfies (A1)′, then

(a) H^{2•}(G_{1}, F) ≅ ⨁_{t=0}^{p−1} ind_{P_{J}}^{G} S^{\frac{2•−(m+1)t(p−t)}{2}} (𝔲_{J}^{∗}) ⊗ ϖ_{t(m+1)};

(b) H^{2•+1}(G_{1}, F) = 0.

7.4. When J consists of a single simple root the assumptions (A1) and (A2) were verified by Thomsen [Th, Prop. 7, proof of Thm. 2, Lemma 14], and (A2) has also been verified by Kumar, Lauritzen and Thomsen [KLT, Thm. 6]. In this case N (Φ_{0}) is the closure of the subregular orbit O_{subreg} which occurs precisely when p = h − 1. Consequently, we have the following corollary.

Corollary 7.4.1. Let G be a reductive algebraic group and assume that p = h − 1.

(a) H^{2•}(G_{1}, F) ≅ F [\overline{O}_{subreg}]

(b) H^{2•+1}(G_{1}, F) = 0.

Using Thomsen’s work [Th] one can also compute the cohomology algebra for more general examples when J consists of pairwise orthogonal simple short roots. In these cases, (A1) and (A2) are satisfied. In particular, Theorem 7.3.1 can be applied to the situation when Φ is of types A_{n}, C_{n}, D_{n} with p > h/2, Φ is of type B_{n} with p ≥ h/2, Φ is of type F_{4} with p ≥ 7,Φ is of type E_{6} with p ≥ 7, Φ is of type E_{7} with p ≥ 11, and Φ is of type E_{8} with p ≥ 17. Recent results of Christophersen [C, Thm. 1, Example 3.15] verify (A1) and (A2) for the group E_{6} when p ≥ 5 for those subsets J ⊆ Π listed in Section 9.1, which gives the following result.

Corollary 7.4.2. Let G be a reductive algebraic group with Φ = E_{6} and p ≥ 5

(a) H^{2•}(G_{1}, F) ≅ F [N (Φ_{0})]
(b) H^{2•+1}(G_{1}, F) = 0.

8. Support varieties of induced modules/Weyl modules

8.1. In this section, we will assume that l satisfies Assumption 1.3.3. Let R = H^{2•}(u_{ζ}(𝔤), ℂ). We have proven that R is a commutative finitely generated ℂ-algebra. Moreover, if M is a finite dimensional u_{ζ}(𝔤)-module, then H^{•}(u_{ζ}(𝔤), M) is a finitely-generated R-module. Let J_{M} be the annhilator of the action of R on H^{•}(u_{ζ}(𝔤), M ⊗ M^{∗}), and set V_{𝔤}(M) equal to the maximum ideal spectrum of R/J_{M}. The variety V_{𝔤}(M) is called the support variety of M. This variety is a conical subvariety of N (Φ_{0}). Moreover, if M is a U_{ζ}(𝔤)-module then V_{𝔤}(M) is stable under the adjoint action of G.

8.2. Let λ ∈ X and let Φ_{λ} = {α ∈ Φ : ⟨λ + ρ, α^{∨}⟩ ∈ lℤ}. We can now present a result which allows one to compute the supports of induced/Weyl modules provided that one can conjugate this stabilizer set into a subroot system which is generated by simple roots. Using the notation of Section 2.10, let H_{ζ}^{0}(λ) denote the U_{ζ}(𝔤)-module induced from the one dimensional U_{ζ}(𝔟)-module ℂ_{λ} determined by the character λ. In what follows, consider H_{ζ}^{0}(λ) to be a u_{ζ}(𝔤)-module by restriction.

Theorem 8.2.1. Let λ ∈ X_{+}. Then (a) dim V_{𝔤}(H_{ζ}^{0}(λ)) ≥ |Φ| − |Φ_{λ}|;

(b) If there exists w ∈ W such that w(Φ_{λ}) = Φ_{J}, then V_{𝔤}(H_{ζ}^{0}(λ)) = G · 𝔲_{J}.

Proof. For part (a) one can use the proof given in [UGA3, §2] by replacing G_{1} with u_{ζ}(𝔤) (resp. B_{1} by u_{ζ}(𝔟)). One can do the same in [NPV, §5 and Cor 6.1.3] to see that V_{𝔤}(H_{ζ}^{0}(λ)) ⊆ G·𝔲_{J}. Part (b) now follows by using the fact that G·𝔲_{J} is irreducible and dim G·𝔲_{J} = |Φ|−|Φ_{λ}|. _{□}

The following result (generalizing [NPV, (6.2.1) Thm.]) provides a description of the support varieties of the modules H_{ζ}^{0}(λ) for λ ∈ X(T)_{+} in terms of closures of Richardson orbits. This is Theorem 1.3.5. There are restrictions on l even in the case when l > h. See the remarks in Sections 3.1 and 8.3.

Corollary 8.2.2. Let 𝔤 be a complex simple Lie algebra and l be an odd positive integer which is good for Φ. If λ ∈ X_{+}, then there exists w ∈ W such that w(Φ_{λ}) = Φ_{J} for some J ⊆ Π, and V_{𝔤}(H_{ζ}^{0}(λ)) = G · 𝔲_{J}.

Proof. This follows immediately from Lemma 3.1.1 and Theorem 8.2.1. _{□}

8.3. The statement of [Ost, Thm. 6.1] is not correct. Corollary 8.2.2 provides an accurate formulation. Even though l > h, one can still have Φ_{λ} not being W conjugate to Φ_{J} for some J ⊆ Π. For example, let Φ be of type F_{4} and let l = 15 with h = 13. Then

Φ_{4ρ} = {α ∈ Φ : ⟨4ρ + ρ, α^{∨}⟩ ∈ 15ℤ}
= {α ∈ Φ : ⟨5ρ, α^{∨}⟩ ∈ 15ℤ}
= {α ∈ Φ : ⟨ρ, α^{∨}⟩ ∈ 3ℤ}.

In [UGA2, §4.2], it is shown that |Φ_{4ρ}| = 12. By a direct calculation with subroot systems in F_{4} given by J ⊆ Π, there is no such J with |Φ_{J}| = 12. Hence, in this case there exists no w ∈ W with w(Φ_{4ρ}) = Φ_{J} where J ⊆ Π. Furthermore, the same methods used in [UGA3, Cor. 2.5] show that dim V_{𝔤}(H_{ζ}^{0}(4ρ)) ≥ |Φ| − |Φ_{4ρ}| = 36. On the other hand, let I ⊆ Π and x_{I} = ∑_{α∈I} x_{α} (i. e., the regular element in the Levi subalgebra determined by I). The arguments in [NPV, §4] [UGA3, §2.6] demonstrate that if w(Φ_{4ρ}) ∩ Φ_{I} ≠ ∅ for all w ∈ W then x_{I} ∉ V_{𝔤}(H_{ζ}^{0}(λ)). This information in addition to the “constrictor” technique provided by [UGA3, §2.8] can be used to prove that

V_{𝔤}(H_{ζ}^{0}(4ρ)) = \overline{O(A_{1} + Ã_{2})}

which is not a Richardson orbit. This is essentially the same calculation as for the support variety of the trivial module over F_{4} when the characteristic of the field is three (see [UGA2, §4.2]).

9. Appendix: Tables

9.1. For each odd integer l > 1 which is not equal to a bad prime for Φ, the following tables give an element w ∈ W and subset J ⊂ Π such that w(Φ_{0}) = Φ_{J}. The element w is chosen to be a distinguished right W_{J}-coset representative, and identified by means of a reduced expression in terms of the simple reflections s_{i} = s_{α_{i}}. Similar short-hand is used to denote the simple roots in J. Also given is the type of Φ_{0} (or equivalently Φ_{J}), the Bala-Carter label for the nilpotent orbit N (Φ_{0}) = G · 𝔲_{J}, and the dimension of that orbit.

Type E_{6}:

l dim N (Φ_{0})  Φ_{0}  J  orbit
5  62  A_{2} × A_{1} × A_{1} {1, 2, 3, 5} A_{4} + A_{1}
7  66  A_{1} × A_{1} × A_{1} {2, 3, 5} E_{6}(a_{3})
9  70  A_{1}  {4}  E_{6}(a_{1})
11  70  A_{1}  {4}  E_{6}(a_{1})
≥ 12  72  ∅  ∅  E_{6}

\begin{matrix}l & w \\ 5 & s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{2}s_{5}s_{6}s_{3}s_{5}s_{6}s_{2}s_{4}s_{1}s_{3}s_{4}s_{3}s_{2}s_{1}s_{4}s_{3}s_{1}s_{4}s_{3}s_{2}s_{1} \\ 7 & s_{4}s_{2}s_{1}s_{3}s_{4}s_{6}s_{5}s_{4}s_{3}s_{6}s_{1}s_{5}s_{3}s_{2}s_{4}s_{6}s_{5} \\ 9 & s_{3}s_{1}s_{2}s_{5}s_{4}s_{6}s_{5}s_{3} \\ 11 & s_{5}s_{6}s_{2}s_{3}s_{4}s_{5}s_{1}s_{3}s_{4}s_{2} \end{matrix}

Type E_{7}:

l dim N_{1}(Φ_{0})  Φ_{0}  J  orbit
5  106  A_{3} × A_{2} × A_{1}  {1, 2, 3, 5, 6, 7} A_{4} + A_{2}
7  114  A_{2} × A_{1} × A_{1} × A_{1} {1, 2, 3, 5, 7}  A_{6}
9  118  A_{1} × A_{1} × A_{1} × A_{1} {2, 3, 5, 7}  E_{6}(a_{1})
11  120  A_{1} × A_{1} × A_{1}  {2, 3, 5}  E_{7}(a_{3})
13  122  A_{1} × A_{1}  {4, 6}  E_{7}(a_{2})
15  124  A_{1}  {1}  E_{7}(a_{1})
17  124  A_{1}  {1}  E_{7}(a_{1})
≥ 18  126  ∅  ∅  E_{7}

COHOMOLOGY FOR QUANTUM GROUPS VIA THE GEOMETRY OF THE NULLCONE 51

\begin{matrix}l & w \\ 5 & s_{4}s_{5}s_{3}s_{2}s_{4}s_{2}s_{3}s_{1}s_{3}s_{6}s_{5}s_{4}s_{3}s_{5}s_{6}s_{5}s_{3}s_{2}s_{4}s_{2}s_{3}s_{2}s_{4}s_{6}s_{5}s_{4}s_{7}s_{6}s_{4}s_{3}s_{5}s_{6} \\ 7 & s_{4}s_{2}s_{3}s_{6}s_{5}s_{4}s_{6}s_{3}s_{1}s_{2}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}s_{2} \\ 9 & s_{4}s_{2}s_{5}s_{4}s_{1}s_{6}s_{5}s_{3}s_{4}s_{5}s_{2}s_{3}s_{4}s_{7}s_{6}s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7} \\ 11 & s_{4}s_{3}s_{2}s_{5}s_{6}s_{5}s_{7}s_{6}s_{1}s_{3}s_{4}s_{2}s_{3}s_{5}s_{4}s_{3}s_{1}s_{2}s_{6}s_{5}s_{4}s_{3} \\ 13 & s_{7}s_{5}s_{6}s_{2}s_{3}s_{4}s_{5}s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{1}s_{2}s_{3}s_{4}s_{5}s_{6} \\ 15 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4} \\ 17 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}s_{2}s_{4}s_{3}s_{1} \end{matrix}

Type E_{8}:

l dim N (Φ_{0})  Φ_{0}  J  orbit
7  212  A_{4} × A_{2} × A_{1}  {1, 2, 3, 5, 6, 7, 8} A_{6} + A_{1}
9  220  A_{3} × A_{2} × A_{1}  {1, 2, 4, 6, 7, 8} E_{8}(b_{6})
11  224  A_{2} × A_{2} × A_{1} × A_{1} {1, 2, 3, 5, 7, 8} E_{8}(a_{6})
13  228  A_{2} × A_{1} × A_{1} × A_{1} {2, 3, 5, 6, 8}  E_{8}(a_{5})
15  232  A_{1} × A_{1} × A_{1} × A_{1}  {1, 4, 6, 8}  E_{8}(a_{4})
17  232  A_{1} × A_{1} × A_{1} × A_{1}  {2, 3, 5, 7}  E_{8}(a_{4})
19  234  A_{1} × A_{1} × A_{1}  {2, 3, 5}  E_{8}(a_{3})
21  236  A_{1} × A_{1}  {6, 8}  E_{8}(a_{2})
23  236  A_{1} × A_{1}  {6, 8}  E_{8}(a_{2})
25  238  A_{1}  {1}  E_{8}(a_{1})
27  238  A_{1}  {1}  E_{8}(a_{1})
29  238  A_{1}  {1}  E_{8}(a_{1})
≥ 30  240  ∅  ∅  E_{8}

\begin{matrix}l & w \\ 7 & s_{4}s_{3}s_{2}s_{4}s_{5}s_{4}s_{2}s_{1}s_{3}s_{1}s_{4}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{7}s_{6}s_{8}s_{7}s_{5} \\ – & s_{4}s_{2}s_{6}s_{5}s_{4}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{8}s_{7}s_{8}s_{1}s_{3}s_{2}s_{4}s_{5}s_{6}s_{4}s_{1}s_{3}s_{2}s_{4}s_{1}s_{2}s_{3} \\ 9 & s_{5}s_{6}s_{4}s_{5}s_{2}s_{4}s_{3}s_{4}s_{5}s_{6}s_{1}s_{2}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{4}s_{2}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}s_{8}s_{7}s_{6}s_{1}s_{3} \\ – & s_{4}s_{2}s_{5}s_{4}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{1}s_{2}s_{3}s_{4}s_{5}s_{1}s_{3}s_{4}s_{7}s_{6}s_{5}s_{2}s_{4}s_{5}s_{6}s_{7}s_{1}s_{2}s_{3}s_{4}s_{5} \\ 11 & s_{4}s_{2}s_{5}s_{3}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{3}s_{5}s_{4}s_{5}s_{6}s_{7}s_{1}s_{3}s_{4}s_{5}s_{2}s_{4}s_{6}s_{5}s_{8} \\ – & s_{7}s_{6}s_{7}s_{1}s_{3}s_{4}s_{5}s_{2}s_{4}s_{2}s_{6}s_{7}s_{3}s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{4}s_{3}s_{1}s_{3}s_{8}s_{7}s_{6}s_{5} \\ 13 & s_{4}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{7}s_{5}s_{4}s_{2}s_{3}s_{4}s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{2}s_{6}s_{5} \\ – & s_{4}s_{3}s_{2}s_{5}s_{4}s_{3}s_{1}s_{7}s_{8}s_{6}s_{5}s_{7}s_{6}s_{4}s_{2}s_{5}s_{3}s_{4}s_{5}s_{2}s_{6}s_{3}s_{1}s_{4}s_{3} \\ 15 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{7}s_{8}s_{6}s_{5}s_{7}s_{6}s_{2}s_{4}s_{5}s_{2}s_{4}s_{6}s_{7}s_{5}s_{6}s_{1}s_{3}s_{8}s_{7} \\ – & s_{4}s_{5}s_{3}s_{4}s_{1}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{8}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4} \\ 17 & s_{4}s_{2}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{8}s_{7}s_{6}s_{5}s_{3}s_{4}s_{2}s_{1}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}s_{4}s_{5}s_{6}s_{7}s_{2}s_{4}s_{5}s_{6}s_{1}s_{3}s_{4} \\ – & s_{2}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{4}s_{5}s_{3}s_{4}s_{1}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1} \\ 19 & s_{4}s_{2}s_{3}s_{4}s_{1}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{4}s_{5}s_{6}s_{7}s_{8}s_{4}s_{5}s_{3}s_{4} \\ – & s_{1}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{8}s_{7}s_{6}s_{5}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7} \\ 21 & s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{1}s_{3}s_{4}s_{5}s_{6}s_{2}s_{4}s_{5}s_{3}s_{4}s_{1}s_{3}s_{2}s_{4}s_{5}s_{6} \\ – & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5} \\ 23 & s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{1}s_{3}s_{4}s_{5}s_{6}s_{2}s_{4}s_{5}s_{3}s_{4}s_{1}s_{3}s_{2}s_{4}s_{5}s_{6}s_{7}s_{8} \\ – & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2} \\ 25 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4} \\ 27 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6} \\ 29 & s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}s_{1}s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8} \end{matrix}

Type F_{4}:

l dim N (Φ_{0})  Φ_{0}  w  J  orbit
5  40  A_{2} × A_{1} s_{2}s_{3}s_{4}s_{2}s_{3}s_{2}s_{1}s_{2}s_{3}s_{1}s_{2}s_{3} {1, 3, 4} F_{4}(a_{3})
7  44  A_{1} × A_{1} s_{2}s_{1}s_{4}s_{3}s_{2}s_{3}s_{4}s_{1}s_{3}s_{2}s_{4}s_{3}s_{2} {1, 3} F_{4}(a_{2})
9  46  A_{1}  s_{4}s_{2}s_{3}s_{1}s_{2}  {3} F_{4}(a_{1})
11  46  A_{1}  s_{4}s_{2}s_{3}s_{1}s_{2}s_{3}s_{4}  {3} F_{4}(a_{1})
≥ 12  48  ∅  –  ∅  F_{4}

Type G_{2}:

l dim N_{1}(Φ_{0}) Φ_{0} w J orbit
5  10  A_{1} s_{2}s_{1} {1} G_{2}(a_{1})
≥ 6  12  ∅ – ∅  G_{2}

9.2. The information listed below for the exceptional Lie algebras is used for justifying the arguments given in Section 4.9. For each l, the element w ∈ W referred to here is the w identified in the tables of Section 9.1. All weights (e.g., w · 0) are listed in the weight (or ϖ) basis.

\begin{matrix}l & J & w · 0 & −w_{0,J}(w · 0) \\ 11 & \{4\} & (0, −6, −6, 10, −6, 0) & (0, −4, −4, 10, −4, 0) \\ 9 & \{4\} & (1, −5, −5, 8, −5, 1) & (−1, −3, −3, 8, −3, −1) \\ 7 & \{2, 3, 5\} & (−3, 6, 6, −11, 6, −6) & (−3, 6, 6, −7, 6, 0) \\ 5 & \{1, 2, 3, 5\} & (4, 4, 4, −10, 4, −5) & (4, 4, 4, −6, 4, 1) \end{matrix}

\begin{matrix}l & ⟨−w_{0,J}(w · 0), δ^{∨}⟩ & λ & ⟨λ, δ^{∨}⟩ \\ 11 & 10 & (0, −4, −4, 10, −4, 0) & 10 \\ 9 & 8 & (0, −4, −4, 10, −4, 0) & 10 \\ 7 & 18 & (−2, 6, 6, −7, 6, −2) & 18 \\ 5 & 16 & (3, 6, 2, −6, 6, −2) & 17 \end{matrix}

\begin{matrix}l & α & ⟨x, α^{∨}⟩ & ⟨−w_{0,J}(w · 0), α^{∨}⟩ \\ 7 & α_{1} & −4 ≤ ∗ ≤ 1 & −3 \\ 7 & α_{4} & −7 ≤ ∗ ≤ −6 & −7 \\ 7 & α_{6} & −4 ≤ ∗ ≤ 1 & 0 \end{matrix}

Type E_{7}:

\begin{matrix}l & J & w · 0 & −w_{0,J}(w · 0) \\ 17 & \{1\} & (16, 0, −11, 0, 0, 0, 0) & (16, 0, −5, 0, 0, 0, 0) \\ 15 & \{1\} & (14, 0, −10, 0, 0, 1, 0) & (14, 0, −4, 0, 0, 0, 0) \\ 13 & \{4, 6\} & (0, −7, −7, 12, −13, 12, −8) & (0, −5, −5, 12, −11, 12, −4) \\ 11 & \{2, 3, 5\} & (−6, 10, 10, −17, 10, −6, 0) & (−4, 10, 10, −13, 10, −4, 0) \\ 9 & \{2, 3, 5, 7\} & (−5, 8, 8, −14, 8, −8, 8) & (−3, 8, 8, −10, 8, −8, 8) \\ 7 & \{1, 2, 3, 5, 7\} & (6, 6, 6, −14, 6, −7, 6) & (6, 6, 6, −10, 6, −5, 6) \\ 5 & \{1, 2, 3, 5, 6, 7\} & (4, 4, 4, −15, 4, 4, 4) & (4, 4, 4, −9, 4, 4, 4) \end{matrix}

\begin{matrix}l & ⟨−w_{0,J}(w · 0), δ^{∨}⟩ & λ & ⟨λ, δ^{∨}⟩ \\ 17 & 16 & (16, 0, −5, 0, 0, 0, 0,) & 16 \\ 15 & 14 & (16, 0, −5, 0, 0, 0, 0,) & 16 \\ 13 & 24 & (0, −5, −5, 12, −11, 12, −4) & 24 \\ 11 & 30 & (−4, 10, 10, −13, 10, −4) & 30 \\ 9 & 32 & (−4, 8, 10, −11, 8, −7, 8) & 34 \\ 7 & 30 & (5, 8, 4, −10, 8, −7, 8) & 33 \\ 5 & 24 & (5, 10, 4, −10, 3, 0, 5) & 27 \end{matrix}

\begin{matrix}l & α & ⟨x, α^{∨}⟩ & ⟨−w_{0,J}(w · 0), α^{∨}⟩ \\ 15 & α_{2}, α_{4}, α_{5}, α_{6}, α_{7} & −10 ≤ ∗ ≤ 12 & 0 \\ 15 & α_{3} & −15 ≤ ∗ ≤ 4 & −4 \\ 9 & α_{1} & −8 ≤ ∗ ≤ 4 & −3 \\ 9 & α_{4} & −13 ≤ ∗ ≤ −6 & −10 \\ 9 & α_{6} & −12 ≤ ∗ ≤ 0 & −8 \end{matrix}

54 C.P. BENDEL, D.K. NAKANO, B.J. PARSHALL, AND C. PILLEN

\begin{matrix}l & J & w · 0 & −w_{0,J}(w · 0) \\ 29 & \{1\} & (28, 0, −17, 0, 0, 0, 0, 0) & (28, 0, −11, 0, 0, 0, 0, 0) \\ 27 & \{1\} & (26, 0, −17, 1, 0, 0, 0, 0) & (26, 0, −9, −1, 0, 0, 0, 0) \\ 25 & \{1\} & (24, 0, −15, 0, 0, 1, 0, 0) & (24, 0, −9, 0, 0, −1, 0, 0) \\ 23 & \{6, 8\} & (0, 0, 0, 0, −13, 22, −25, 22) & (0, 0, 0, 0, −9, 22, −19, 22) \\ 21 & \{6, 8\} & (2, 0, 0, 0, −13, 20, −21, 20) & (−2, 0, 0, 0, −7, 20, −19, 20) \\ 19 & \{2, 3, 5\} & (−10, 18, 18, −29, 18, −13, 4, 0) & (−8, 18, 18, −25, 18, −5, −4, 0) \\ 17 & \{2, 3, 5, 7\} & (−14, 16, 16, −25, 16, −16, 16, −11) & (−2, 16, 16, −23, 16, −16, 16, −5) \\ 15 & \{1, 4, 6, 8\} & (14, −10, −17, 14, −13, 14, −13, 14) & (14, −4, −11, 14, −15, 14, −15, 14) \\ 13 & \{2, 3, 5, 6, 8\} & (−7, 12, 12, −26, 12, 12, −21, 12) & (−5, 12, 12, −22, 12, 12, −15, 12) \\ 11 & \{1, 2, 3, 5, 7, 8\} & (10, 10, 10, −22, 10, −18, 10, 10) & (10, 10, 10, −18, 10, −12, 10, 10) \\ 9 & \{1, 2, 4, 6, 7, 8\} & (8, 8, −13, 8, −24, 8, 8, 8) & (8, 8, −11, 8, −16, 8, 8, 8) \\ 7 & \{1, 2, 3, 5, 6, 7, 8\} & (6, 6, 6, −25, 6, 6, 6, 6) & (6, 6, 6, −17, 6, 6, 6, 6) \end{matrix} \begin{matrix}l & ⟨−w_{0,J}(w · 0), δ^{∨}⟩ & λ & ⟨λ, δ^{∨}⟩ \\ 29 & 28 & (28, 0, −11, 0, 0, 0, 0, 0) & 28 \\ 27 & 26 & (28, 0, −11, 0, 0, 0, 0, 0) & 28 \\ 25 & 24 & (28, 0, −11, 0, 0, 0, 0, 0) & 28 \\ 23 & 44 & (0, 0, 0, 0, −8, 22, −21, 22) & 44 \\ 21 & 40 & (0, 0, 0, 0, −8, 22, −21, 22) & 44 \\ 19 & 54 & (−8, 18, 18, −25, 18, −8, 0, 0) & 54 \\ 17 & 64 & (−7, 16, 16, −22, 16, −15, 16, −6) & 64 \\ 15 & 56 & (16, −5, −15, 16, −15, 16, −15, 16) & 64 \\ 13 & 60 & (−7, 16, 16, −21, 7, 8, −14, 16) & 63 \\ 11 & 60 & (8, 16, 7, −21, 16, −14, 8, 7) & 62 \\ 9 & 48 & (18, 7, −16, 10, −14, 6, 0, 10) & 51 \\ 7 & 42 & (9, 18, 8, −21, 6, 0, 0, 9) & 50 \end{matrix}

\begin{matrix}l & α & ⟨x, α^{∨}⟩ & ⟨−w_{0,J}(w · 0), α^{∨}⟩ \\ 27 & α_{2}, α_{5}, α_{6}, α_{7}, α_{8} & −18 ≤ ∗ ≤ 20 & 0 \\ 27 & α_{3} & −27 ≤ ∗ ≤ 4 & −9 \\ 27 & α_{4} & −18 ≤ ∗ ≤ 20 & −1 \\ 25 & α_{2}, α_{4}, α_{5}, α_{7}, α_{8} & −20 ≤ ∗ ≤ 22 & 0 \\ 25 & α_{3} & −27 ≤ ∗ ≤ 6 & −9 \\ 25 & α_{6} & −20 ≤ ∗ ≤ 22 & −1 \\ 23 & α_{1}, α_{2}, α_{3}, α_{4} & −10 ≤ ∗ ≤ 12 & 0 \\ 23 & α_{5} & −20 ≤ ∗ ≤ 1 & −9 \\ 23 & α_{7} & −26 ≤ ∗ ≤ −14 & −19 \\ 21 & α_{1} & −14 ≤ ∗ ≤ 16 & −2 \\ 21 & α_{2}, α_{3}, α_{4} & −14 ≤ ∗ ≤ 16 & 0 \\ 21 & α_{5} & −24 ≤ ∗ ≤ 6 & −7 \\ 21 & α_{7} & −26 ≤ ∗ ≤ −9 & −19 \\ 19 & α_{1} & −16 ≤ ∗ ≤ 1 & −8 \\ 19 & α_{4} & −25 ≤ ∗ ≤ −24 & −25 \\ 19 & α_{6} & −16 ≤ ∗ ≤ 1 & −5 \\ 19 & α_{7} & −10 ≤ ∗ ≤ 12 & −4 \\ 19 & α_{8} & −10 ≤ ∗ ≤ 12 & 0 \\ 17 & α_{1} & −14 ≤ ∗ ≤ 1 & −2 \\ 17 & α_{4} & −25 ≤ ∗ ≤ −18 & −23 \\ 17 & α_{6} & −20 ≤ ∗ ≤ −8 & −16 \\ 17 & α_{8} & −14 ≤ ∗ ≤ 1 & −5 \\ 13 & α_{1} & −17 ≤ ∗ ≤ 5 & −5 \\ 13 & α_{4} & −24 ≤ ∗ ≤ −13 & −22 \\ 13 & α_{7} & −22 ≤ ∗ ≤ −3 & −15 \\ 11 & α_{4} & −24 ≤ ∗ ≤ −15 & −18 \\ 11 & α_{6} & −21 ≤ ∗ ≤ −5 & −12 \end{matrix}

Type F_{4}:

\begin{matrix}l & J & w · 0 & −w_{0,J}(w · 0) & ⟨−w_{0,J}(w · 0), δ^{∨}⟩ & λ & ⟨λ, δ^{∨}⟩ \\ 11 & \{3\} & (0, −6, 10, −6) & (0, −4, 10, −4) & 10 & (0, −4, 10, −4) & 10 \\ 9 & \{3\} & (1, −5, 8, −5) & (−1, −3, 8, −3) & 8 & (0, −4, 10, −4) & 10 \\ 7 & \{1, 3\} & (6, −7, 6, −6) & (6, −5, 6, 0) & 12 & (4, −5, 9, −3) & 13 \\ 5 & \{1, 3, 4\} & (4, −8, 4, 4) & (4, −4, 4, 4) & 12 & (4, −4, 4, 4) & 12 \end{matrix}

Type G_{2}:

\begin{matrix}l & J & w · 0 & −w_{0,J}(w · 0) & ⟨−w_{0,J}(w · 0), δ^{∨}⟩ & λ & ⟨λ, δ^{∨}⟩ \\ 5 & \{1\} & (4, −3) & (4, −1) & 4 & (4, −1) & 4 \end{matrix}

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Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, USA

E-mail address: bendelc@uwstout.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: nakano@math.uga.edu

Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA

E-mail address: bjp8w@virginia.edu

Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA

E-mail address: pillen@jaguar1.usouthal.edu