SUPPORT VARIETIES FOR ALGEBRAIC GROUPS

DANIEL K. NAKANO, BRIAN J. PARSHALL, AND DAVID C. VELLA

1 Introduction

1.1 Over the last twenty years, considerable attention in representation theory has focused on the study of the cohomological spectrum for finite groups and restricted Lie algebras over fields of positive characteristic. Carlson [Ca1] [Ca2] first defined affine algebraic varieties associated to modules over group algebras (called support varieties) which are subvarieties of the cohomological spectrum. The structure of these varieties was later related to “rank varieties” for elementary abelian subgroups [Q] [AS] via a stratification theorem. For restricted enveloping algebras, or equivalently infinitesimal k-groups of height at most one, Friedlander and the second author [FP1] [FP2] first investigated the theory of support varieties. More recently, Bendel, Friedlander and Suslin [SFB1] [SFB2] have extended this work to more general infinitesimal group schemes. The methods and results in the infinitesimal case often differ considerably from the corresponding theory in the finite group case. Despite much progress in using these varieties for conceptual as well as constructive purposes, there exist few explicit computations of support varieties for specific modules, with the exception of small rank cases [LN1] and induced modules for GL_{n} (type A) [Jan4].

The present paper provides new results of both a theoretical and a explicit computational nature relative to the determination of support varieties for the infinitesimal subgroups G_{r} (r ≥ 1) of a reductive group G. These results involve the induced modules H^{0}(λ) = ind_{B}^{G} λ and Z_{r}(λ) = ind_{B_{r}}^{G_{r}} λ (see (1.3) below for notation). In particular, when r = 1 we succeed in understanding the relationship between the support varieties of these modules. This leads to the computation of the support variety V_{G_{1}}(H^{0}(λ)) of H^{0}(λ) in all types when the underlying field has good characteristic relative to G; see Theorem 6.2.1 for a precise statement of results. This theorem completely answers a question raised by Jantzen in 1987 [Jan4, (2.7)(1)] (now commonly known as the “Jantzen conjecture”). Even when the characteristic is not good, our methods provide some new information. Although we obtain the sharpest results in the r = 1 case, many results (described more fully below) have been presented for arbitrary r ≥ 1. We hope our approach will lead to further progress towards explicit computations in the r > 1 case.

1.2 The paper is organized as follows. Section 2 provides some basic results which will be used throughout the paper. The complexity c_{G_{r}}(M) of a G_{r}-module M is the dimension of its support variety V_{G_{r}}(M). Section 3 develops the analytic methods for obtaining lower bounds on c_{G_{r}}(M). To this end, we make considerable use of the notion dim_{t} M, the generic dimension of M, as developed in [PW]. To be technically precise, dim_{t} M is only defined when M lies in the subcategory G_{r}T -mod of G_{r}-mod having a compatible action of a maximal torus T of G. Letting Ẑ_{r}(λ) = ind_{B_{r}T}^{G_{r}T} λ and h_{r}(t) = dim_{t} Ẑ_{r}(0), the rational function dim_{t} M/h_{r}(t) has poles located at roots of unity. Further, the order of any such pole is bounded above by c_{G_{r}}(M). In cases when the character of M can be given, this approach provides an explicit lower bound for c_{G_{r}}(M). For example, when M = H^{0}(λ), we obtain in Corollary 3.4.3 such a bound using the Weyl character formula. For irreducible G-modules L(λ) having characters given by Lusztig’s formula, the same methods would apply to estimate c_{G_{r}}(L(λ)), though we do not pursue this line in the present paper. We record here the influence to our approach in Section 3 of the work of V. Ostrik [Ost] in the context of quantum enveloping algebras at an ℓth root of unity, for ℓ > h (the Coxeter number). Although we work in a somewhat different setting, we hope that our exposition in Section 3 will make his methods more widely known.

2000 Mathematics Subject Classification. Primary 20G, 20C, Secondary 20G10, 20G05.

Research of the first author partially supported by NSF grant DMS-9800960.

Research of the second author partially supported by NSF grant DMS-9700965.

Our approach makes heavy use of the theory of “relative” support varieties to provide a bridge linking the theory for the H^{0}(λ) with that for the Z_{r}(μ). For finite groups, these relative support varieties have already been used to study the complete vanishing of cohomology (see, e. g., [BCRi]). Thus, these methods play an important role in Theorem 4.4.1b which shows that

V_{G_{r}}(H^{0}(λ)) ⊆ ⋃_{μ∈Γ_{r}(λ)} G · V_{G_{r}}(Z_{r}(μ)),  (1.2.1)

where Γ_{r}(λ) denotes the weights corresponding to the G_{r}-block containing Z_{r}(λ). Also, Theorem 4.6.1 provides necessary and sufficient conditions for (1.2.1) to be an equality of varieties. The end of Section 4 presents an alternate proof of the computation, given in [Jan4], of V_{G_{1}}(H^{0}(λ)) for G = GL_{n}. In our argument, the inclusion (1.2.1) substitutes for the “weak BGG resolution” method in [Jan4].

Section 5 investigates the behavior of induction on varieties. Using a key relationship on relative support varieties established in Theorem 5.4.1, Theorem 5.6.1 proves in the r = 1 case, for any dominant weight λ and any w in the Weyl group W, the containment

V_{G_{1}}(H^{0}(λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)).  (1.2.2)

These two theorems are the technical heart of our paper. By using these theorems and some results in Sections 4 and 5, we answer affirmatively in Section 6 the conjecture raised by Janzten mentioned above.^{1} In addition, this section contains further applications of our approach.

This paper provides new information concerning the support varieties V_{G_{r}}(Z_{r}(λ)). In some special cases, e. g., when λ is regular and r = 1, these varieties can be calculated; see Theorem 4.1.1. Finally, Theorem 7.3.1 provides a basic link with H^{0}(λ), namely, that in good characteristics we have

V_{G_{1}}(H^{0}(λ)) = G · V_{B_{1}}(Z_{1}(w · λ))  (1.2.3)

1We mention here that [Ost] also claims to provide an answer to this question when p is larger than the Coxeter number h of G; however, a number of readers (including the authors of this paper) are not convinced that the proof is correct.

for all λ ∈ X(T)_{+} and all w ∈ W. Since Z_{1}(w · λ) is naturally a G_{1}B-module, V_{B_{1}}(Z_{1}(λ)) carries an action of the Borel subgroup B. For w = 1, we have ind_{G_{1}B}^{G} Z_{1}(λ) ≅ H^{0}(λ), so the identity (1.2.3) can be regarded as an “induction formula” for certain support varieties. As we note in the example given in (7.1), the varieties V_{B_{1}}(Z_{1}(w · λ)) vary with w. While a determination of the V_{B_{1}}(Z_{1}(w · λ)) does not yet exist, (1.2.3) does tie these varieties together with that for the H^{0}(λ).

1.3 Notation. Throughout this paper, let k be an algebraically closed field of positive characterstic p. Let Φ be a finite root system for a Euclidean space 𝔼. The inner product on 𝔼 will be denoted by ( , ). For α ∈ Φ, let α^{∨} = 2α/(α, α) be the corresponding coroot. Fix a set Π = {α_{1}, ⋯, α_{ℓ}} of simple roots, and let Φ^{+} be the corresponding set of positive roots. The Weyl group W ⊂ O(𝔼) is the group generated by the reflections s_{α} : 𝔼 → 𝔼, α ∈ Φ, given by s_{α}(x) = x − 2(x, α^{∨})α. Let T_{p} be the group generated by translations t_{pα} : 𝔼 → 𝔼, α ∈ Φ, defined by x ↦ x + pα. The affine Weyl group W_{p} is W ⋉ T_{p}.

Unless otherwise stated, G will denote a reductive algebraic group defined and split over the prime field 𝔽_{p}. We will always assume that the derived group G′ is simply connected. Also, assume that G has root system Φ with respect to maximal split torus T. Let B ⊃ T be the Borel subgroup defined by −Φ^{+}. The positive Borel subgroup containing T will be denoted B^{+}. Moreover, let X(T) = X(B) be the group of integral characters on T or, equivalently, B. Given λ ∈ X(T), we will let λ also denote the one-dimensional B-module defined by regarding λ as a character on B. Then the set of dominant integral weights is defined by

X(T)_{+} = {λ ∈ X(T) | 0 ≤ (λ, α_{i}^{∨}) 1 ≤ i ≤ ℓ}.

(If G is not semisimple, then λ ∈ X(T)_{+} is not determined by its “values” at the coroots α^{∨}.) Furthermore, for r ≥ 1, the set of p^{r}-restricted weights is

X_{r}(T) = {λ ∈ X(T) | 0 ≤ (λ, α_{i}^{∨}) < p^{r} 1 ≤ i ≤ ℓ}.

The group W_{p} acts on X(T) via the “dot action” given by w · λ = w(λ + ρ) − ρ, w ∈ W_{p}, λ ∈ X(T). Here ρ is the half sum of the positive roots. We partially order X(T) by setting λ ≥ μ if and only if λ − μ ∈ ∑_{α∈Π} ℕα. Let h be the Coxeter number of G. Thus, if G′ is simple, h = (ρ, α_{0}^{∨})+1 where α_{0} is the maximal short root in Φ; otherwise, h is the maximal of the Coxeter numbers for the simple factors of G′. Let

C = {λ ∈ 𝔼 | 0 < (λ + ρ, α_{0}^{∨}) < p}

denote the bottom p-alcove in 𝔼. Thus, W_{p} is generated by the reflections in the walls of C. We set C_{ℤ} = C ∩ X(T).

The prime p is good for Φ (or G) provided that for any (integrally) closed subsystem Φ′ of Φ, ℤΦ/ℤΦ′ has no p-torsion. Equivalently, p is good provided: p > 2 when G has a component of type B, C or D; p > 3 when G has a component of type G_{2}, F_{4}, E_{6} or E_{7}; and p > 5 when G has a component of type E_{8}.

Let F : G → G be the Frobenius morphism on G induced by its 𝔽_{p}-structure. For r ≥ 1, put G_{r} = ker(F^{r}). If H is an F -stable subgroup of G, write similarly H_{r} = ker(F^{r}|_{H})—e. g., B_{r} = ker(F^{r}|_{B}). In general, let G_{r}H = F^{−r}(H), the pull-back of H through F^{r}—e. g., G_{r}T = F^{−r}(T). The group scheme G_{r} is a finite k-group, i. e., an affine algebraic group scheme over k with finite dimensional coordinate algebra k[G_{r}]. Also, it has height ≤ r. In what follows, all affine k-groups K will, by definition, be assumed to be algebraic, i. e., the coordinate algebra k[K] is Noetherian over k.

For an (affine) k-group H, H-mod denotes the category of rational H-modules. For G above, the dominant weights λ ∈ X(T)_{+} index the simple modules L(λ) by their highest weight. If ind_{B}^{G} : B-mod → G-mod is the induction functor, let H^{0}(λ) = ind_{B}^{G} λ for λ ∈ X(T). If λ ∈ X(T)_{+}, H^{0}(λ) has socle L(λ). For j ≥ 0, let H^{j}(−) = R^{j}ind_{B}^{G}(−) be the jth right derived functor of ind_{B}^{G}.

For λ ∈ X(T) set Z_{r}(λ) = ind_{B_{r}}^{G_{r}} λ. Then Z_{r}(λ) has irreducible socle, denoted L_{r}(λ). Of course, if μ ∈ X(T), Z_{r}(λ) ≅ Z_{r}(λ + p^{r}μ) and L_{r}(λ) ≅ L_{r}(λ + p^{r}μ). Thus, the set X_{r}(T) indexes the distinct simple modules L_{r}(λ) and the distinct Z_{r}(λ). For λ ∈ X_{r}(T), L(λ)|_{G_{r}} ≅ L_{r}(λ).

Any λ ∈ X(T), defines a one-dimensional B_{r}T -module, and Ẑ_{r}(λ) = ind_{B_{r}T}^{G_{r}T} λ will be the corresponding induced G_{r}T -module. Then Ẑ_{r}(λ)|_{G_{r}} ≅ Z_{r}(λ). If P̂_{r}(λ) = ind_{B_{r}^{+}T}^{G_{r}T} (λ−2(p^{r} −

1)ρ), then P̂_{r}(λ)|_{B_{r}T} (resp., P̂_{r}(λ)|_{B_{r}}) is the projective cover in B_{r}T -mod (resp., B_{r}-mod) of the one-dimensional module λ. As T -modules, Ẑ_{r}(λ) and P̂_{r}(λ) have identical characters given by

ch Ẑ_{r}(λ) = ch P̂_{r}(λ) = e^{λ} ∏_{α∈Φ^{+}} \frac{1 − e^{−p^{r}α}}{1 − e^{−α}}  (1.3.1)

For I ⊆ Π, let L_{I} be the reductive subgroup of G generated by T and the root groups U_{α}, ±α ∈ I. Let Φ_{I} be the root system of T in L_{I}, i. e., Φ_{I} = ℤI ∩ Φ. The subgroup L_{I} is a Levi factor for a parabolic subgroup P_{I} ⊇ B. Let P_{I}^{+} be the opposite parabolic subgroup. The unipotent radical of P_{I} (resp., P_{I}^{+}) will be denoted U_{I} (resp., U_{I}^{+}). For r ≥ 1 and λ ∈ X(T), let Z_{r}^{I}(λ) = ind_{(B_{I})_{r}}^{(L_{I})_{r}} λ and let Z_{r}^{I}(λ)^{e} be the (P_{I})_{r}-module obtained from Z_{r}^{I}(λ) by making (P_{I})_{r} act through the quotient map (P_{I})_{r} → (P_{I})_{r}/(U_{I})_{r} ≅ (L_{I})_{r}. Then ind_{(P_{I})_{r}}^{G_{r}} Z_{r}^{I}(λ)^{e} ≅ Z_{r}(λ).

In this paper, we work almost exculsively with algebraic varieties in the classical sense. In particular, any commutative, Noetherian k-algebra A defines an affine variety X which, as a set, can be identified with the set of all algebra homomorphisms f : A → k. If A_{red} denotes the quotient of A by its nilpotent radical, then X has coordinate algebra k[X] = A_{red}. If I is an ideal in A, let V (I) be the closed subvariety of X defined by I, consisting of all f ∈ X such that f(I) = 0.

Often the algebra A will carry an action of a group G. Thus, G acts on the variety X defined by A. Given an ideal I in A, we let ℓ_{G}(I) denote the largest ideal contained in I which is G-stable. Clearly,

\overline{G · V (I)} = V (ℓ_{G}(I)).  (1.3.2)

In particular, suppose G is a reductive group acting rationally on A and Y = V (I) is a P -stable, closed subvariety of X for some parabolic subgroup P = P_{I}. Let Z be the closed subvariety of G/P × X consisting of all (gP, x) such that g · x ∈ Y. Since G/P is complete, the projection G · Y of Z to X is closed. Thus, G · Y = V (ℓ_{G}(I)).

For example, let I ⊆ Π and let 𝔲_{I} = Lie U_{I} ⊆ 𝔤. Regard 𝔤 as a G-variety by means of the adjoint action. Then G · 𝔲_{I} is a closed subvariety of 𝔤. Also, at least when p is good for G, we have

dim G · 𝔲_{I} = 2 dim 𝔲_{I} = |Φ| − |Φ_{I}|.  (1.3.3)

See [Car, (5.2.3)]. In addition, for I, J ⊆ Π, we have

w(Φ_{J}) ∩ Φ_{I} ≠ ∅ for all w ∈ W ⟺ x_{I} ∉ G · 𝔲_{J},  (1.3.4)

where x_{I} = ∑_{α∈I} x_{α}, with 0 ≠ x_{α} ∈ 𝔤_{α} for all α ∈ I. The proof is a simple application of the Bruhat decomposition; see [Jan4, (2.5)(1)] for details.

2 Cohomology and support varieties.

In this section, we introduce some preliminary results which will be used through the paper.

2.1 Rates of growth. Suppose that {s_{n}}_{n≥0} is a sequence of complex numbers. The rate of growth r(s_{n}) of this sequence is the smallest non-negative integer d for which there exists a positive real number C such that

|s_{n}| ≤ C · n^{d−1}, ∀n ≥ 1.

If no such d exists, set r(s_{n}) = ∞.

Suppose s_{n} is a polynomial in n of degree d − 1 (at least for large n). In this case, there exist positive integers C, C′ such that

C′n^{d−1} ≤ |s_{n}| ≤ Cn^{d−1}, ∀n sufficiently large.

In particular, r(s_{n}) = d. If S_{n} = |s_{0}| + ⋯ + |s_{n}|, then it also follows that r(S_{n}) = d + 1.

In practice, {s_{n}} arises as the sequence of coefficients in a power series expansion ∑_{n} s_{n}t^{n} of a rational function p(t). In general, such coefficients s_{n} are not polynomial functions in n, though the following lemma describes a situation in which {s_{n}} can be broken down into “polynomial subsequences.”

(2.1.1) Lemma . Let p(t) = ∑_{n=0}^{∞} s_{n}t^{n} ∈ ℂ[[t]] be a rational function.

(a) If p(t) = \frac{f(t)}{(1−t)^{d}} for some positive integer d and f(t) ∈ ℂ[t] with f(1) ≠ 0, then s_{n} is a polynomial in n of degree d − 1. Hence, r(s_{n}) = d.

(b) Assume the poles of p(t) are roots of unity. If e^{iθ} is a pole of order γ, then

γ ≤ r(s_{n}).

(c) Assume p(t) = \frac{f(t)}{(1−t^{a})^{b}} for positive integers a, b and f(t) ∈ ℂ[t]. Fix i, 0 ≤ i < a. Then s_{i+aj} is a polynomial in j (say of degree d_{i} − 1).

(d) In (c), let d = max d_{i}. Then d = r(s_{n}). If S_{n} = |s_{0}| + ⋯ + |s_{n}|, then r(S_{n}) = d + 1.

Proof. (a) Expanding f(t) as a polynomial in 1 − t with non-zero constant term, we reduce immediately to the case p(t) = \frac{1}{(1−t)^{d}} = ∑_{n=0}^{∞} \binom{n+d−1}{d−1}t^{n}. Now (a) follows.

(b) Suppose that d = r(s_{n}). For a pole e^{iθ}, consider z = re^{iθ}, 0 < r < 1,

|p(z)| ≤ ∑_{n=0}^{∞} Cr^{n}n^{d−1} ≤ \frac{C′}{(1 − r)^{d}}

for some C, C′ > 0. Thus, the order of the pole of p(t) at e^{iθ} is at most d.

(c) It is enough to consider the special case in which f(t) = t^{e} for some non-negative integer e. If e = i + ℓ_{0}a, 0 ≤ i < a, then p(t) = ∑_{n=0}^{∞} \binom{n+b−1}{b−1}t^{i+(n+ℓ_{0})a}.

Finally, (d) follows directly from the remarks before the statement of the lemma. _{□}

Finally, if {V_{n}}_{n} is a sequence of finite dimensional vector spaces over k, define its rate of growth r(V_{n}) to be r(dim V_{n}).

2.2 Support varieties. Let K be a finite k-group. Set

H(K, k) = \begin{cases}H^{2•}(k, k) & if char k ≠ 2 \\ H^{•}(K, k) & if char k = 2. \end{cases}

The cohomology algebra H(K, k) is a commutative, Noetherian k-algebra [FS]; thus, we will work with the variety V_{K} = Spec H(K, k)_{red}.

Given M, M′ ∈ K-mod, define the relative support variety V_{K}(M, M′) as follows. First, the functor − ⊗ M defines an algebra homomorphism Φ_{M} : H(K, k) → Ext_{K}^{•}(M, M). Then

Ψ : H(K, k) ⊗ Ext_{K}^{•}(M, M′) → Ext_{K}^{•}(M, M′), a ⊗ b ↦ Φ_{M}(a) ⊔ b,

where ⊔ is Yoneda composition, defines an action of H(K, k) on Ext_{K}^{•}(M, M′). Alternatively, we have Ψ(a ⊗ b) = ±b ⊔ Φ_{M′}(a), see [Ben, §5.7]. Let J_{K}(M, M′) be the annihilator ideal in H(K, k) for its action on Ext_{K}^{•}(M, M′) and set V_{K}(M, M′) equal to the closed subvariety of V_{K} defined by J_{K}(M, M′). In particular, the support variety V_{K}(M) is defined by setting V_{K}(M) = V_{K}(M, M).

The algebraic variety V_{K} has coordinate algebra H(K, k)/J, where J is the radical of H(K, k). Since J is necessarily a graded ideal, V_{K} is a homogeneous variety. Any V_{K}(M, M′) is a homogeneous closed subvariety of V_{K}, and

V_{K}(M, M′) ⊆ V_{K}(M) ∩ V_{K}(M′)  (2.2.1)

Let ⋯ → P_{2} → P_{1} → P_{0} → M → 0 be a minimal projective resolution in K-mod, and define the complexity c_{K}(M) of M to be r(P_{n}). The following theorem explains the relationship between c_{K}(M) and V_{K}(M). A proof is given in [Ben, §5.7].

(2.2.2) Theorem . Let K be a finite k-group, and let {L_{i} | i = 1, 2, … m} be the complete set of non-isomorphic simple objects in K -mod. For M ∈ K -mod the following numbers are equal:

(a) c_{K}(M);

(b) dim V_{K}(M);

(c) r(Ext_{K}^{n}(M, M));

(d) r(Ext_{K}^{n}(M, ⊕_{i=1}^{m}L_{i})).

(e) r(Ext_{K}^{n}(⊕_{i=1}^{m}L_{i}, M))

We have listed below some fundamental properties of complexity and relative support varieties for finite k-groups K [SFB1] [SFB2] [Ben].

(2.2.3) If M ∈ K-mod then V_{K}(M) is a closed, conical subvariety of V_{K}.

(2.2.4) If M_{1}, M_{2} ∈ K-mod then V_{K}(M_{1} ⊕ M_{2}) = V_{K}(M_{1}) ∪ V_{K}(M_{2}).

(2.2.5) If M_{1}, M_{2} ∈ K-mod then V_{K}(M_{1} ⊗ M_{2}) = V_{K}(M_{1}) ∩ V_{K}(M_{2}) (as varieties).

(2.2.6) Assume that K is a closed normal subgroup of an algebraic k-group L. There is a natural rational action of L on V_{K}. If M, N ∈ L-mod, the subvariety V_{K}(M, N) of V_{K} is stable under the action of L.

(2.2.7) Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence in K-mod. If 𝔖_{3} is the symmetric group on three letters and σ ∈ 𝔖_{3}, then V_{K}(M_{σ(1)}) ⊆ V_{K}(M_{σ(2)}) ∪ V_{K}(M_{σ(3)}).

(2.2.8) For any finite-dimensional K-module M, we have V_{K}(M) ⊆ ⋃ V_{K}(L_{i}), where the union runs over the set of composition factors L_{i} of M.

(2.2.9) If {L_{1}, L_{2}, …, L_{m}} is a complete set of non-isomorphic simple modules in K-mod then V_{K}(M) = V_{K}(M, ⊕_{i=1}^{m}L_{i}) = V_{K}(⊕_{i=1}^{m}L_{i}, M).

Let H be a closed subgroup of a finite k-group K. Assume K has height ≤ r (e. g., K = G_{r}). Then the image of the restriction map H(K, k)_{red} → H(H, k)_{red} contains all p^{r}th powers x^{p^{r}} of elements x ∈ H(H, k)_{red} [SFB2, (5.4)]. In particular, the induced morphism V_{H} → V_{K} maps V_{H} homeomorphically onto its image as a closed subvariety of V_{K}. Thus, we often “identify” V_{H} with its image in V_{K}. With this identification, we have the following naturality result [SFB2, §7].

(2.2.10) Assume that H is a closed subgroup of a finite k-group K. Identify V_{H} as a closed subvariety of V_{K}. Then V_{H} ∩ V_{K}(M) = V_{H}(M).

Now let H be an affine algebraic group defined over 𝔽_{p}. Let N (H) be the closed subvariety of nilpotent elements in the Lie algebra 𝔥 of H. Let x ↦ x^{[p]} be the [p]-operator for the restricted Lie algebra structure on 𝔥. We will often work with the closed subvariety N_{1}(H) = {x ∈ 𝔥 | x^{[p]} = 0} of N (H). By [SFB1, (1.6), (5.11)], V_{H_{1}} is homeomorphic to N_{1}(H), and we often identify these two spaces. (When H = G and p > h, this homeomorphism is in fact an isomorphism [AJ].) Given M ∈ H_{1}-mod, any x ∈ N_{1}(H) defines an operator X on M satisfying X^{p} = 0. We then say that M is x-projective if M is a projective module for the algebra k[X] ⊆ End(M). We have the following result [FP1]:

(2.2.11) Let M ∈ H_{1}-mod. Under the identification of V_{H_{1}} with N_{1}(H), V_{H_{1}}(M) identifies with the set {x ∈ N_{1}(H) | M is not x–projective} ∪ {0}.

When G is reductive and the prime p is good, N (G) is G-isomorphic to the irreducible variety U(G) of unipotent elements of G; e. g., see [SS, p. 229] [BR, (9.3.2)]. In this case, many of the nice properties, such as existence of regular, subregular, etc. elements, of U(G) transfer to N (G). This fact will be used in the sequel without mention.

Finally, consider the k-groups G_{r}T, r ≥ 1. Given a G_{r}T -module M, let P_{•} → M be a minimal projective resolution of M in G_{r}T -mod, define its complexity c_{G_{r}T} (M) to be the rate of growth of the sequence {P_{n}}. Restricting to G_{r}, it follows easily from [Jan2, II, (11.3)(3)] that P_{•} → M is also a minimal projective resolution in G_{r}-mod. This gives the following result for G_{r}T -mod; a similar argument applies for B_{r}T -mod. (See also [DNP, (7.2)].)

(2.2.12) For any M ∈ G_{r}T -mod, we have c_{G_{r}T} (M) = c_{G_{r}}(M). Similarly, for M ∈ B_{r}T -mod, we have c_{B_{r}T} (M) = c_{B_{r}}(M).

The reader should observe that the version of (2.2.2), interpreting complexity in terms of the rate of growth of Ext-groups, fails in general for the categories G_{r}T -mod and B_{r}T -mod, e. g., c_{G_{r}T} (M) ≠ r(Ext_{G_{r}T}^{n} (M, M)) for general M ∈ G_{r}T -mod.

2.3 We conclude this section with the following result concerning how support varieties behave relative to certain induction functors. We will return to this topic in Section 5.

(2.3.1) Proposition . Assume the hypotheses of (2.2.10). For M ∈ H-mod, we have

(a) V_{K}(ind_{H}^{K} M) ⊆ V_{H}(M).

(b) V_{K}(ind_{H}^{K} M) = V_{H}(ind_{H}^{K} M).

In particular, for any positive integer r, we have V_{G_{r}}(Z_{r}(λ)) = V_{B_{r}}(Z_{r}(λ)) for all λ ∈ X(T). Also, let N ∈ K–mod. Then the following statement holds for relative support varieties:

(c) V_{K}(N, ind_{H}^{K} M) = V_{H}(N, M).

Proof. By (2.2.10), (a) ⇒ (b). By (2.2.1), (c) ⇒ (a), taking N = ind_{H}^{K} M. Now (c) follows easily from the remarks following (2.2.9), using the fact that since K/H is affine, Ext_{K}^{•}(N, ind_{H}^{K} M) ≅ Ext_{H}^{•}(N, M). We leave further details to the reader. _{□}

3 Complexity and the graded dimension.

Throughout this section G will denote a fixed simple, simply connected algebraic group. The assumption that G is simple is merely one of convenience, and the application of results later to a general reductive group is routine.

3.1 In what follows, we often identity the category of finite dimensional rational T -modules with the category of finite dimensional vector spaces graded by X(T). For an object V in this category, we wish to associate to V a natural ℤ-grading V = ⨁ V_{i} and consider the associated Laurent polynomial dim_{t} V = ∑ dim V_{i} t^{i} ∈ ℤ[t, t^{−1}]. In our presentation, we follow closely the procedure described in detail in [PW, §1].^{2}

2Our method for grading rational T -modules differs from the so-called principal grading discussed in [K, (10.10)] which assigns a ℤ-grading to any highest weight module L(Λ) for a Kac-Moody Lie algebra 𝔤(A).

This grading depends on the highest weight Λ, though a definition could be given for weight modules having all their weights lying in a cone of weights. Such gradings could be adopted in our current framework, and, in fact, [Ost] appears to use some version of the principal grading in the similar context of quantum groups,

Let C = ((α_{i}, α_{j}^{∨})) be the Cartan matrix of G. Let α_{0} ∈ Φ^{+} be the maximal short root in Φ, and, for any root α, set d_{α} = (α, α)/(α_{0}, α_{0}) ∈ {1, 2, 3}. For 1 ≤ i ≤ ℓ, let d_{i} = d_{α_{i}}, so that the matrix C · diag(d_{1}, ⋯, d_{ℓ}) is symmetric. Given λ = ∑_{i=1}^{ℓ} n_{i}α_{i} ∈ X(T) (so that each n_{i} ∈ ℚ), its weighted height is defined to be wht(λ) = ∑ d_{i}n_{i}. A direct calculation shows that wht(λ) = 2(λ, ρ)/(α_{0}, α_{0}) = \frac{1}{2} ∑_{α∈Φ^{+}} d_{α}(λ, α^{∨}), so that 2wht(λ) ∈ ℤ. Now given a finite dimensional X(T)-graded vector space V (i. e., a rational T -module) put

dim_{t} V = ∑_{λ∈X(T)} dim V_{λ} t^{−2 wht(λ)} ∈ ℤ[t, t^{−1}].  (3.1.1)

In [PW], dim_{t}V is called the generic dimension of V (and accordingly denoted dim_{gen} V there). We record several useful properties of this concept. First, if λ ∈ X(T)_{+}, then

dim_{t} H^{0}(λ) = ∏_{α∈Φ^{+}} \frac{t^{d_{α}(λ+ρ,α^{∨})} − t^{−d_{α}(λ+ρ,α^{∨})}}{t^{d_{α}(ρ,α^{∨})} − t^{−d_{α}(ρ,α^{∨})}} .  (3.1.2)

See [PW, (1.3)]. Observe that d_{α}(ρ, α^{∨}) = wht(α). Second, if V, W are X(T)-graded finite dimensional vector spaces, then V ⊗ W and V^{∗} have natural X(T)-gradings and

dim_{t}(V ⊗ W) = dim_{t} V · dim_{t} W  (3.1.3)
dim_{t^{−1}} V = dim_{t} V^{∗}.  (3.1.4)

Third, define

h_{r}(t) = ∏_{α∈Φ^{+}} \frac{1 − t^{2p^{r} wht(α)}}{1 − t^{2 wht(α)}} ∈ ℤ[t].  (3.1.5)

Then for any λ ∈ X(T), (1.3.1) implies

dim_{t} P̂_{r}(λ) = dim_{t} Ẑ_{r}(λ) = t^{−2 wht(λ)}h_{r}(t).  (3.1.6)

For any finite dimensional T -module M, define e(M) to be the smallest integer such that t^{e(M)} dim_{t} M ∈ ℤ[t]. Observe that for every weight λ of M, we have e(M) − 2 wht(λ) ≥ 0. In particular, e(M) ≥ 0.

3.2 Consider the algebra A = H(B_{r}, k) for some fixed integer r ≥ 1. The argument in [CPSK, (2.1), (2.2)] with the normalized bar resolution shows that as a T -module A has all weights in p^{r}ℤΦ^{+}. Thus, for all integers n ≥ 0,

dim_{t} H^{n}(B_{r}, k) ∈ ℤ[t^{2p^{r}}, t^{−2p^{r}}].  (3.2.1)

For M ∈ B_{r}T –mod, let P_{•} → M → 0 be a minimal projective resolution in B_{r}T –mod. Restricting to B_{r}, P_{•} → M → 0 still provides a minimal projective resolution in B_{r}-mod. Because Hom_{B_{r}}(P_{n}, μ) ≅ Ext_{B_{r}}^{n}(M, μ), we have

P_{n} ≅ ⨁_{μ∈X_{r}(T)} Ext_{B_{r}}^{n}(M, μ)^{∗} ⊗ P̂_{r}(μ)  (3.2.2)

though the gradings there are not explicitly defined. The use of dim_{t} avoids any difficulty of working with a cone of weights and presents an explicit ℤ-grading for any rational T -module.

as T -modules. Of course, Ext_{B_{r}}^{n}(M, μ) is a direct sum of one-dimensional T -modules of the form p^{r}ν.

Let λ ∈ X_{r}(T) and ν ∈ X(T) be such that P̂_{r}(λ) ⊗ p^{r}ν is a B_{r}T -direct summand of P_{n}. Then τ_{n} =^{def}λ + p^{r}ν appears in the head of P_{n}. Since P_{•} is a minimal projective resolution, it follows that τ_{n} is a weight in the radical of P_{n−1}. Thus, τ_{n−1} =^{def}τ_{n} + β appears in the head of P_{n−1}, where β is some non-trivial sum of positive roots. In particular,

−2 wht(τ_{n}) ≥ −2 wht(τ_{n−1}) + 2.

Repeating this argument n times, we conclude that −2 wht(λ + p^{r}ν) ≥ −2 wht(τ) + 2n for some weight τ = τ_{0} in M. Since e(M) − 2 wht(τ) ≥ 0 for all weights τ in M, we finally conclude that

−2 wht(λ + p^{r}ν) ≥ 2n − e(M).  (3.2.3)

3.3 For a positive integer d let Ψ_{d}(t) ∈ ℤ[t] be the dth cyclotomic polynomial over ℚ. The polynomials Ψ_{d}(t) are irreducible and t^{n} − 1 = ∏_{d|n} Ψ_{d}(t). Let M ∈ B_{r}T –mod. The following result (influenced by [Ost]) shows the complexity c_{B_{r}}(M) = dim V_{B_{r}}(M) is related to the orders of the poles of the rational function dim_{t} M/h_{r}(t).

(3.3.1) Theorem . For M ∈ B_{r}T -mod, write

q(t) = \frac{dim_{t} M}{h_{r}(t)} = \frac{f(t)}{(1 − t^{p^{r}})^{γ}g(t)}

where Ψ_{p^{r}}(t) ∤ f(t)g(t). Then γ ≤ c_{B_{r}}(M).

Proof. We can assume γ ≥ 0, so that γ is the order of the pole of any primitive p^{r}th root of unity in q(t). Let P_{•} → M be a minimal projective resolution in B_{r}T -mod, and write

dim_{t} P_{n}/h_{r}(t) = ∑_{m} a(m, n)t^{m},

where each a(m, n) is a non-negative integer, and, using (3.2.2),

∑_{m} a(m, n) = dim Ext_{B_{r}}^{n}(M, N) ≤ Cn^{c_{Br} (M)−1}.  (3.3.2)

Here N = ⨁_{μ∈X_{r}(T)} μ and C > 0. If a(m, n) ≠ 0, then for some λ ∈ X_{r}(T) and ν ∈ X(T), we have that P̂_{r}(λ+p^{r}ν) appears as a B_{r}T -direct summand of P_{n} with m = −2 wht(λ+p^{r}ν). Thus, (3.2.3) implies that if a(m, n) ≠ 0 then m ≥ 2n − e(M). In particular, if a(m, n) ≠ 0, then n ≤ m + e(M). Thus, q(t) = ∑_{n=0}^{∞}(−1)^{n} \frac{dim_{t} P_{n}}{h_{r}(t)} . Express q(t) = ∑_{i=0}^{∞} s_{i}t^{i} as a power series, so for any i > 0, (3.3.2) gives that

S_{i} =| s_{0} | + ⋯ + | s_{i} | ≤ ∑_{m=0}^{i} ∑_{n=0}^{∞} a(m, n) ≤ ∑_{m=0}^{i} ∑_{n=0}^{i+e(M)} a(m, n)

≤ ∑_{n=0}^{i+e(M)} Cn^{c_{Br} (M)−1} ≤ C′i^{c_{Br} (M)}  (3.3.3)

for some C′ > 0.

By (3.1.5), the poles of q(t) are roots of unity. So if a is the least common multiple of the orders of these poles, q(t) = \frac{m(t)}{(1−t^{a})^{d}} for some positive integer d and some m(t) ∈ ℂ[t]. By (2.1.1c), q(t) = q_{0}(t) + ⋯ + q_{a−1}(t), with q_{i}(t) = ∑_{j} a_{ij}t^{i+ja} so that a_{ij} is a polynomial in j of degree d_{i} − 1. Let d = max d_{i}. By (2.1.1d), r(S_{i}) = d + 1 and r(s_{i}) = d. Thus, by (3.3.3), r(s_{i}) ≤ c_{B_{r}}(M). Now (3.3.1) follows from (2.1.1b). _{□}

3.4 For the root system Φ, define

d(Φ^{+}, p^{r}) = |{α ∈ Φ^{+} | d_{α}(ρ, α^{∨}) = wht(α) ∈ p^{r}ℤ}|.

If p^{r} ≥ h, the tables in [B, pp. 250–275] can be used to readily check that d(Φ^{+}, p^{r}) = 0. Set d(Φ, p^{r}) = 2d(Φ^{+}, p^{r}).

By using (3.3.1), we obtain a relationship between the complexity c_{B_{r}}(M) of a B_{r}T - module M with the multiplicity of the cyclotomic polynomial Ψ_{p^{r}}(t) as a divisor of dim_{t} M.

(3.4.1) Theorem . Let M ∈ B_{r}T -mod.

(a) Let s be a positive integer such that Ψ_{p^{r}}(t)^{s} ∤ dim_{t} M (resp., Ψ_{2^{r+1}}(t)^{s} ∤ dim_{t} M) if p > 2 (resp., p = 2). Then

c_{B_{r}}(M) ≥ |Φ^{+}| − d(Φ^{+}, p^{r}) − s + 1.

(b) Now suppose that r = 1. If M ∈ G-mod and s is as in (a), then

c_{G_{1}}(M) = 2c_{B_{1}}(M) ≥ |Φ| − d(Φ, p) − 2s + 2.

Proof. Assume p ≠ 2. By (3.1.5), Ψ_{p^{r}}(t) is a divisor of h_{r}(t) with multiplicity |Φ^{+}| − d(Φ^{+}, p^{r}). Thus, (3.3.1) implies that Ψ_{p^{r}}(t) is a divisor of dim_{t} M with multiplicity at least |Φ^{+}| − d(Φ^{+}, p^{r}) − c_{B_{r}}(M). Hence, if Ψ_{p^{r}}(t)^{s} ∤ dim_{t} M, s ≥ |Φ^{+}| − c_{B_{r}}(M) − d(Φ^{+}, p^{r}) + 1. This proves (a), while (b) follows from this and [LN2, (3.4)]: if M ∈ G–mod then \frac{1}{2}c_{G_{1}}(M) = c_{B_{1}}(M). Finally, if p = 2, the above argument for (a) works using Ψ_{2^{r+1}}(t). _{□}

For λ ∈ X(T) and r ≥ 1, let Φ_{λ,p^{r}} = {α ∈ Φ | d_{α}(λ + ρ, α^{∨}) ∈ p^{r}ℤ}. Set Φ_{λ,p^{r}}^{+} = Φ_{λ,p^{r}} ∩ Φ^{+}, so that |Φ_{λ,p^{r}}| = 2|Φ_{λ,p^{r}}^{+}|. For any w ∈ W_{p}, if \overline{w} denotes the image of w under the quotient map W_{p} → W_{p}/T_{p} = W, then

Φ_{w·λ,p^{r}} = \overline{w}(Φ_{λ,p^{r}}).  (3.4.2)

(3.4.3) Corollary . Let λ ∈ X(T)_{+}. Then

(a) c_{B_{r}}(H^{0}(λ)) ≥ |Φ^{+}| − |Φ_{λ,p^{r}}^{+}|.

(b) For r = 1, c_{G_{1}}(H^{0}(λ)) ≥ |Φ| − |Φ_{λ,p}|.

Proof. We give the proof for p ≠ 2, leaving the modification for p = 2 to the reader. By (3.1.2), |Φ_{λ,p^{r}}^{+}| − d(Φ^{+}, p^{r}) is the multiplicity of Ψ_{p^{r}}(t) as a divisor of dim_{t} H^{0}(λ). Thus, (3.4.1a) implies that

c_{B_{r}}(H^{0}(λ)) ≥ |Φ^{+}| − d(Φ^{+}, p^{r}) − (|Φ_{λ,p^{r}}^{+}| − d(Φ^{+}, p^{r}))
= |Φ^{+}| − |Φ_{λ,p^{r}}^{+}|.

This proves (a), while (b) follows from (a) and (3.4.1b). _{□}

3.5 The following result relates the p-divisibility of the module with the dimension of the support variety.

(3.5.1) Theorem . We have

(a) If M ∈ B_{r}T -mod and a = |Φ^{+}| − d(Φ^{+}, p^{r}) − c_{B_{r}}(M) ≥ 0, then p^{a} | dim M.

(b) Suppose r = 1. If M ∈ G-mod and b = |Φ| − d(Φ, p) − c_{G_{1}}(M), then p^{b/2} | dim M.

Proof. First, assume p > 2. By (3.4.1a), Ψ_{p^{r}}(t)^{a}| dim_{t} M. Since Ψ_{p^{r}}(t) = 1+t^{q} +⋯+t^{q(p−1)} for q = p^{r−1}, p^{a} = Ψ_{p^{r}}(1)^{a} divides dim M, proving (a). Next, \frac{b}{2} = \frac{1}{2}(|Φ| − c_{G_{1}}(M)) = |Φ^{+}| − c_{B_{1}}(M) = a, so (a) ⇒ (b). If p = 2, replace Ψ_{p^{r}}(t) by Ψ_{2^{r+1}}(t). _{□}

4 Comparing supports of Verma and induced modules

4.1 A weight λ ∈ X(T) is p-regular if (λ + ρ, α^{∨}) ∉ pℤ for all α ∈ Φ. The condition p ≥ h is equivalent to the existence of p-regular weights. In particular, by the definition of h, the zero weight 0 is p-regular if and only if 0 ∈ X(T)_{+} ∩ C_{ℤ} if and only if p ≥ h. If p > h, then V_{G_{1}} is isomorphic to N_{1}(G) which coincides with the variety N (G) of nilpotent elements in 𝔤. If N_{1}(G) = N (G), it is true that p ≥ h.^{3}

For α ∈ Π, let 0 ≠ x_{α} ∈ 𝔤_{α}. For I ⊆ Π, let x_{I} = ∑_{α∈I} x_{α}. Assume that p ≥ h, so that x_{Π} ∈ N_{1}(G). In this section, we will make several uses of [Jan1, (4.14)(1)], which says the following: Assume that p ≥ h and assume that λ ∈ X(T)_{+} is not p-regular. Then L(λ) is x_{Π}-projective. Thus, for all p, if λ ∈ X(T)_{+} is not p-regular, then x_{Π} ∉ V_{G_{1}}(L(λ)); in particular, V_{G_{1}}(L(λ)) ⊊ N (G). In (6.4.1), we will give another proof of this result.

The next result computes V_{G_{1}}(Z_{1}(λ)) when λ is p-regular.

(4.1.1) Theorem . Let p ≥ h. Then λ ∈ X(T) is p-regular if and only if V_{G_{1}}(Z_{1}(λ)) = N_{1}(U).

Proof. By [AJ, (2.9)], Ext_{B_{1}}^{n}(k, w · 0) ≅ Ext_{B_{1}}^{n−l(w)}(k, k), so

r(Ext_{B_{1}}^{n}(k, w · 0)) = r(Ext_{B_{1}}^{n−l(w)}(k, k))
= r(Ext_{B_{1}}^{n}(k, k))
= dim N_{1}(U).

3This fact can be checked in a case-by-case basis. It is obvious in type A. Otherwise, one shows that, if p < h, then x^{[p]} ≠ 0 for x a regular nilpotent element (e. g., x is the sum of the simple root vectors).

In classical types B, C, D and type G_{2}, this check is easy. A more laborious verification is required in the remaining exceptional cases. On the other hand, the claim follows immediately from (6.3.1) below.

Furthermore, the universal mapping property of induction and (2.2.2) imply that

c_{G_{1}}(Z_{1}(w · 0)) = r(Ext_{G_{1}}^{n}(⊕_{λ∈X_{1}(T)}L_{1}(λ), Z_{1}(w · 0))
≥ r(Ext_{G_{1}}^{n}(k, Z_{1}(w · 0)))
= r(Ext_{B_{1}}^{n}(k, w · 0))
= dim N_{1}(U)

However, by (2.3.1), V_{G_{1}}(Z_{1}(w · 0)) = V_{B_{1}}(Z_{1}(w · 0)) ⊆ N_{1}(U). It follows that

c_{G_{1}}(Z_{1}(w · 0)) = dim V_{G_{1}}(Z_{1}(w · 0)) ≤ dim N_{1}(U).

The variety N_{1}(U) is irreducible, so V_{G_{1}}(Z(w · 0)) = N_{1}(U).

For any p-regular weight λ, there exist ν ∈ X(T) and w ∈ W such that μ = w · 0 + pν lies in the same alcove as λ. The translation functor T_{λ}^{μ} : G_{1}T -mod → G_{1}T -mod carries Ẑ_{1}(λ) to Ẑ_{1}(μ), defining isomorphisms

Ext_{G_{1}T}^{•} (Ẑ_{1}(λ), ⊕_{σ∈X(T)}L̂_{1}(σ)) ≅ Ext_{G_{1}T}^{•} (Ẑ_{1}(μ), ⊕_{σ∈X(T)}L̂_{1}(σ))).

See [Jan2, II, §9]. It follows by [DNP, (7.2)] that c_{G_{1}T} (Ẑ_{1}(λ)) = c_{G_{1}T} (Ẑ_{1}(μ)). Thus, by (2.2.12), c_{G_{1}}(Z_{1}(λ)) = c_{G_{1}}(Z_{1}(μ)), and again V_{G_{1}}(Z_{1}(λ)) = N_{1}(U).

On the other hand, suppose that V_{G_{1}}(Z_{1}(λ)) = N_{1}(U), but that λ is not p-regular. By [Jan1, (4.14)(1)], V_{B_{1}}(L_{1}(w · λ)) ⊊ N_{1}(U) for all w ∈ W. The module Z_{1}(λ) has a composition series with factors all of the form L_{1}(w · λ) where w ∈ W. Therefore, by (2.2.8),

V_{G_{1}}(Z_{1}(λ)) ⊆ ⋃_{w∈W} V_{G_{1}}(L_{1}(w · λ)) ⊊ N_{1}(U).

This is a contradiction, so λ is p-regular. _{□}

For λ ∈ X(T), let Γ_{r}(λ) = W · λ + p^{m}ℤΦ + p^{r}X(T), where m be the smallest integer such that there exists a root α with (λ + ρ, α^{∨}) ∉ p^{m}ℤ. Then Γ_{r}(λ) equals the set of weights μ that L_{r}(μ) belongs to the block B_{r}(λ) in G_{r}-mod containing L_{r}(λ); see [Jan2, II, (9.19)]. For r = 1, Γ_{1}(λ) = W · λ + pX(T).

The following result is a variation of the previous theorem.

(4.1.2) Proposition . Let p ≥ h and λ ∈ X(T).

(a) The weight λ ∈ X(T)_{+} is p-regular if and only if V_{G_{r}}(H^{0}(λ)) = V_{G_{r}}.

(b) If the weight λ ∈ X(T) is not p-regular, then V_{G_{r}}(Z_{r}(λ)) ⊊ V_{U_{r}}.

Proof. If λ is p-regular, then p ∤ dim H^{0}(λ). This means that V_{G_{r}}(H^{0}(λ)) = V_{G_{r}}. Conversely, if V_{G_{r}}(H^{0}(λ)) = V_{G_{r}}, then, by [SFB2, (7.1)], V_{G_{1}}(H^{0}(λ)) = V_{G_{1}}∩V_{G_{r}}(H^{0}(λ)) = V_{G_{1}}. But, if λ is not p-regular then V_{G_{1}}(H^{0}(λ)) ⊊ V_{G_{1}} by [Jan1, (4.14)(1)]. This proves (a).

To see (b), suppose λ is not p-regular, but that V_{G_{r}}(Z_{r}(λ)) = V_{U_{r}}. By [SFB2, (7.1)] again, V_{G_{1}}(Z_{r}(λ)) = V_{U_{1}}. But Ẑ_{r}(λ)|_{G_{1}T} is a projective U_{1}^{+}-module, so it has a G_{1}T - module filtration with sections Ẑ_{1}(μ) for various μ ∈ X(T). Since the G_{r}-compostion factors L_{r}(μ) of Z_{r}(λ) satisfy μ ∈ Γ_{r}(λ) since L_{r}(μ)|_{G_{1}} is a direct sum of copies of L_{1}(μ), we see that the G_{1}-composition factors L_{1}(μ) of Z_{r}(λ) all satisfy μ ∈ Γ_{1}(λ). Thus, if Ẑ_{1}(μ) appears as a section in the filtration of Ẑ_{r}(λ)|_{G_{1}T}, then μ ∈ Γ_{1}(λ). Now (4.1.1) implies that V_{G_{1}}(Z_{r}(λ)) ⊊ V_{U_{1}}, a contradiction. _{□}

4.2 Support varieties over Levi subgroups. Let I ⊆ Π and set L = L_{I}, P = U_{I}.L, and P^{+} = U_{I}^{+}.L.

(4.2.1) Proposition . If λ ∈ X(T), then V_{L_{r}}(Z_{r}(λ)) = V_{L_{r}}(Z_{r}^{I}(λ)).

Proof. By Mackey theory for G_{r}T [CPS2, (4.1)] and remarks following (1.3.1),

Ẑ_{r}(λ)|_{P_{r}^{+}T} ≅ ind_{P_{r}T}^{G_{r}T} Ẑ_{r}^{I}(λ)^{e}|_{P_{r}^{+}T} ≅ ind_{L_{r}T}^{P_{r}^{+}T} Ẑ_{r}^{I}(λ).

Since Ẑ_{r}^{I}(λ) inflates to a P_{r}^{+}T -module (which we still denote by Ẑ_{r}^{I}(λ)), the tensor identity yields

Ẑ_{r}(λ)|_{P_{r}^{+}T} ≅ ind_{L_{r}T}^{P_{r}^{+}T} k ⊗ Ẑ_{r}^{I}(λ) ≅ k[U_{r}^{+}] ⊗ Ẑ_{r}^{I}(λ).

Thus,

Z_{r}(λ) ≅ k[U_{r}^{+}] ⊗ Z_{r}^{I}(λ)

as L_{r}-modules, so V_{L_{r}}(Z_{r}(λ)) ⊆ V_{L_{r}}(Z_{r}^{I}(λ)).

To see the reverse inclusion, let N = k[U_{r}^{+}]/k. The short exact sequence 0 → k → k[U_{r}^{+}] → N → 0 of L_{r}T -modules splits because the weights of T in N do not lie in ℤΦ_{I}. Hence, Z_{r}^{I}(λ) is an L_{r}-direct summand of Z_{r}(λ), so V_{L_{r}}(Z_{r}^{I}(λ)) ⊆ V_{L_{r}}(Z_{r}(λ)). _{□}

4.3 As an application of the above results, we have

(4.3.1) Theorem . Let I ⊆ Π and assume that p is good for G.

(a) x_{I} ∈ V_{(L_{I})_{1}}(Z_{1}(λ)) if and only if Φ_{λ,p} ∩ Φ_{I} = ∅.

(b) x_{I} ∉ V_{(L_{I})_{1}}(⊕_{w∈W} Z_{1}(w · λ)) if and only if w(Φ_{λ,p}) ∩ Φ_{I} ≠ ∅ for all w ∈ W.

Proof. We first prove (a). By (4.2.1), we can assume that L = G. First, assume that Φ_{λ,p} = ∅, i. e., λ is p-regular. Then p ≥ h and by (4.1.1), V_{G_{1}}(Z_{1}(λ)) = N_{1}(U) = N (G), so x_{Π} ∈ V_{G_{1}}(Z_{1}(λ)). Conversely, suppose that x_{Π} ∈ V_{G_{1}}(Z_{1}(λ)) = V_{B_{1}}(Z_{1}(λ)). Since p is good, the B-orbit of x_{Π} is dense in N_{1}(B), so V_{B_{1}}(Z_{1}(λ)) = N_{1}(U) = N (U). By (4.1.1), Φ_{λ,p} = ∅ as required. Finally, (b) now follows from (3.4.2). _{□}

4.4 The following theorem provides an upper bound on the support of a module in B_{r}(λ).

(4.4.1) Theorem . Let λ ∈ X(T) and M ∈ B_{r}(λ)-mod. Then

(a) V_{B_{r}}(M) ⊆ ⋃_{μ∈Γ_{r}(λ)} V_{G_{r}}(Z_{r}(μ));

(b) If M is a G-module, then V_{G_{r}}(M) ⊆ ⋃_{μ∈Γ_{r}(λ)} G · V_{G_{r}}(Z_{r}(μ)).

Proof. We first prove (a). Let Γ′_{r}(λ) = X_{r}(T) ∩ Γ_{r}(λ). Given μ ∈ Γ_{r}(λ), Z_{r}(μ) ≅ Z_{r}(μ′) for some μ′ ∈ Γ′_{r}(μ). Let M′ = ⊕_{μ∈Γ′_{r}(λ)}Z_{r}(μ) and M′′ = ⊕_{μ∈Γ′_{r}(λ)} μ ∈ B_{r}–mod. Form the commutative diagram

H(G_{r}, k) ⊗ Ext_{G_{r}}^{•}(M, M′) ⟶ Ext_{G_{r}}^{•}(M, M′)
↓^{res⊗δ}  ↓^{σ}
H(B_{r}, k) ⊗ Ext_{B_{r}}^{•}(M, M′′) ⟶ Ext_{B_{r}}^{•}(M, M′′)

where δ and σ are isomorphisms implied by the universal mapping property of induction. If I is the annihilator of H(G_{r}, k) on Ext_{G_{r}}^{•}(M, M′) and J is the annihilator of H(B_{r}, k) on Ext_{B_{r}}^{•}(M, M′′), then res(I) ⊆ J. Thus, V_{B_{r}}(M, M′′) = V (J) ⊆ V_{G_{r}}(M, M′). If Ext_{B_{r}}^{•}(M, μ) ≠ 0 for some μ ∈ X(T), then Ext_{G_{r}}^{•}(M, Z_{r}(μ)) ≠ 0, so μ ∈ Γ_{r}(λ). Thus, by (2.2.9), V (J) = V_{B_{r}}(M). Hence, (a) follows from (2.2.1) and the containments

V_{B_{r}}(M) = V_{B_{r}}(M, M′′) ⊆ V_{G_{r}}(M, M′) ⊆ V_{G_{r}}(M′).

Now assume that M is a G-module. Since G acts on V_{G_{r}}(M) and G · V_{B_{r}}(M) = V_{G_{r}}(M) [Bend, (4.5.2)], (b) follows also. _{□}

4.5 We will now indicate how Theorem 4.4.1 fits into the context of Jantzen’s earlier work. The next corollary is the key proposition ([Jan4, (2.4)]) used in his calculation of the supports varieties for H^{0}(λ) in type A.

(4.5.1) Corollary . Let r = 1 and I ⊆ Π. If w(Φ_{λ,p}) ∩ Φ_{I} ≠ ∅ for all w ∈ W, then x_{I} ∉ V_{G_{1}}(H^{0}(λ)).

Proof. Suppose that w(Φ_{λ,p})∩Φ_{I} ≠ ∅ for all w ∈ W. Since Γ_{1}(λ) = W ·λ+pX(T), (4.3.1a) and (3.4.2) imply that x_{I} ∉ V_{(L_{I})_{1}}(⊕_{μ∈Γ_{1}(λ)}Z_{1}(μ)). Hence, by (4.4.1a), x_{I} ∉ V_{B_{1}}(H^{0}(λ)) = V_{G_{1}}(H^{0}(λ)) ∩ V_{B_{1}}, so x_{I} ∉ V_{G_{1}}(H^{0}(λ)). _{□}

Let G = GL_{n}(k). Any subroot system Φ′ of Φ is isomorphic to A_{d_{1}−1} × A_{d_{2}−1} × … A_{d_{s}−1} where d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{s} > 0 is a partition of n. This defines a bijective correspondence between non-isomorphic subroot systems of Φ and partitions of n. Given J ⊆ Π, let σ(J) be the partition of n corresponding to the subsystem Φ_{J}. In addition, given λ ∈ X(T), let σ(λ) be the partition of n defined by the subroot system Φ_{λ,p} of Φ. For a partition σ of n, let x_{σ} ∈ GL_{n}(k) be the nilpotent matrix in upper triangular Jordan canonical having blocks of the sizes corresponding to the partition σ. Also, let σ^{t} be the dual partition. From [Kr],

\overline{G · x_{σ(J)^{t}}} = G · 𝔲_{J}.  (4.5.2)

We can now recover Jantzen’s calculations of the supports of H^{0}(λ) for G_{1} where G = GL_{n}.

(4.5.3) Corollary . Let G = GL_{n}. For λ ∈ X(T)_{+}, choose J ⊆ Π such that Φ_{λ,p} = y(Φ_{J}) for some y ∈ W. Then then V_{G_{1}}(H^{0}(λ)) = G · 𝔲_{J} = \overline{G · x_{σ(λ)^{t}}}.

Proof. The G-stable variety V_{G_{1}}(H^{0}(λ)) is is a finite union of closures of G-orbits in N_{1}(G). Let x_{I} ∈ V_{G_{1}}(H^{0}(λ)) for some I ⊆ Π. By (4.5.1), w(Φ_{λ,p}) ∩ Φ_{I} = ∅ for some w ∈ W; equivalently, w(Φ_{J}) ∩ Φ_{I} = ∅ for some w ∈ W. Consequently, x_{I} ∈ G · 𝔲_{J} = \overline{G · x_{σ(λ)^{t}}} by (1.3.4) and (4.5.2). Hence, V_{G_{1}}(H^{0}(λ)) ⊆ \overline{G · x_{σ(λ)^{t}}}. By (3.4.3b), dim V_{G_{1}}(H^{0}(λ)) ≥ dim G· 𝔲_{J} = |Φ| − |Φ_{λ,p}|. The variety \overline{G · x_{σ(λ)^{t}}} is irreducible, thus V_{G_{1}}(H^{0}(λ)) = \overline{G · x_{σ(λ)^{t}}}. _{□} 4.6 Independence of weights. Fix r ≥ 1. For λ ∈ X(T)_{+}, choose s ≥ r so that the composition factors L(μ) of H^{0}(λ) all satisfy μ ∈ X_{s}(T). Then [Jan2, II, (9.21)] implies that each such μ lies in Γ_{s}(λ). Since Γ_{s}(λ) ⊆ Γ_{r}(λ), we conclude that H^{0}(λ) ∈ B_{r}(λ). Thus, (4.4.1b) implies that for all σ ∈ Γ_{r}(λ) ∩ X(T)_{+}, V_{G_{r}}(H^{0}(σ)) ⊆ G · [⋃_{μ∈Γ_{r}(λ)} V_{G_{r}}(Z_{r}(μ))].

(4.6.1) Theorem . Let G be a reductive algebraic group and let λ ∈ X(T)_{+}. The following statements are equivalent:

(a) V_{G_{r}}(H^{0}(σ)) = G · [⋃_{μ∈Γ_{r}(λ)} V_{G_{r}}(Z_{r}(μ))] for all σ ∈ Γ_{r}(λ) ∩ X(T)_{+};

(b) V_{G_{r}}(H^{0}(λ)) = V_{G_{r}}(H^{0}(σ)) for all σ ∈ Γ_{r}(λ) ∩ X(T)_{+}.

Proof. Clearly (a) ⇒ (b) . Conversely, suppose that (b) holds, and let V = V_{G_{r}}(H^{0}(λ)). We claim that V_{G_{r}}(L(μ)) ⊆ V for all μ ∈ Γ_{r}(λ) ∩ X(T)_{+}. If μ ∈ C_{ℤ}, then L(μ) = H^{0}(μ), so the claim holds in this case. Otherwise, form the short exact sequence 0 → L(μ) → H^{0}(μ) → N → 0. If L(τ) is a composition factor of N, then τ < μ, so the evident induction and (2.2.8) imply that V_{G_{r}}(N) ⊆ V. By (2.2.7), V_{G_{r}}(L(μ)) ⊆ V. This proves our claim. Given μ ∈ Γ_{r}(λ), L_{r}(μ) = L(μ′) for some μ′ ∈ Γ_{r}(λ) ∩ X(T)_{+}. Also, L(μ′)|_{G_{r}} is a direct sum of copies of L_{r}(μ′). Thus, by (2.2.7) again, V_{G_{r}}(M) ⊆ V for all M ∈ B_{r}(λ). In particular, V_{G_{r}}(⊕_{μ∈Γ_{r}(λ)}Z_{r}(μ)) ⊆ V. Since V is G-stable, we have G · [⋃_{μ∈Γ_{r}(λ)} V_{G_{r}}(Z_{r}(μ))] ⊆ V_{G_{r}}(H^{0}(λ)) = V_{G_{r}}(H^{0}(σ)). The reverse inclusion follows by (4.4.1b). Thus, (b) ⇒ (a). _{□}

5 Varieties and Induction

5.1 Let H be an affine k-group. By an H-algebra, we mean a commutative, Noetherian k-algebra S upon which H acts as rational k-algebra automorphisms. Thus, the structure maps ν : S ⊗ S → S, a ⊗ b ↦ ab, and η : k → S, 1 ↦ 1_{S}, are morphisms of H-modules. Then a left S-module M is called an S.H-module provided that M is a rational H-module and the map S ⊗ M → M, s ⊗ m ↦ sm, is a morphism of H-modules. Let S.H-mod be the category of S.H-modules.

(5.1.1) Proposition . Suppose that M is an S.H-module for an affine k-group H and H-algebra S. Suppose H is a closed subgroup of an affine k-group K. Then:

(a) The induced module ind_{H}^{K} M is an (ind_{H}^{K} S).K-module.

(b) Let R be a K-algebra. For any R.H-module M, ind_{H}^{K} M has the (unique) structure of an R.K-module compatible with the R.H-module structure of M. Let I be a K-stable ideal in R which annhilates M. Then I annihilates ind_{H}^{K} M.

(c) In (b), suppose that R → S is a morphism of H-algebras. Then ind_{H}^{K} defines a functor S.H-mod → R.K-mod.

Proof. The assertions follow from the universal mapping properties of induction. Let Ev_{S} : ind_{H}^{K} S → S be the evaluation map. Then ν : S ⊗ S → S lifts to a unique map μ of rational K-modules making the following diagram commutative

ind_{H}^{K} S ⊗ ind_{H}^{K} S ⟶^{μ} ind_{H}^{K} S
Ev_{S} ⊗ Ev_{S} ↓  ↓^{Ev}_{S}
S ⊗ S  ⟶_{ν} S

and defining a multiplication on ind_{H}^{K} S. Similarly, the unit map η : k → S lifts to define a unit map k → ind_{H}^{K} S. It is straightforward to check that the axioms for an (associative) algebra hold for ind_{H}^{K} S. In the same way, the module structure map S ⊗ M → M lifts to define the structure of an ind_{H}^{K} S-module on ind_{H}^{K} M. This proves (a). To see (b), we consider the commutative diagram

I ⊗ ind_{H}^{K} M ⟶ R ⊗ ind_{H}^{K} M ⟶ ind_{H}^{K} M
↓  ↓  ↓
I ⊗ M ⟶ R ⊗ M ⟶ M

obtained by applying ind_{H}^{K} to the bottom row. Here we use the fact that, for any Kmodule V, ind_{H}^{K}(V ⊗ M) ≅ V ⊗ ind_{H}^{K}. The right-hand square in the above diagram proves that ind_{H}^{K} M has the unique structure of a R.K-module which is compatible with the R.Hmodule structure of M. Since I annihilates M, the universal mapping property of induction implies that I annihilates ind_{H}^{K} M. Finally, (c) is clear. _{□}

5.2 Let H be a connected algebraic group over k, and let S be an H-algebra. It will be convenient to recast the category S.H-mod in terms of the module category of a certain algebra of “differential operators”. Thus, let Dist(H) be the distribution Hopf algebra of H [DG, II; §4, (6.1)] [T, Ch. 3]. Let 𝔣 : H − mod → Dist(H) -mod be the natural functor which assigns to any rational B-module its associated Dist(B)-module structure. Then 𝔣 is a full embedding, and so defines an equivalence of categories from H-mod to its strict image Dist(H) -mod′ in Dist(H) -mod.

Since η : k → S and ν : S ⊗ S → S are H-module morphisms, they induce morphisms of Dist(H)-modules. Equivalently, S is a left “module algebra” for Dist(H), in the sense that, for h ∈ Dist(B), a, b ∈ S, we have

\begin{cases}h(ab) = ∑(h_{(1)}a)(h_{(2)}b) \\ h1_{S} = ϵ(h)1_{S} \end{cases}

if Δ(h) = ∑ h_{(1)} ⊗ h_{(2)} is the image of h under the comultiplication Δ : Dist(H) → Dist(H) ⊗ Dist(H) and ϵ : Dist(H) → k is the counit on Dist(H).

Let D = S ♯ Dist(H) be the smash product of S and Dist(H) [M, (4.1.1)]. As a vector space, D = S ⊗ Dist(H), with multiplication given by

(a ⊗ h)(a′ ⊗ h′) = ∑ a(h_{(1)}a′) ⊗ h_{(2)}h′.

A D-module M is merely a module simultaneously for both S and Dist(H) such that

h(am) = ∑(h_{(1)}a)(h_{(2)}m)

for all h ∈ Dist(H), a ∈ S, and m ∈ M.

Consider the functor 𝔉 : Dist(H) -mod → D-mod

defined by 𝔉(−) = D ⊗_{Dist(H)} −. Alternatively, for M ∈ Dist(H) -mod, 𝔉(M) = S ⊗ M as vector spaces with action as follows: for a, b ∈ S, h ∈ Dist(H), and m ∈ M, we have

a(b ⊗ m) = ab ⊗ m; h(b ⊗ m) = ∑ h_{(1)}b ⊗ h_{(2)}m.

The functor 𝔉 is exact and provides a left adjoint to the natural restriction functor 𝔊 : D-mod → Dist(H) -mod defined by restricting any D-module to the subalgebra Dist(H) of D. Thus, G takes injective D-modules to injective Dist(H)-modules. Given M ∈ D-mod, we often denote 𝔊(M) simply by just M again.

If D-mod′ is the full subcategory of D-mod with objects V satisfying 𝔊(V) ∈ Dist(H) -mod′, then D-mod′ is equivalent to the category S.H-mod. Since S is a rational H-module, the functor F carries Dist(H) -mod′ to D-mod′ and it provides a left adjoint to G (restricted to D-mod′). The definitions imply that if ψ : V → W is a morphism in D-mod with V ∈ D-mod′, then the image ψ(V) is a submodule of W lying in D-mod′.

The category D-mod′ contains enough injectives. In fact, given V ∈ D-mod′, let Ĩ be a D-injective module containing V (which exists because D-mod is the module category for a ring). Let I be the sum (or union) of all submodules of Ĩ which lie in D-mod′. Then V ⊆ I. From remarks at the end of the previous paragraph, it follows directly that I is an injective in the category D-mod′. Also, 𝔊 : D-mod′ → Dist(H) -mod′ carries injectives to injectives.

Now let H = B the fixed Borel subgroup in the reductive group G.^{4} Let R be a Galgebra and let R → S be a B-algebra morphism. For M ∈ D-mod′, let 0 → M → I^{•} be an injective resolution in D-mod′ (and hence an injective resolution in B-mod). By (5.1.1c), 0 → ind_{B}^{G} M → ind_{B}^{G} I^{•} is a complex of R.G-modules so that, for n ≥ 0, R^{n} ind_{B}^{G} M inherits an R.G-module structure which is functorial in M.

The next result relates the annihilators of the higher right derived functors of R^{n}ind_{B}^{G} M with the annihilator on M. Recall the notation of (1.3.2): if I is an ideal in a G-algebra R, we let ℓ_{G}(I) be the largest G-stable ideal of R contained in I.

(5.2.1) Proposition . Let R be a G-algebra and let S be a B-algebra. Let π^{∗} : R → S be a B-algebra morphism. For M ∈ S ♯ Dist(B)- mod′ ≅ S.B-mod and n ≥ 0, we have

ℓ_{G}(π^{∗−1}(Ann_{S} M)) ⊆ Ann_{R} R^{n} ind_{B}^{G} M.

4In this case, the category Dist(B)-mod′ can be explicitly described. Locally finite modules for the distribution algebra Dist(T) of the torus T are direct sums of one-dimensional submodules defined by elements in the character group X(Dist(T)). Now X(T) ⊆ X(Dist(T)), and Dist(B)-mod′ is the full subcategory of Dist(B)-mod with objects consisting of those V which are Dist(B)-locally finite and which have Dist(T)- weights lying in X(T). See [CPS1, §9.4] for details.

Proof. By (5.1.1b), ℓ_{G}(π^{∗−1}(Ann_{S} M)) annihilates ind_{B}^{G} M. Now M ∈ D-mod′, where D = S/ Ann_{S} M ♯ Dist(B). Consider a short exact sequence

0 → M → I → L → 0,  (5.2.2)

in D-mod′ in which I is injective. By construction, Ann_{S} M ⊆ Ann_{S} T for T = M, I, L. Because M, I, and L are rational B-modules, (5.2.2) determines an exact sequence of R.G-modules

0 → ind_{B}^{G} M → ind_{B}^{G} I → ind_{B}^{G} L → R^{1} ind_{B}^{G} M → 0  (5.2.3)

as well as R.G-module isomorphisms

R^{n} ind_{B}^{G} L ≅ R^{n+1} ind_{B}^{G} M, ∀n ≥ 1.  (5.2.4)

By (5.2.3) and (5.1.1b), ℓ_{G}(π^{∗−1}(Ann_{S} M)) ⊆ Ann_{R} R^{1} ind_{B}^{G} M, while (5.2.4) and induction imply that

ℓ_{G}(π^{∗−1}(Ann_{S} M)) ⊆ ℓ_{G}(π^{∗−1}(Ann_{S} L))
⊆ Ann_{R} R^{n} ind_{B}^{G} L
= Ann_{R} R^{n+1} ind_{B}^{G} M

for all n ≥ 1. _{□}

5.3 Now fix an affine k-group G. In the next section, we will work with a pull-back diagram of closed subgroups of G:

\begin{matrix}K & ⟶^{◁} & G \\ ↑ & & ↑ \\ H ∩ K & ⟶_{◁} & H \end{matrix}  (5.3.1)

in which we have written H ∩ K for H ×_{G} K. Here K (resp. H ∩ K) is a normal subgroup of G (resp. H). Now let M be a rational G-module, and consider the commutative diagram

G-mod ⟶^{Hom_{K}(M,−)}  G/K-mod
ind_{H}^{G} ↑  ↑^{ind}_{H/H∩K}^{G/K}
H-mod ⟶_{Hom_{H∩K}(M,−)} H/H ∩ K-mod

(5.3.2)

of functors. Write

F(M, −) = Hom_{K}(M, ind_{H}^{G} −) = ind_{H/H∩K}^{G/K} Hom_{H∩K}(M, −)

for the composite functors. For N ∈ H-mod N, this gives two spectral sequences

Ê_{2}^{m,n}(M, N) = Ext_{K}^{m}(M, R^{n} ind_{H}^{G} N) ⇒ (R^{m+n}F)(M, N)  (5.3.3)
E_{2}^{m,n}(M, N) = R^{m} ind_{H/H∩K}^{G/K} Ext_{H∩K}^{n}(M, N) ⇒ (R^{m+n}F)(M, N)  (5.3.4)

both converging to R^{•}F(M, N); see [Jan2, I, (6.12)]. Also, given M, N above, the spectral sequence E_{2}^{•,•}(M, N) is a differential graded module for E_{2}^{•,•}(k, k).

5.4 Fix r ≥ 1 and set R = H(G_{r}, k) and S = H(B_{r}, k) for some fixed integer r ≥ 1. Let π^{∗} : R → S be the restriction map on cohomology. The morphism π : V_{B_{r}} → V_{G_{r}} induced by π^{∗} maps V_{B_{r}} homeomorphically onto its image. As discussed in (2.2.10), we will always identify V_{B_{r}} with its image in V_{G_{r}}.

(5.4.1) Theorem . Let M be a rational B-module such that R^{m} ind_{B}^{G} M = 0 for m ≠ t, where t is some fixed integer. Then

V_{G_{r}}(R^{t} ind_{B}^{G} M) = G · V_{B_{r}}(R^{t} ind_{B}^{G} M, M).

Proof. By (2.2.1), V_{B_{r}}(R^{t}ind_{B}^{G}M, M) ⊆ V_{B_{r}}(R^{t}ind_{B}^{G}M) ⊆ V_{G_{r}}(R^{t} ind_{B}^{G} M), so that

G · V_{B_{r}}(R^{t} ind_{B}^{G} M, M) ⊆ V_{G_{r}}(R^{t} ind_{B}^{G} M).

We consider the spectral sequences (5.3.3) and (5.3.4) for (H, K) = (B, G_{r}) and (M, N) = (R^{t} ind_{B}^{G} M, M). Since R^{n} ind_{B}^{G} M = 0 for all n ≠ t, the spectral sequence (5.3.3) collapses, so we finally obtain a spectral sequence:

E_{2}^{m,n} = R^{m} ind_{B/B_{r}}^{G/G_{r}} Ext_{B_{r}}^{n}(R^{t} ind_{B}^{G} M, M) ⇒ Ext_{G_{r}}^{m+n−t}(R^{t} ind_{B}^{G} M, R^{t} ind_{B}^{G} M).

Let M = Ext_{B_{r}}^{•}(R^{t} ind_{B}^{G} M, M). By (5.2.1), the ideal I = ℓ_{G}(π^{∗−1}(Ann_{S} M)) of R = H(G_{r}, k), which defines

G · V (Ann_{S} M) = G · V_{B_{r}}(R^{t} ind_{B}^{G} M, M),

annihilates E_{2}^{m,n} = R^{m} ind_{B/B_{r}}^{G/G_{r}} M for all non-negative integers m. Thus, I ⊆ Ann_{R} E_{∞}^{•,•}.

Since R^{m} ind_{B/B_{r}}^{G/G_{r}} = 0 for m > N = dim G/B, Ext_{G_{r}}^{•}(R^{t} ind_{B}^{G} M, R^{t} ind_{B}^{G} M) has a finite increasing R-stable filtration 0 = F^{0} ⊆ F^{1} ⊆ ⋯ ⊆ F^{N} = Ext_{G_{r}}^{•}(R^{t} ind_{B}^{G} M, R^{t} ind_{B}^{G} M) in which F^{i}/F^{i−1} = ⨁_{j} E_{∞}^{N−i+1,j}. Since IF^{i} ⊆ F^{i−1}, I^{N} annihilates Ext_{G_{r}}^{•}(R^{t} ind_{B}^{G} M, R^{t} ind_{B}^{G} M). Thus, V_{G_{r}}(R^{t} ind_{B}^{G} M) ⊆ G · V_{B_{r}}(R^{t} ind_{B}^{G} M, M), as required. _{□}

5.5 There are several applications of Theorem 5.4.1 worth mentioning. The first gives a characterization of V_{G_{r}}(H^{0}(λ)) when λ ∈ X_{+}(T) in terms of relative support varieties.

(5.5.1) Corollary . Let G a reductive algebraic group. For λ ∈ X(T)_{+},

V_{G_{r}}(H^{0}(λ)) = G · V_{B_{r}}(H^{0}(λ), λ).

Now let \overline{C}_{ℤ} = \overline{C} ∩ X(T) be the set of integral weights in the closure of the bottom p-alcove. We can prove an inclusion of supports for H^{0}(λ) for λ ∈ \overline{C}_{ℤ}.

(5.5.2) Corollary . Let λ ∈ \overline{C}_{ℤ} and w ∈ W. Then

V_{G_{r}}(L(λ)) = V_{G_{r}}(H^{0}(λ)) ⊆ G · V_{G_{r}}(Z_{r}(w · λ)).

Proof. For w ∈ W, L(λ) = H^{0}(λ) = H^{l(w)}(w · λ). Furthermore, H^{i}(w · λ) = 0 for i ≠ l(w). Therefore, by (5.4.1), (2.3.1c), and finally (2.2.1), we obtain

V_{G_{r}}(H^{l(w)}(w · λ)) = G · V_{B_{r}}(H^{l(w)}(w · λ), w · λ)
= G · V_{G_{r}}(H^{l(w)}(w · λ), Z_{r}(w · λ))
⊆ G · V_{G_{r}}(Z_{r}(w · λ)),

as required. _{□}

5.6 The following key result generalizes Corollary 5.5.2.

(5.6.1) Theorem . Let λ ∈ X(T)_{+}. Then for all w ∈ W,

(a) V_{G_{1}}(L(λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ));

(b) V_{G_{1}}(H^{0}(λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)).

Proof. If (a) holds, then (b) follows by the linkage principle, together with (2.2.7), so we prove (a). We will use induction on the ordering of dominant weights to prove that

V_{G_{1}}(L(λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ))  (5.6.2)

for all w ∈ W. For λ ∈ \overline{C}_{ℤ}, the inclusion (5.6.2) follows by (5.5.2). Now let λ ∈ X(T)_{+} be fixed and assume that that (5.6.2) holds whenever λ is replaced by any dominant weight μ satisfying μ < λ. By using (2.2.7), it suffices to prove that V_{G_{1}}(H^{l(w)}(w · λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)) because L(λ) is a composition factor of H^{l(w)}(w · λ) of multiplicity one and all other composition factors L(μ) have the property that μ < λ and μ is linked to λ [Jan2, II, (6.16)]. Here we are using the fact that since λ and μ are linked (i. e., W_{p}-conjugate), there exists x ∈ W such that Z_{1}(x · μ) ≅ Z_{1}(w · λ).

Applying (5.3.3) and (5.3.4) for (H, K) = (B, G_{1}) and (M, N) = (H^{l(w)}(w · λ), w · λ) yields spectral sequences

Ê_{2}^{m,n} = Ext_{G_{1}}^{m}(H^{l(w)}(w · λ), H^{n}(w · λ)) ⇒ (R^{m+n}F)(M, N)  (5.6.3)
E_{2}^{m,n} = R^{m}ind_{B/B_{1}}^{G/G_{1}} Ext_{B_{1}}^{n}(H^{l(w)}(w · λ), w · λ) ⇒ (R^{m+n}F)(M, N)  (5.6.4)

which have the same E_{∞}-term. Let R = H(G_{1}, k). The differentials in (5.6.4) are R-module homomorphisms, so Ann_{R} E_{2} ⊆ Ann_{R} E_{∞}. Hence, by (5.2.1) and (2.3.1c),

V (Ann_{R} Ê_{∞}) = V (Ann_{R} E_{∞})
⊆ V (Ann_{R} E_{2})
⊆ G · V_{B_{1}}(H^{l(w)}(w · λ), w · λ)
⊆ G · V_{G_{1}}(Z_{1}(w · λ)).  (5.6.5)

Let id ∈ Hom_{G_{1}}(H^{l(w)}(w · λ), H^{l(w)}(w · λ)) be the identity homomorphism. Then

R/ Ann_{R} id ≅ R · id ⊆ Ê_{2}^{•,l(w)} = Ext_{G_{1}}^{•}(H^{l(w)}(w · λ), H^{l(w)}(w · λ)).

Set K_{2} = R · id and inductively define K_{r} = ker δ̂_{r−1}|_{K_{r−1}}/(Im(δ̂_{r−1}) ∩ K_{r−1}), where δ̂_{r−1} is the differential on Ê_{r−1}. Then K_{r} ↪ Ê_{r}^{•,l(w)}, so

V (Ann_{R} K_{r}) ⊆ V (Ann_{R} Ê_{r}^{•,l(w)}).  (5.6.6)

Set J_{2} = R and J′_{2} = Ann_{R} id. There exists a decreasing chain J_{2} ⊇ J_{3} ⊇ J_{4} ⊇ ⋯ and increasing chain J′_{2} ⊆ J′_{3} ⊆ ⋯ of ideals in R such that ⋃ J′_{r} ⊆ ⋂ J_{r} and J_{r}/J′_{r} ≅ K_{r} for all r ≥ 2. For r ≥ 3, there is a short exact sequence of R-modules

0 → J_{r}/J′_{r−1} → K_{r−1} → Im(δ̂_{r−1}|_{K_{r−1}}) → 0.

Thus, Im(δ̂_{r−1}|_{K_{r−1}}) ≅ J_{r−1}/J_{r}. Now Im(δ̂_{r−1}|_{K_{r−1}}) is an R-submodule of

F_{r−1} =^{def} ⨁_{n≠l(w)} Ê_{r−1}^{•,n},

and thus

V (Ann_{R} J_{r−1}/J_{r}) ⊆ V (Ann_{R} F_{r−1}) ⊆ V (Ann_{R} F_{2})
= V_{G_{1}}(H^{l(w)}(w · λ), ⨁_{n≠l(w)} H^{n}(w · λ))

⊆ ⋃_{n≠l(w)} V_{G_{1}}(H^{n}(w · λ)).

For n ≠ l(w), the composition factors of H^{n}(w · λ) have high weights which are both linked to and strictly less than λ [Jan2, II, (6.15)], so by our induction hypothesis V_{G_{1}}(H^{n}(w·λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)). We conclude that

V (Ann_{R} F_{r}) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)),  (5.6.7)

and, for r ≥ 3,

V (Ann_{R}(J_{r−1}/J_{r})) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)).  (5.6.8)

In particular, when r = 3, V (J_{3}) ⊆ G·V_{G_{1}}(Z_{1}(w·λ)). For r > 3, (J_{r−1}∩Ann_{R}(J_{r−1}/J_{r}))^{2} ⊆ J_{r}, so

V (J_{r}) ⊆ V (J_{r−1}) ∪ V (Ann_{R} (J_{r−1}/J_{r}))
⊆ G · V (Z_{r}(w · λ))  (5.6.9)

by (5.6.8) and induction on r.

For n > dim G/B, Ê_{2}^{•,n} = 0, so Ê_{s} = Ê_{∞} for some s. The inclusion

J_{s} · (Ann_{R} (J_{s}/ Ann_{R} id)) ⊆ Ann_{R} id = J′_{2}

of ideals gives that V_{G_{1}}(H^{l(w)}(w · λ)) = V (J′_{2}) ⊆ V (J_{s}) ∪ V (Ann_{R} (J_{s}/J′_{2})). Therefore, by (5.6.9), the theorem will follow once the inclusion

V (Ann_{R} (J_{s}/J′_{2})) ⊆ G · V_{G_{1}}(Z_{1}(w · λ))  (5.6.10)

is established. Since K_{s} ⊆ Ê_{s} = Ê_{∞}, (5.6.5) implies that

V (Ann_{R}(J_{s}/J′_{s})) = V (Ann_{R} K_{s}) ⊆ V (Ann_{R} Ê_{∞}) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)).

Furthermore, V (Ann_{R} (J′_{s}/J′_{2})) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)) because

Ann_{R} (J′_{s}/J′_{2}) ⊇ (Ann_{R} (J′_{s}/J′_{s−1}))(Ann_{R}(J′_{s−1}/J′_{s−2})) ⋯ (Ann_{R} (J′_{3}/J′_{2})).

and, by (5.6.7),

V (Ann_{R} (J′_{s}/J′_{2})) ⊆ V (Ann_{R} (J′_{s}/J′_{s−1})) ∪ ⋯ ∪ V (Ann_{R} (J′_{3}/J′_{2}))
⊆ G · V_{G_{1}}(Z_{1}(w · λ))  (5.6.11)

since J′_{j}/J′_{j−1} is a section of F_{j}. Finally, we observe that

(Ann_{R} K_{s})(Ann_{R} (J′_{s}/J′_{2})) = (Ann_{R} (J_{s}/J′_{s}))(Ann_{R} (J′_{s}/J′_{2}))
⊆ Ann_{R} (J_{s}/J′_{2}).

Thus, by (5.6.6) and (5.6.11),

V (Ann_{R} (J_{s}/J′_{2})) ⊆ V (Ann_{R} K_{s}) ∪ V (Ann_{R} (J′_{s}/J′_{2})) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)).

This proves (5.6.10) as required. _{□}

(5.6.12) Remark . When r > 1, the above proof can be easily modified to give the following equality

V_{G_{r}}(H^{0}(λ)) = G · [  ⋃_{μ↑λ, μ∈X(T)_{+}} V_{G_{r}}(Z_{r}(w · μ))]  (5.6.13)

for any w ∈ W. Here ↑ is as in [Jan2, II,§6].

6 Applications

Throughout this section, G will be a fixed reductive group over k. We will identify V_{G_{1}} with its image in N_{1}(G); see the discussion around (2.2.10).

6.1 The following result provides a condition which guarantees that V_{G_{r}}(Z_{r}(λ)) lies in the nilpotent radical 𝔲_{I} of a parabolic subalgebra.

(6.1.1) Proposition . Let L_{I} be a Levi subgroup of G corresponding to I ⊆ Π. Let r ≥ 1 and λ ∈ X(T) be given such that Z_{r}^{I}(λ) is a projective L_{r}-module (i. e., (λ + ρ, α^{∨}) ∈ p^{r}ℤ for all α ∈ I). Then

V_{G_{r}}(Z_{r}(λ)) ⊆ V_{(U_{I})_{r}}.

When r = 1, V_{(U_{I})_{1}} ⊆ 𝔲_{I}.

Proof. In this proof, we write L_{r} for (L_{I})_{r}, P_{r} for (P_{I})_{r}, and U_{r} for (U_{I})_{r}. If Z_{r}^{I}(λ)^{e} denotes the inflation of Z_{r}^{I}(λ) from L_{r} to P_{r}, then Z_{r}(λ) = ind_{P_{r}}^{G_{r}} Z_{r}^{I}(λ)^{e}, as noted in (1.3). By (2.3.1a), V_{G_{r}}(Z_{r}(λ)) ⊆ V_{P_{r}}(Z_{r}^{I}(λ)^{e}). For M = Z_{r}^{I}(λ)^{e}, consider the commutative diagram

\begin{matrix}H(P_{r}, k) & ⟶^{α} & H(U_{r}, k) \\ ^{γ}↓ & & ↓_{δ} \\ Ext_{P_{r}}^{•}(M, M) & ⟶_{β} & Ext_{U_{r}}^{•}(M, M) \end{matrix}  (6.1.2)

in which α, β are the natural restriction maps on cohomology, while γ = − ⊗ M and δ = −⊗M. Because the action of U_{r} on M = Z_{r}^{I}(λ)^{e} is trivial, Ext_{U_{r}}^{•}(M, M) ≅ Ext_{U_{r}}^{•}(k, k)⊗ M^{∗} ⊗ M and δ is an injection. On the other hand, because M is projective for L_{r}, the Hochschild-Serre spectral sequence for Ext_{P_{r}}^{•}, using the normal subgroup U_{r} of P_{r} and quotient group L_{r} = P_{r}/U_{r}, shows that β maps Ext_{P_{r}}^{•}(M, M) isomorphically into its image Ext_{U_{r}}^{•}(M, M)^{L_{r}}. Therefore, ker(α) = ker(γ). Since V_{P_{r}}(Z_{r}^{I}(λ)^{e}) = V (ker(γ)) and Im(V_{U_{r}} → V_{P_{r}}) = V (ker(α)), we conclude that V_{G_{r}}(Z_{r}(λ)) ⊆ V_{P_{r}}(Z_{r}^{I}(λ)^{e}) = V_{U_{r}}, identifying V_{U_{r}} with its image in V_{G_{r}}. _{□}

We have the following consequence when r = 1.

(6.1.3) Corollary . For λ ∈ X(T)_{+} and p good, choose I ⊆ Π so that w(Φ_{λ,p}) = Φ_{I} for some w ∈ W. Then

V_{G_{1}}(H^{0}(λ)) ⊆ G · V_{(U_{I})_{1}}.

Proof. By (5.6.1), V_{G_{1}}(H^{0}(λ)) ⊆ G · V_{G_{1}}(Z_{1}(w · λ)). By (3.4.2), Φ_{w·λ,p} = Φ_{I}, so by (6.1.1) V_{G_{1}}(Z_{1}(w · λ)) ⊆ V_{(U_{I})_{1}}. _{□}

6.2 Jantzen conjecture. As another application of our results, we answer affirmatively and completely the question raised by Jantzen [Jan4, (2.7)(1)]. First, recall the definition of Φ_{λ,p} for r = 1 given in (3.4.2). For any α ∈ Φ, suppose α = ∑ n_{β}β_{i}, where β_{1}, ⋯, ∈ Φ and n_{i} ∈ ℚ. Then d_{α}α^{∨} = ∑ n_{i}d_{i}α_{i}^{∨}. In particular, this implies that if α, β ∈ Φ satisfy α + β ∈ Φ, then d_{α+β}(α + β)^{∨} = d_{α}α^{∨} + d_{β}β^{∨}. It follows that Φ_{λ,p} is a (integrally) closed subroot system of Φ. Moreover, Φ_{λ,p}^{∨} is also a closed subroot system of the dual root system Φ^{∨}.

In addition, if p is good, then Φ_{λ,p} = {α ∈ Φ | (λ + ρ, α^{∨}) ∈ pℤ}. The definition of “good prime” now implies that Φ_{λ,p} = ℚΦ_{λ,p} ∩ Φ, so, by [B, Prop. 24, p. 165], there exists a subset I ⊆ Φ and an element w ∈ W such that Φ_{I} = w(Φ_{λ,p}). (See also [Jan4, (2.7)].)

(6.2.1) Theorem . Let G be a reductive algebraic group and assume that p is good. Let λ ∈ X(T)_{+}. Choose w ∈ W such that w(Φ_{λ,p}) = Φ_{I} for some I ⊆ Π. Then V_{G_{1}}(H^{0}(λ)) = G · 𝔲_{I}.

Proof. By (6.1.3), V_{G_{1}}(H^{0}(λ)) ⊆ G · V_{(U_{I})_{1}}. Under our identifications, V_{(U_{I})_{1}} is identified with the subvariety N_{1}(U_{I}) of N (U_{I}) = 𝔲_{I}. By (3.4.3b) and (1.3.3), c_{G_{1}}(H^{0}(λ)) ≥ |Φ| − |Φ_{I}| = dim G · 𝔲_{I}. The result follows by the irreducibility of G · 𝔲_{I}. _{□}

6.3 Taking λ = 0 in the above theorem gives the following result, V_{G_{1}} is being identified with N_{1}(G). Recall the notation d(Φ, p) = |Φ_{0,p}|.

(6.3.1) Corollary . Assume that p is a good prime for the reductive group G. The variety N_{1}(G) is irreducible of dimension |Φ| − d(Φ, p).

It is unknown (to the authors at least) whether N_{1}(G) is irreducible in general. However, even when p is a bad prime, the methods of this paper do give information on dim N_{1}(G). For example, suppose that G has type G_{2}. In case p = 3, Φ_{0,3} consists of all long roots in Φ.

In this case, we can modify the argument in §3, replacing the cyclotomic polynomial Ψ_{3}(t) by Ψ_{9}(t), to conclude that dim N_{1}(G) ≥ |Φ| − |{α ∈ Φ | (ρ, α^{∨}) ∈ 3ℤ}| = 10. For x, y ∈ 𝔤,

(x + y)^{[3]} = x^{[3]} + [[y, x], x] + [[x, y], y] + y^{[3]}.

For any root α, 𝔤_{α}^{[3]} = 0. Thus, using [St, Satz 3′], it follows that N_{1}(G) is the union of five G-conjugacy classes defined by the elements u ∈ {x_{β} + x_{2α+β}, x_{β} + x_{α+β}, x_{α}, x_{β}, 0}, where α (resp., β) is the simple short (resp., long) root. For such u, it is easy to determine that dim Z_{G}(u) ≥ 4, so we conclude that dim N_{1}(G) = 10; in fact, it must be the closure of the G-orbit of x_{β} + x_{2α+β}.^{5} In case p = 2, we have

(x + y)^{[2]} = x^{[2]} + [x, y] + y^{[2]}.

Using [St, Satz 2′], N_{1}(G) = \overline{G · x_{β}}, where β is a short root. Now dim N_{1}(G) = 8 = |Φ| − d(Φ, 2), and the estimate is exact.

6.4 Now assume that p is good. Since N_{1}(G) is an irreducible variety and since G has only finitely many orbits in N (G), N_{1}(G) contains a dense, open orbit O_{[p]−reg}. Any element x ∈ O_{[p]−reg} is called a [p]-regular element.

(6.4.1) Corollary . Let G be a reductive group and assume p is good. Let λ ∈ X(T)_{+} be such that |Φ_{λ,p}| > d(Φ, p). Then the G-modules H^{0}(λ) and L(λ) are x-projective modules for any x ∈ O_{[p]−reg}.

Proof. Suppose that w(Φ_{λ,p}) = Φ_{I}. Then

dim G · 𝔲_{I} = |Φ| − |Φ_{λ,p}| < dim N_{1}(G).

Thus, if x ∈ O_{[p]−reg}, then x does not belong to V_{G_{1}}(H^{0}(λ)). It follows that H^{0}(λ) is xprojective. If μ ∈ X(T)_{+} satisfies μ = w ·λ for some w ∈ W_{p}, then |Φ_{μ,p}| = |Φ_{λ,p}| > d(Φ, p). Since H^{0}(λ) = L(λ) if λ ∈ C̄_{ℤ}, the evident induction on λ proves that L(λ) is x-projective also. _{□}

When p ≥ h, the condition that |Φ_{λ,p}| > d(Φ, p) amounts to saying that λ is not p-regular. Thus, the result above extends [Jan1, (4.14)(1)].

6.5 It is interesting to observe that the results above can be used to give “homological proofs” of several well known results on nilpotent (and unipotent) classes. We give two examples.

(6.5.1) Example . Assume that p ≥ h. Given I ⊆ Π, let λ ∈ X(T)_{+} satisfy

(λ, α^{∨}) = \begin{cases}p − 1, a ∈ I \\ 0, α ∉ I \end{cases} .

5Although [St, Satz 3′] gives no indication of the orbit closures, they are easy to work out. First, conjugating x_{β} + x_{2α+β} by root group elements u_{−α}(t) and then by a torus element yields, in characteristic 3, that x_{β} + sx_{2α+β} + x_{α+β} ∈ G · (x_{β} + x_{2α+β}) for nonzero s ∈ k. Taking closures gives that x_{β} + x_{α+β} ∈ \overline{G · (x_{β} + x_{2α+β})}. Similarly, using just the torus action, gives that x_{α}, x_{β} ∈ \overline{G · (x_{β} + x_{α+β})}. Also, \overline{G · x_{α}} ∩ \overline{G · x_{β}} = \{0\}. The orbit dimensions in N_{1}(G) are (10, 8, 6, 6, 0).

Then Φ_{λ,p} = Φ_{I} and V_{G_{1}}(H^{0}(λ)) = G · 𝔲_{I}. It is well-known that the nilpotent radical 𝔲_{I} contains a dense P_{I}-orbit O_{I}. If x ∈ O_{I}, then any element y ∈ G · x is called a Richardson element of type P_{I}. Now suppose that P_{J} is another parabolic subgroup of G which is associated to P_{I}, in the sense that the Levi factors L_{I} and L_{J} are conjugate in G. This means there exists y ∈ W such that y(I) = J. It follows that Φ_{λ,p} is W -conjugate to Φ_{J} also. Hence, G · 𝔲_{J} = V_{G_{1}}(H^{0}(λ)) = G · 𝔲_{I}, so that Richardson elements of type P_{I} identify with Richardson elements of type P_{J}. Hence, Richardson elements for any two associated parabolic subgroups are G-conjugate (Johnston-Richardson theorem [JR]). Since our assumption p ≥ h guarantees that p is good, N_{1}(G) is G-equivariantly homeomorphic to the unipotent variety of G [SS, (3.12)], so the corresponding result for unipotent Richardson elements follows also.

(6.5.2) Example . While not difficult to prove directly, the equality (4.5.2) also has a support variety explanation at least when p ≥ n = h. Given J ⊆ Π, choose λ ∈ X(T)_{+} so that w(Φ_{λ,p}) = Φ_{J} for some w ∈ W.^{6} Then [Jan3, pp. 105-106] proves directly that GL_{n}(k) · x_{σ(J)^{t}} ⊆ V_{(GL_{n})_{1}}(H^{0}(λ)). The argument involves showing via a Weyl character argument that, for I ⊆ Π satisfying σ(I) = σ(J)^{t}, H^{0}(λ)|_{L_{I}} has a direct summand with dimension prime to p. An essential step involves the fact that, for Young subgroups W′ and W′′ of W = 𝔖_{n} corresponding to dual partitions, there exists a unique double coset W′wW′′ with trivial intersection property, i. e., W′ ∩ wW′′w^{−1} = ∅—a fact closely related to the isomorphism S_{μ}^{∗} ≅ S_{μ^{t}} ⊗ sign for complex Specht modules. On the other hand, dim GL_{n}(k) · x can be easily calculated for any nilpotent matrix x in Jordan normal form (see [SS, p. 251]), viz., dim GL_{n}(k) · x_{σ(λ)^{t}} = dim GL_{n}(k) · 𝔲_{J}. Now (4.5.2) follows.

6.6 Using (3.5.1) and (5.6.1), we can obtain information on the support varieties when G has type A_{2}, B_{2} or G_{2} and p is good. If I = {α} ⊆ Π, then G · 𝔲_{I} has codimension 2 in the variety N (G) of nilpotent elements in 𝔤. Hence, G · 𝔲_{I} = \overline{O}_{sub-reg}, the closure of the subregular class in N (G). In type A_{2} (resp., B_{2}; G_{2}) the dimensions of the G-orbits in N (G) are 0, 4, 6 (resp., 0, 4, 6, 8; 0, 6, 8, 10, 12). The orbit closures are linearly ordered. (See [Car].)

In the following, let {α, β} be the simple roots for the root system of G. If there are two root lengths, let α be the short root. Given non-negative integers r, s, let [r, s] denote the dominant weight λ satisfying (λ, α^{∨}) = r and (λ, β^{∨}) = s.

(6.6.1) Corollary . Let G be a simple algebraic group of rank 2 with p good.

(a) Let λ ∈ X(T)_{+} and assume that λ − (p − 1)ρ ∉ pX(T). If G has type B_{2} or G_{2}, assume that p ≥ h. Then

\overline{O}_{sub-reg} ⊆ V_{G_{1}}(L(λ)).

If Φ_{λ,p} ≠ ∅ (i. e., λ is singular), this containment is an equality.

(b) If λ is regular, then V_{G_{1}}(L(λ)) = V_{G_{1}}, except possibly when G has type G_{2}, p = 17 (resp., p = 29) and λ lies in the same (restricted) alcove as λ = [10, 14] (resp., λ = [25, 25]).

6Depending on J, the requirement that p ≥ n can be relaxed: the required weight λ will exist precisely when σ(J) has at most p nonzero parts.

(c) In types B_{2} (for p > 3) and G_{2}, there does not exist an irreducible G-module L(λ) such that V_{G_{1}}(L(λ)) is the closure of the unique non-trivial G-orbit of minimal dimension.

Proof. We can assume that λ ∈ X_{1}(T). By (5.6.1a) and (6.1.1), V_{G_{1}}(L(λ)) ⊆ G · 𝔲_{I}, where w(Φ_{λ,p}) = Φ_{I} for some I ⊆ Π and some w ∈ W.

First, suppose that Φ_{λ,p} = ∅. Then p ∤ dim L(μ) for some μ ∈ X_{1}(T) lying in the same alcove as λ, except in the exceptional cases for G_{2} cited in the statement of (b) [JJ, §§4,5]. Thus, assume λ is not one of these exceptional weights. Then using [Jan2, II,§7], (2.2.12) implies that

c_{G_{1}}(L(λ)) = c_{G_{1}T} (L(λ)) = c_{G_{1}T} (L(μ)) = c_{G_{1}}(L(μ)).

Since p ∤ dim L(μ), we find, using the irreducibity of V_{G_{1}}, that V_{G_{1}}(L(μ)) = V_{G_{1}}. This proves (b).

For arbitrary λ ∈ X_{1}(T), p^{2} ∤ dim L(λ), except when p = 17 (resp., p = 29) with λ = [10, 14] (resp., [24, 26] or [25, 25] which lie in the same alcove). In each of these cases, there is a weight μ lying in the same alcove as these regular weights and which satisfies p^{2} ∤ dim L(μ); see [JJ, §§4,5]. Thus, using a translation argument, we can replace λ with μ to assume p^{2} ∤ dim L(λ). Hence, in (3.5.1b), taking M = L(λ), we obtain

|Φ| − d(Φ, p) − c_{G_{1}}(L(λ)) ≤ 2.

In all cases, d(Φ, p) ≤ 2. Thus, in type G_{2}, c_{G_{1}}(L(λ)) ≥ |Φ| − 4 > 6. In type B_{2} with p > 3, d(Φ, p) = 0, so c_{G_{1}}(L(λ)) ≥ |Φ| − 2 > 4. This proves (c). Finally, (a) follows from the same calculation, since d(Φ, p) = 0 for p ≥ h. _{□}

7 Varieties for Z_{r}(λ)

7.1 A_{2}-example. We begin with an example which will provide some motivation and insight for the forthcoming results. For G = GL_{3}(k), we compare the support varieties of Z_{1}(λ) and H^{0}(λ). Write Φ^{+} = {α_{1}, α_{2}, α_{1}+α_{2}}. For each postive root α, let y_{α} = x_{−α} be the negative root vector associated to α. Recall that \overline{G · y_{α_{1}}} = \overline{G · y_{α_{2}}} = \overline{G · y_{α_{1}+α_{2}}} = \overline{O}_{sub-reg}.

(7.1.1) Proposition . Let G = GL_{3}(k), λ ∈ X_{1}(T) with p ≥ h = 3.

(a) If λ is p-regular then V_{G_{1}}(H^{0}(λ)) = N_{1}(G) and V_{G_{1}}(Z_{1}(λ)) = N_{1}(U).

(b) If (λ+ρ, α^{∨}) ∈ pℤ for α = α_{1}, but not for α = α_{2}, α_{1} +α_{2}, then V_{G_{1}}(H^{0}(λ)) = \overline{G · y_{α_{1}}} and V_{G_{1}}(Z_{1}(λ)) = \overline{B · y_{α_{2}}}.

(c) If (λ+ρ, α^{∨}) ∈ pℤ for α = α_{2}, but not for α = α_{1}, α_{1} +α_{2}, then V_{G_{1}}(H^{0}(λ)) = \overline{G · y_{α_{1}}} and V_{G_{1}}(Z_{1}(λ)) = \overline{B · y_{α_{1}}}.

(d) If (λ+ρ, α^{∨}) ∈ pℤ for α = α_{1} +α_{2}, but not for α = α_{1}, α_{2}, then V_{G_{1}}(H^{0}(λ)) = \overline{G · y_{α_{1}}} and V_{G_{1}}(Z_{1}(λ)) = \overline{B · y_{α_{1}}} ∪ \overline{B · y_{α_{2}}}.

(e) If (λ + ρ, α^{∨}) ∈ pℤ for α ∈ Φ^{+}, then V_{G_{1}}(H^{0}(λ)) = {0} = V_{G_{1}}(Z_{1}(λ)).

Proof. The computation of the V_{G_{1}}(H^{0}(λ)) follows from (6.6.1). Next, (a) follows from by (4.1.1); (e) holds because, under the hypotheses, Z_{1}(λ) is a projective G_{1}-module.

The proper B-orbits in N_{1}(U) are given by \{0\}, \overline{B · y_{α_{1}}}, \overline{B · y_{α_{2}}}, and \overline{B · y_{α_{1}+α_{2}}} with \overline{B · y_{α_{1}+α_{2}}} ⊂ \overline{B · y_{α_{j}}} for j = 1, 2. Suppose that (λ + ρ, α^{∨}) ∈ pℤ for α = α_{1}, but not for α = α_{2}, α_{1} + α_{2}. By (4.3.1), y_{α_{1}} ∉ V_{G_{1}}(Z_{1}(λ)) and y_{α_{2}} ∈ V_{G_{1}}(Z_{1}(λ)). Also, by (4.1.1), V_{G_{1}}(Z_{1}(λ)) ≠ N_{1}(U). Hence, V_{G_{1}}(Z_{1}(λ)) = \overline{B · y_{α_{2}}}, so (b) holds. Now (c) follows in a similar manner. The hypotheses of (d) imply that \overline{B · y_{α_{1}}} ∪ \overline{B · y_{α_{2}}} ⊆ V_{G_{1}}(Z_{1}(λ)). Once again by (4.1.1), V_{G_{1}}(Z_{1}(λ)) ≠ N_{1}(U), thus \overline{B · y_{α_{1}}} ∪ \overline{B · y_{α_{2}}} = V_{G_{1}}(Z_{1}(λ)). _{□}

7.2 The results from the preceding section indicate that number of irreducible components of V_{G_{r}}(Z_{r}(λ)) can vary depending on the position of λ. Let σ be the involutory graph automorphism of G such that σ|_{T} induces the opposition involution −w_{0} on X(T) where w_{0} ∈ W is the long word. We can assume that σ is defined over 𝔽_{p}, so it induces an automorphism on G_{r}, B_{r}, V_{G_{r}}, etc. The following result indicates how σ permutes the support varieties of the Z_{r}(λ).

(7.2.1) Proposition . If λ ∈ X(T) then V_{G_{r}}(Z_{r}(λ)) = σ · V_{G_{r}}(Z_{r}(w_{0} · λ)).

Proof. For a G_{r}-module M, let M^{σ} be the G_{r}-module with action twisted by σ. Then Z_{r}(λ)^{σ} ≅ Z_{r}(−w_{0}λ). We have Z_{r}(λ)^{∗} ≅ ind_{B_{r}}^{G_{r}}(−λ + 2(p^{r} − 1)ρ); cf. [Jan2, II, (3.5)]. Consequently,

(Z_{r}(λ)^{σ})^{∗} ≅ Z_{r}(−w_{0}λ)^{∗}
≅ Z_{r}(w_{0}λ + 2(p^{r} − 1)ρ)
≅ Z_{r}(w_{0} · λ) ⊗ 2p^{r}ρ,

since w_{0}(ρ) = −ρ. Since V_{G_{r}}(M) = V_{G_{r}}(M^{∗}) for any finite dimensional G_{r}-module, we have (using [NP, (2.1)] for the last equality):

V_{G_{r}}(Z_{r}(w_{0} · λ)) = V_{G_{r}}(Z_{r}(λ)^{σ}) = σ^{−1} · V_{G_{r}}(Z_{r}(λ)).

This completes the proof. _{□}

7.3 We conclude this paper with the following result when r = 1.

(7.3.1) Theorem . Let G be a reductive algebraic group and let λ ∈ X(T)_{+}. Assume that p is good. Then

V_{G_{1}}(H^{0}(λ)) = G · V_{G_{1}}(Z_{1}(w · λ))

for all w ∈ W. Equivalently

G · V_{G_{1}}(Z_{1}(λ)) = G · V_{G_{1}}(Z_{1}(w · λ))

for all w ∈ W.

Proof. According to (6.2.1), we have V_{G_{1}}(H^{0}(λ)) = V_{G_{1}}(H^{0}(μ)) for all μ ∈ Γ_{1}(λ) ∩ X(T)_{+}. Therefore, by (4.6.1),

V_{G_{1}}(H^{0}(λ)) = G · [∪_{μ∈Γ_{1}(μ)}V_{G_{1}}(Z_{r}(μ))] = G · [∪_{w∈W} V_{G_{1}}(Z_{1}(w · λ))].

It follows that G · V_{G_{1}}(Z_{1}(w · λ)) ⊆ V_{G_{1}}(H^{0}(λ)) for all w ∈ W. The other inclusion follows by (5.6.1). _{□}

One should observe that even though G · V_{G_{1}}(Z_{1}(μ_{1})) = G · V_{G_{1}}(Z_{1}(μ_{2})) for all μ_{1}, μ_{2} ∈ Γ_{1}(λ), one may have V_{G_{1}}(Z_{1}(μ_{1})) ≠ V_{G_{1}}(Z_{1}(μ_{2})) for some μ_{1}, μ_{2} ∈ Γ_{1}(λ) from the example given in (7.1.1).

Acknowledgements. The authors would like to express their gratitude to the Department of Mathematics at the University of Virginia, the Department of Mathematics and Statistics at Utah State University and the Department of Mathematics and Computer Science at Skidmore College for their hospitality and financial support. We would also like to thank Chris Bendel for several useful discussions about support varieties for higher Frobenius kernels.

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Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322-3900 U.S.A.

   E-mail address: nakano@sunfs.math.usu.edu

Department of Mathematics, University of Virginia, Charlottesville, VA 22903-3199 U.S.A

   E-mail address: bjp8w@fermi.clas.virginia.edu

Department of Mathematics and Computer Science, Skidmore College, Saratoga Springs, NY 12866 U.S.A.

   E-mail address: dvella@skidmore.edu