SUPPORT VARIETIES FOR WEYL MODULES OVER BAD PRIMES

UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP^{1}

1. Introduction

1.1. Support varieties were introduced in the pioneering work of Alperin [A] and Carlson [Ca1, Ca2] nearly 25 years ago as a method to study complexes and resolutions of modules over group algebras. Since that time these ideas have been extended to encompass restricted Lie algebras by Friedlander and Parshall [FP], finite-dimensional sub Hopf algebras of the Steenrod algebra by Nakano and Palmieri [NPal], infinitesimal group schemes by Suslin, Friedlander and Bendel [SFB1, SFB2], and arbitrary finite-dimensional cocommutative Hopf algebras by Friedlander and Pevtsova [FPe]. Further recent attempts to generalize the theory have been made to finite-dimensional algebras by Solberg and Snashall [SS] via Hochschild cohomology and to Hecke algebras by Erdmann and Holloway [EH].

For a finite-dimensional cocommutative Hopf algebra A over an algebraically closed field k the cohomology ring H^{•}(A, k) is a graded commutative finitely generated algebra [FS]. For any finitely generated A-module M, one can assign a conical subvariety V_{A}(M) inside the spectrum of the cohomology ring. These support varieties provide a method to introduce the geometry of the spectrum of H^{•}(A, k) into the representation theory of A.

The geometric implications become evident when one considers a reductive algebraic group G over an algebraically closed field k of characteristic p > 0 and the rth Frobenius kernels G_{r}, r ≥ 1. It is well-known that representations for G_{1} are equivalent to modules for the restricted enveloping algebra A := u(𝔤) where 𝔤 = Lie G. In this situation the spectrum of the cohomology ring is homeomorphic to the restricted nullcone [SFB1, SFB2]

N_{1}(𝔤) = {x ∈ 𝔤 | x^{[p]} = 0}.

Moreover, N_{1}(𝔤) is a G-stable conical subvariety inside the cone of nilpotent elements (nullcone) N (𝔤). The nullcone N (𝔤) has been well studied (see [Car, CM, Hum2]) because of its beautiful geometric properties with deep connections to Lie and representation theory.

1.2. Although support varieties are easy to define once the finite generation of cohomology is established, they are often very difficult to compute. In 1987, Jantzen [Jan2] stated a conjecture for reductive groups that the support varieties of the Weyl modules V (λ) over G_{1} are closures of certain Richardson orbits (depending on λ) when the characteristic of the underlying field is good. Nakano, Parshall and Vella [NPV, Thm. 6.2.1] proved this conjecture. The verification of this conjecture provides a bridge linking the representation theory, cohomology theory and conjugacy class theory of 𝔤. As an immediate corollary, it is shown that the restricted nullcone is an irreducible variety when k is of good characteristic. In [CLNP], Carlson, Lin, Nakano and Parshall, using techniques in [NPV], calculated N_{1}(𝔤) for good primes. This computation was replicated by the authors in [UGA1] using more elementary methods. The authors in [UGA2] determined the restricted nullcone over fields of bad characteristic. It was shown that N_{1}(𝔤) is always irreducible but need not be the closure of a Richardson orbit when p is bad. Calculating the support varieties of Weyl modules is a fundamental result in the theory. The calculation for good primes has advanced our understanding for both restricted and non-restricted representations of 𝔤 (see [GP, BN, CLN, CLNP, NT]).

1The members of the UGA VIGRE Algebra Group are David J. Benson, Philip Bergonio, Brian D. Boe, Leonard Chastkofsky, Bobbe Cooper, Jeremiah Hower, Jo Jang Hyun, Jonathan Kujawa, Nadia Mazza, Daniel K. Nakano, Kenyon J. Platt and Caroline Wright.

For the sake of consistency with [NPV], we will compute the support varieties of the induced modules H^{0}(λ) := ind_{B}^{G}λ. Since V (λ) is the contragredient dual of H^{0}(−w_{0}λ) it follows that the support variety of V (λ) is the same as the support variety of H^{0}(−w_{0}λ). In this paper we will compute the support varieties of induced/Weyl modules over fields of bad characteristic.

There are several differences between the good characteristic and bad characteristic settings that should be emphasized. First, unlike the good characteristic situation the orbits/conjugacy classes in bad characteristic need not coincide with the classes in characteristic zero—interest in the unipotent conjugacy classes over fields of small characteristic has recently been revived by Lusztig [Lu2]. Second, the stabilizer subset Φ_{λ} (defined in Section 1.4) need not be conjugate under the action of the Weyl group W to a subroot system generated by simple roots. We manage to deal with these issues by introducing a new technique involving “constrictors” of orbits (Section 2.7) which works remarkably well with the closure ordering of orbits in bad characteristic. We can also handle the cases when the support varieties are not closures of Richardson orbits.

Our results in conjunction with [NPV, Thm. 6.2.1] give a complete description of the support varieties of induced/Weyl modules over algebraically closed fields of arbitrary positive characteristic.

1.3. Notation. The notation and conventions of this paper will follow those given in [Jan3]. Let k be an algebraically closed field of characteristic p > 0, and G a simple algebraic group defined over k with T a maximal torus of G. The root system associated to the pair (G, T) is denoted by Φ and identified with a subset of the set of weights X(T). Let Φ^{+} be a set of positive roots and Φ^{−} be the corresponding set of negative roots. The set of simple roots determined by Φ^{+} is Δ = {α_{1}, …, α_{l}}. We will use throughout this paper the ordering of simple roots given in [Hum1] following Bourbaki. Let B be the Borel subgroup relative to (G, T) given by the set of negative roots and let U be the unipotent radical of B. More generally, if J ⊆ Δ, let P_{J} be the parabolic subgroup relative to −J and let U_{J} be the unipotent radical of P_{J}. Let Φ_{J} be the root subsystem in Φ generated by the simple roots in J. Set 𝔤 = Lie G, 𝔟 = Lie B, 𝔲 = Lie U, 𝔭_{J} = Lie P_{J}, and 𝔲_{J} = Lie U_{J}.

Let 𝔼 be the Euclidean space associated with Φ, and the inner product on 𝔼 will be denoted by ⟨ , ⟩. Let α̌ be the coroot corresponding to α ∈ Φ. Set α_{0} to be the highest short root. Moreover, let ρ be the half sum of positive roots. The Coxeter number associated to Φ is h = ⟨ρ, α̌_{0}⟩ + 1.

We denote the set of positive (resp. nonnegative) integers by ℕ (resp. ℤ_{≥0}). Let X(T) be the integral weight lattice spanned by the fundamental weights {ω_{1}, …, ω_{l}}. The set X(T) has a partial ordering defined as follows: λ ≥ μ if and only if λ − μ ∈ ∑_{α∈Δ} ℤ_{≥0} α. The set of dominant integral weights is denoted by X(T)_{+} and the set of p^{r}-restricted weights is X_{r}(T). Set H^{0}(λ) = ind_{B}^{G}λ where λ is the one-dimensional B-module obtained from the character λ ∈ X(T)_{+} by letting U act trivially. The Weyl group corresponding to Φ is W and acts on X(T) via the dot action (w · λ = w(λ + ρ) − ρ where w ∈ W, λ ∈ X(T)).

In this paper we will consider the case when p is a bad prime for Φ. A list of bad primes is provided below.

• Φ of type A_{l}, no primes;

• Φ of type B_{l}, C_{l}, D_{l}, p = 2;

• Φ of type E_{6}, E_{7}, F_{4}, G_{2}, p = 2, 3;

• Φ of type E_{8}, p = 2, 3, 5. If the prime p does not appear on this list then p is a good prime relative to Φ.

Let N (𝔤) be the irreducible variety (of dimension |Φ|) of nilpotent elements of 𝔤, which is often called the nullcone. The group G acts on N (𝔤) via the adjoint representation, and N (𝔤) has finitely many G-orbits [Lu1]. For good primes the nilpotent orbits are classified in exactly the same way as over the complex numbers. On the other hand for bad primes the nilpotent orbits are determined in [He, Sp1, Sp2, Sp3, HS]. The conventions in [Law1, Law2] will be employed for the exceptional groups. If X is the Bala-Carter label for an orbit in characteristic zero which “splits” in characteristic p, then X^{(p)} will denote the new orbit that arises.

Now view G as an algebraic group scheme defined over 𝔽_{p} and let F : G → G be the Frobenius morphism. Set G_{r} = Ker F^{r} where F^{r} is the Frobenius map iterated with itself r times. Let F |_{B} : B → B be the restriction of F to B, let B_{r} = Ker (F |_{B})^{r} and let B_{r}T be the inverse image of T under (F |_{B})^{r}. Set

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

According to [FS], R is a finitely generated commutative k-algebra. Furthermore, if M is a finite-dimensional module for G_{r} then Ext_{G_{r}}^{•}(M, M) is a finitely generated module over R. Let J_{r}(M) denote the set of elements in R which annihilate Ext_{G_{r}}^{•}(M, M). Define V_{G_{r}}(M) = Maxspec(R/J_{r}(M)). The affine homogeneous variety V_{G_{r}}(M) is called the support variety of M. Similarly, if M is a B_{r}-module one can define V_{B_{r}}(M). We will mainly be interested in the case when r = 1, but will state results for arbitrary r when appropriate.

1.4. Motivation. For λ ∈ X(T), define

Φ_{λ} = {α ∈ Φ | ⟨λ + ρ, α̌⟩ ∈ pℤ}.

Then Φ_{λ} is either empty or a root subsystem of Φ, and when p is a good prime, there exist w ∈ W and I ⊆ Δ such that w(Φ_{λ}) = Φ_{I}. The following result was proved by Nakano, Parshall and Vella for good primes [NPV, Thm. 6.2.1, Cor. 6.2.2].

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}) = Φ_{I} for some I ⊆ Δ. Then

(a) V_{G_{1}}(H^{0}(λ)) = G · 𝔲_{I}.

(b) dim V_{G_{1}}(H^{0}(λ)) = |Φ| − |Φ_{λ}|.

The main issue relevant for bad primes is the fact that Φ_{λ} need not be conjugate to a root subsystem Φ_{I}. This makes formulating a precise conjecture on the support varieties of induced modules in this case somewhat intractable. Nevertheless, one might conjecture that

(1.4.1)  dim V_{G_{1}}(H^{0}(λ)) = |Φ| − |Φ_{λ}|.

for all primes. Indeed this is exactly what we will prove. The first step will be to prove the lower bound

(1.4.2)  dim V_{G_{1}}(H^{0}(λ)) ≥ |Φ| − |Φ_{λ}|

for all primes. The next step is to prove a general version of the upper bound [NPV, Cor. 4.5.1]. These results along with our information about the restricted nullcone for bad primes will allow us to compute the support varieties of the induced modules while simultaneously verifying the dimension equality (1.4.1).

Note that our definition of Φ_{λ} differs from the stabilizer given in [NPV, §3.4]. For good primes, these two definitions agree, but for bad primes this will lead us to adapt several graded dimension results in [NPV, §3] to our setting.

2. General results

2.1. Restricted Nullcones. In [UGA2] the authors computed the restricted nullcone N_{1}(𝔤) over fields of bad characteristic. This result will be used throughout this paper because the support variety of any G_{1}-module is contained in N_{1}(𝔤).

Theorem (A). Let G be a simple classical connected algebraic group over k with p = 2 a bad prime.

(i) If Φ is of type B_{l} then N_{1}(𝔤) = \overline{O(2_{2}^{l}, 1_{1})}.

(ii) If Φ is of type C_{l} then N_{1}(𝔤) = \overline{O(2_{1}^{l})}.

(iii) If Φ is of type D_{l} then N_{1}(𝔤) = \overline{O(2_{2}^{l})}.

Theorem (B). Let G be an exceptional algebraic group with p a bad prime.

(i) If Φ is of type E_{6} then N_{1}(𝔤) = \overline{O(X)} where X = 2A_{2} + A_{1} (p = 3), 3A_{1} (p = 2). (ii) If Φ is of type E_{7} then N_{1}(𝔤) = \overline{O(X)} where X = 2A_{2} + A_{1} (p = 3), 4A_{1} (p = 2). (iii) If Φ is of type E_{8} then N_{1}(𝔤) = \overline{O(X)} where X = A_{4} + A_{3} (p = 5), 2A_{2} + 2A_{1}

(p = 3), 4A_{1} (p = 2). (iv) If Φ is of type F_{4} then N_{1}(𝔤) = \overline{O(X)} where X = A_{1} + Ã_{2} (p = 3), A_{1} + Ã_{1} (p = 2).

(v) If Φ is of type G_{2} then N_{1}(𝔤) = \overline{O(X)} where X = G_{2}(a_{1}) (p = 3), Ã_{1} (p = 2).

The inclusion relations among orbit closures in N_{1}(𝔤) are provided in Section 6.

2.2. Poles and rates of growth. Our first objective is to prove the inequality in (1.4.2). We will adapt the methods used in [NPV, §3] in order to accomplish this. Several of the arguments are also provided here for the sake of completeness and to assist the reader. We should also mention that our argument was inspired by Ostrik [Ost] who proved the inequality in the quantum setting when p > h.

Recall that if {a_{n}}_{n≥0} is a sequence of complex numbers then the rate of growth r(a_{n}) of this sequence is the smallest non-negative integer d such that there exists a positive number C with the property that

|a_{n}| ≤ C · n^{d−1}

for all n > 0. By convention, if no such d exists, then r(a_{n}) = ∞.

If there is a polynomial f(t) of degree d − 1 such that a_{n} = f(n) for sufficiently large n then r(a_{n}) = d. Furthermore, if S_{n} = |a_{0}| + ⋯ + |a_{n}|, then r(S_{n}) = d + 1. We state a useful proposition from [NPV] which relates the rate of growth of {a_{n}}_{n≥0} with the poles of the corresponding Poincaré series.

Proposition. Let p(t) = ∑_{n=0}^{∞} a_{n}t^{n} ∈ ℂ[[t]].

(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 a_{n} is a polynomial in n of degree d − 1. Hence, r(a_{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(a_{n}).

(c) Assume p(t) = \frac{f(t)}{(1−t^{r})^{b}} for positive integers b, r and f(t) ∈ ℂ[t], with f(1) ≠ 0. Fix i, 0 ≤ i < r. For j sufficiently large, a_{i+rj} is a polynomial in j (say of degree d_{i} − 1).

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

2.3. Fix a weight Λ and suppose that M ∈ mod(B_{r}T) (the set of finite-dimensional B_{r}T -modules) and all weights of M satisfy μ ≤ Λ. That is, all weights of M lie in the cone defined by Λ. If μ ≤ Λ then 0 ≤ Λ − μ = ∑_{α∈Δ} n_{α}α where n_{α} ∈ ℤ_{≥0} for α ∈ Δ. Set ht(Λ − μ) = ∑_{α∈Δ} n_{α}. We define the graded dimension of M as

(2.3.1)  dim_{t} M = ∑_{μ≤Λ}(dim M_{μ})t^{ht(Λ−μ)}

where M_{μ} is the μ weight space with respect to T. This definition coincides with the principal grading given in [K, §10.8]. According to [K, Prop. 10.10],

(2.3.2)  dim_{t} H^{0}(Λ) = ∏_{α∈Φ^{+}} \frac{1 − t^{⟨Λ+ρ,α̌⟩}}{1 − t^{⟨ρ,α̌⟩}} .

Set

(2.3.3)  h_{r}(t) = ∏_{α∈Φ^{+}} \frac{1 − t^{p^{r}⟨ρ,α̌⟩}}{1 − t^{⟨ρ,α̌⟩}} .

Let P_{r}(μ) be the projective cover of the simple B_{r}T -module L_{r}(μ) of highest weight μ ∈ X(T). Let M ∈ mod(B_{r}T) and let P_{•} be a minimal projective resolution of M in mod(B_{r}T). Upon restriction to B_{r}, this still provides a minimal projective resolution of M as a B_{r}-module. Since P_{•} is a minimal resolution we have

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

as T -modules. Note that dim_{t} P_{r}(Λ) = h_{r}(t) and

(2.3.5)  dim_{t} P_{r}(μ) = t^{ht(Λ−μ)}h_{r}(t)

for μ ≤ Λ, because P_{r}(μ) ≅ coind_{T}^{B_{r}T} μ (cf. [NPV, (3.1.6)]).

Let P_{r}(σ + p^{r}ν) be a B_{r}T -direct summand of P_{n} and set τ_{n} = σ + p^{r}ν (with σ ∈ X_{r}(T) and ν ∈ X(T)). Since P_{•} is a minimal resolution it follows that τ_{n} is a weight in the radical of P_{n−1}. Consequently, τ_{n−1} := τ_{n} + β appears in the head of P_{n−1}, where β is some nontrivial sum of positive roots. Therefore, Λ − τ_{n} = Λ − τ_{n−1} + β. Applying heights we see that ht(Λ − τ_{n}) ≥ ht(Λ − τ_{n−1}) + 1 and iterating this process n times yields

(2.3.6)  ht(Λ − τ_{n}) ≥ ht(Λ − τ) + n

where τ is a weight in the head of M.

Now assuming that all weights μ of M satisfy μ ≤ Λ, then the smallest possible height ht(Λ − τ), where τ is a weight of M, is zero. Hence, in this case ht(Λ − τ_{n}) ≥ n.

2.4. For a positive integer d let Ψ_{d}(t) ∈ ℤ[t] be the dth cyclotomic polynomial in ℚ[t]. The polynomials Ψ_{d}(t) are irreducible over ℚ. The following theorem extends [NPV, Thm. 3.3.1] which shows that dim V_{B_{r}}(M) is related to the order of the poles of the rational function (dim_{t} M)/h_{r}(t).

Theorem. Let M be a finite-dimensional B_{r}T module with all weights of M being less than or equal to some Λ ∈ X(T)_{+}. Let

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

where f(t), g(t) ∈ ℚ[t] and Ψ_{p^{r}}(t) ∤ f(t)g(t). Then γ ≤ dim V_{B_{r}}(M).

Proof. Under our assumptions one can assume γ ≥ 0 and γ is the order of the pole of any primitive p^{r}th root of unity in q(t). Let P_{•} be a minimal projective resolution of M in the category of B_{r}T modules. Now express

\frac{dim_{t} P_{n}}{h_{r}(t)} = ∑_{m} b(m, n)t^{m},

where each b(m, n) is a non-negative integer.

Since dim V_{B_{r}}(M) is the rate of growth of {Ext_{B_{r}}^{n}(M, N)} where N = ⨁_{μ∈X_{r}(T)} μ is the direct sum of simple B_{r}-modules (cf. [Ben]), it follows from (2.3.4) and (2.3.5) that for some C > 0,

(2.4.1)  ∑_{m} b(m, n) = dim Ext_{B_{r}}^{n}(M, N) ≤ Cn^{dim V_{Br} (M)−1}.

Suppose that b(m, n) ≠ 0. Then for some σ ∈ X_{r}(T) and ν ∈ X(T), P_{r}(σ + p^{r}ν) appears as a direct summand of P_{n} with m = ht(Λ − (σ + p^{r}ν)). From (2.3.6) one can conclude that if b(m, n) ≠ 0 then m ≥ n. Hence,

q(t) = ∑_{n=0}^{∞}(−1)^{n} \frac{dim_{t} P_{n}}{h_{r}(t)} .

Now we are in the position to follow the line of reasoning given in [NPV, Thm. 3.3.1] directly. We include the argument for the convenience of the reader.

Expand q(t) = ∑_{i=0}^{∞} a_{i}t^{i} into a power series. From (2.4.1), it follows that for i > 0,

(2.4.2)  \begin{aligned}S_{i} = |a_{0}| + ⋯ + |a_{i}| & ≤ ∑_{m=0}^{i} ∑_{n=0}^{∞} b(m, n) = ∑_{m=0}^{i} ∑_{n=0}^{m} b(m, n) \\ & ≤ ∑_{n=0}^{i} Cn^{dim V_{Br} (M)−1} ≤ Di^{dim V_{Br} (M)} \end{aligned}

for some D > 0.

From the definition of h_{r}(t) one can deduce that the poles of q(t) are roots of unity. Let b be the least common multiple of the orders of these poles. Then q(t) = \frac{m(t)}{(1−t^{b})^{c}} for some positive integer c and m(t) ∈ ℂ[t]. According to Proposition 2.2(c), q(t) = q_{0}(t) + ⋯ + q_{a−1}(t), with q_{i}(t) = ∑_{j} b_{ij}t^{i+jb} with the property that b_{ij} is a polynomial in j of degree d_{i} − 1. Set d = max d_{i}. Then by Proposition 2.2(d), r(S_{i}) = d + 1 and r(a_{i}) = d. Thus, by (2.4.2), r(a_{i}) ≤ dim V_{B_{r}}(M). The statement of the theorem now follows by Proposition 2.2(b). _{□}

2.5. The following corollary establishes the inequality described in (1.4.2), which improves the lower bound given in [NPV, Cor. 3.4.3].

Corollary. Let λ ∈ X(T)_{+}. Then dim V_{G_{1}}(H^{0}(λ)) ≥ |Φ| − |Φ_{λ}|.

Proof. Recall that for any positive integer n, t^{n} − 1 = ∏_{d|n} Ψ_{d}(t). Set Λ = λ and M := H^{0}(λ). Consider

(2.5.1)  q(t) = \frac{dim_{t} M}{h_{1}(t)} = ∏_{α∈Φ^{+}} \frac{t^{⟨λ+ρ,α̌⟩} − 1}{t^{p⟨ρ,α̌⟩} − 1}

Observe that Ψ_{p}(t) divides the numerator exactly |Φ_{λ}^{+}| times and Ψ_{p}(t) divides the denominator exactly |Φ^{+}| times. Therefore, by Theorem 2.4 we have

(2.5.2)  dim V_{B_{1}}(H^{0}(λ)) ≥ |Φ^{+}| − |Φ_{λ}^{+}|.

According to [LN], if M is a rational G-module then dim V_{G_{1}}(M) = 2 dim V_{B_{1}}(M). Hence,

dim V_{G_{1}}(H^{0}(λ)) = 2 dim V_{B_{1}}(H^{0}(λ))
≥ 2 (|Φ^{+}| − |Φ_{λ}^{+}|)
= |Φ| − |Φ_{λ}|.

□

2.6. An upper bound. In this section we provide an effective method for forcing the inclusion of the support variety of an induced module into the closure of certain nilpotent orbits; that is, an upper bound on the support variety. The technique involves checking a combinatorial condition which allows one to deduce that the module is projective over regular elements for Levi subalgebras.

For J ⊆ Δ, let x_{J} = ∑_{α∈J} x_{α}, where x_{α} is a root vector in the root space 𝔤_{α}. The theorem below was proved under the assumption that p is a good prime (see [NPV, Thm. 4.3.1]). The statement of this theorem still remains valid over arbitrary primes because if the regular element x_{Δ} is contained in N_{1}(𝔤) then p ≥ h by [CLNP, UGA2] (the determination of the restricted nullcone). This implies that the B-orbit of x_{Δ} is dense in N_{1}(𝔟). The arguments provided in [NPV, 4.4, 4.5] can then be used to prove the following statement. Theorem. Let J ⊆ Δ and λ ∈ X(T)_{+}. If w(Φ_{λ}) ∩ Φ_{J} ≠ ∅ for all w ∈ W, then x_{J} ∉ V_{G_{1}}(H^{0}(λ)). 2.7. Constrictors. Let O be an orbit in N_{1}(𝔤). The constrictors of O are the orbits contained in N_{1}(𝔤) − \overline{O} which are minimal with respect to the closure ordering of orbits in N (𝔤). The following result will be used in the next two sections to compute the support varieties of the induced modules. We will utilize Theorem 2.6 in addition to information about the constrictors of orbits. Theorem. Let O be an orbit in N_{1}(𝔤) and {O_{1}, O_{2}, …, O_{s}} be the set of constrictors of O. Let λ ∈ X(T)_{+} and assume that the following conditions are satisfied:

(i) |Φ| − |Φ_{λ}| ≥ dim O; (ii) for i = 1, 2, …, s, O_{i} = G · x_{J_{i}} for some J_{i} ⊆ Δ; (iii) for i = 1, 2, …, s, w(Φ_{λ}) ∩ Φ_{J_{i}} ≠ ∅ for all w ∈ W.

Then V_{G_{1}}(H^{0}(λ)) = \overline{O}. Proof. Conditions (ii) and (iii) along with Theorem 2.6 show that x_{J_{i}} ∉ V_{G_{1}}(H^{0}(λ)) for i = 1, 2, …, s. In particular V_{G_{1}}(H^{0}(λ)) ∩ O_{j} = ∅ for j = 1, 2, …, s (by G-invariance). If O′ is an orbit in N_{1}(𝔤) − \overline{O} and V_{G_{1}}(H^{0}(λ)) ∩ O′ ≠ ∅ then \overline{O′} ⊆ V_{G_{1}}(H^{0}(λ)). But, this is impossible since every orbit closure in N_{1}(𝔤) − \overline{O} must contain a constrictor, hence V_{G_{1}}(H^{0}(λ)) ∩ O′ = ∅. This shows that V_{G_{1}}(H^{0}(λ)) ⊆ \overline{O}.

Since \overline{O} is an irreducible variety and dim V_{G_{1}}(H^{0}(λ)) ≥ dim O by (i) and Corollary 2.5 one can deduce that V_{G_{1}}(H^{0}(λ)) = \overline{O}. _{□}

2.8. Strategy. Suppose that λ ∈ X(T)_{+}. We first observe that Φ_{λ} = Φ_{λ+pν} for any ν ∈ X(T) and Φ_{w·λ} = w^{−1}(Φ_{λ}). This implies that if the conditions of Theorem 2.7 hold for λ then they also hold for any w ·λ+pν ∈ X(T)_{+} where w ∈ W and ν ∈ X(T). Therefore, it suffices to show that the conditions of Theorem 2.7 hold for representatives in X(T)_{+} under the dot action of the extended affine Weyl group. These representatives can be chosen in X_{1}(T).

Given such a representative the first step in our computation is to calculate |Φ| − |Φ_{λ}| and find all orbits in N_{1}(𝔤) of that dimension. Many times there will be only one such orbit. Certainly the orbit(s) that we consider will satisfy (i) of Theorem 2.7. For each orbit we look at the constrictors (as defined in Section 2.7). It turns out that all such constrictors are of the form G · x_{J} for some J ⊆ Δ (i.e., an orbit representative is given by a regular Levi element), so condition (ii) is satisfied. For the set of constrictors of a given orbit of dimension |Φ|−|Φ_{λ}| we check (using computer calculations in the exceptional cases) whether condition (iii) holds in Theorem 2.7. It turns out that all three conditions of Theorem 2.7 hold for a unique orbit O and this allows us to deduce that V_{G_{1}}(H^{0}(λ)) = \overline{O}. In particular, V_{G_{1}}(H^{0}(λ)) is uniquely determined by Φ_{λ}. Once this is verified for all such representatives, we have by the above remarks,

V_{G_{1}}(H^{0}(μ)) = V_{G_{1}}(H^{0}(w · μ + pν))

where μ ∈ X(T)_{+}, w ∈ W, ν ∈ X(T) and w · μ + pν ∈ X(T)_{+}.

3. Classical Lie Algebras

In this section we compute the support varieties of the induced modules H^{0}(λ) for the classical groups in bad characteristic (i.e., types B, C, D with p = 2). We follow the strategy of Section 2.8. We first determine orbit representatives x_{J} for the constrictor orbits which will arise later. We next show that every orbit of weights under the extended affine Weyl group has a fundamental dominant weight ω_{k} (or 0) as a representative. This significantly simplifies the remaining computations. We use this information to compute the lower bound |Φ| − |Φ_{λ}| of Corollary 2.5, and identify a candidate orbit of that dimension, together with its associated constrictor orbits. Finally we verify condition (iii) of Theorem 2.7.

Throughout this section we let N = 2l + 1 (resp. 2l, 2l) in type B_{l} (resp. C_{l}, D_{l}).

3.1. Nilpotent orbit representatives. The nilpotent orbits in 𝔤 have been classified by Hesselink [He]. They are parametrized by pairs consisting of a partition μ of N and an index function χ: I → ℤ, where I is the set of (non-zero) parts of μ. The partition gives, as usual, the sizes of the Jordan blocks for any representative of the orbit.

There is a one-to-one correspondence between nilpotent orbits and pairs μ, χ satisfying the following conditions. Write n(m) for the multiplicity of m ∈ I as a part in μ. (3.1.1) (1) For m > k in I, χ(m) ≥ χ(k) and m − χ(m) ≥ k − χ(k).

(2) For G = Sp(N) and m ∈ I:

(a) 0 ≤ χ(m) ≤ m/2;
(b) χ(m) = m/2 if n(m) is odd.

(3) For G = O(N) and m ∈ I:

(a) m/2 ≤ χ(m) ≤ m;
(b) χ(m) = m if n(m) is odd;
(c) { m ∈ I | n(m) is odd } = {i, i − 1} ∩ ℕ for some i ∈ ℤ .

If I = {m_{1} > m_{2} > … }, Hesselink displays the pair μ, χ as a symbol

μ_{χ} = (m_{1}_{χ(m_{1})}^{n(m_{1})}, m_{2}_{χ(m_{2})}^{n(m_{2})}, …) .

Write O(μ_{χ}) for the corresponding nilpotent orbit.

Proposition. Let G = O(N) or Sp(N) over a field k of characteristic 2, and let 𝔤 = Lie(G) have rank l. The table lists certain regular nilpotent elements for Levi subalgebras of 𝔤 having type A_{1} × ⋯ × A_{1}, and the nilpotent orbits to which they belong.

Type Representative Orbit
B_{l} x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} (2r − 1 < l) O(2_{1}^{2r}, 1_{1}^{N−4r})
B_{l}  x_{α_{l}} + x_{α_{l−2}} + ⋯ + x_{α_{l−2r}} (r ≥ 0) O(2_{2}^{2r+1}, 1_{1}^{N−4r−2})
C_{l} x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} (2r − 1 < l) O(2_{0}^{2r}, 1_{0}^{N−4r})
C_{l}  x_{α_{l}} + x_{α_{l−2}} + ⋯ + x_{α_{l−2r}} (r ≥ 0) O(2_{1}^{2r+1}, 1_{0}^{N−4r−2})
D_{l} x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} (2r − 1 < l) O(2_{1}^{2r}, 1_{1}^{N−4r})
D_{l} x_{α_{l−1}} + x_{α_{l}} O(2_{2}^{2}, 1_{1}^{N−4})

In type D_{l} when l is even, x_{α_{1}} +x_{α_{3}} +⋯+x_{α_{l−3}} +x_{α_{l−1}} and x_{α_{1}} +x_{α_{3}} +⋯+x_{α_{l−3}} +x_{α_{l}} are both representatives for the O(2l)-orbit with symbol (2_{1}^{l}), which lie in distinct SO(2l)-orbits. (Here SO(2l) is by definition the component of the identity in O(2l).)

Proof. We first treat the case G = Sp(N) with N = 2l. Following Hesselink, let V be an Ndimensional vector space over k equipped with a non-degenerate bilinear form β satisfying β(v, v) = 0 for all v ∈ V. Then G is the group of automorphisms of V leaving β invariant. Fix x ∈ 𝔤 with x nilpotent. For each i ∈ ℕ ∪ {0} define a quadratic form q_{i}: V → k by q_{i}(v) = β(x^{i+1}v, x^{i}v). An extension to ℕ of the index function given at the beginning of Section 3.1 is defined by

(3.1.2)  χ(m) = min{ i ≥ 0 : q_{i}|_{Ker(x^{m})} = 0 }.

In particular, χ(1) = 0, and if x^{2} = 0 then q_{1} = 0, and

(3.1.3)  χ(2) = \begin{cases}0 & if q_{0} = 0; \\ 1 & otherwise. \end{cases}

We take the matrix of β to be

K = (\begin{matrix}& & & & & 1 \\ & & & & 1 \\ & . & . & . \\ 1 \end{matrix})

so that x ∈ 𝔤 ⟺ x^{t}K = Kx ⟺ Kx^{t}K = x. The matrix x′ := Kx^{t}K is obtained from x by transposing across the antidiagonal, so x ∈ 𝔤 if and only if x is symmetric about the antidiagonal.

We can take a Cartan subalgebra to be the diagonal matrices in 𝔤. Then the simple root vectors can be chosen to be

x_{α_{i}} = \begin{cases}E_{i,i+1} + E_{N−i,N+1−i} & if i < l; \\ E_{l,l+1} & if i = l; \end{cases}

(where E_{ij} are the matrix units). Consider

x = x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} where 2r − 1 < l.

Then x is already in Jordan canonical form, with block sizes given by the partition (2^{2r}, 1^{N−4r}). It is elementary to check that β(xv, v) = 0 for all v ∈ V, so that q_{0} ≡ 0 and hence by (3.1.3), χ(2) = 0. Thus x belongs to the orbit O(2_{0}^{2r}, 1_{0}^{N−4r}). If

x = x_{α_{l}} + x_{α_{l−2}} + ⋯ + x_{α_{l−2r}}

then again x is in Jordan form, with block sizes (2^{2r+1}, 1^{N−4r−2}), but now q_{0}(v) = β(xv, v) = v_{n+1}^{2} ≠ 0 so χ(2) = 1. Thus x belongs to the orbit O(2_{1}^{2r+1}, 1_{0}^{N−4r−2}).

For the orthogonal case, let V be an N-dimensional vector space over k, equipped with a quadratic form q and a bilinear form β satisfying β(v, w) = q(v + w) − q(v) − q(w) for all v, w ∈ V. We assume that V is non-degenerate in the sense of [He]. Then G = O(N) is the algebraic group of automorphisms of V leaving q invariant. Fix x ∈ 𝔤 with x nilpotent. For each i ∈ ℕ ∪ {0} define a quadratic form q_{i}: V → k by q_{i}(v) = q(x^{i}v). The index function χ is defined as in (3.1.2). Then χ(1) = 1, and when x^{2} = 0, χ(2) is given by

(3.1.4)  χ(2) = \begin{cases}1 & if q_{1} = 0; \\ 2 & otherwise. \end{cases}

For v = (v_{1}, …, v_{N}), we may take q(v) = v_{1}v_{N} + v_{2}v_{N−1} + ⋯, so that the matrix of β is K when N = 2l is even, and is K_{0} = K − E_{l+1,l+1} when N = 2l + 1 is odd. A matrix x belongs to 𝔤 if and only if tr x = 0 and β(xv, v) = 0 for all v ∈ V. When N is even the second condition is equivalent to x being symmetric about the antidiagonal, with antidiagonal elements all zero. If N is odd, say N = 2l + 1, then x has the block form

(3.1.5)  x = (\begin{matrix}A & 0 & C \\ w & 0 & u \\ B & 0 & A′ \end{matrix})

where A, B, C are l × l, w and u are 1 × l, B and C are symmetric about the antidiagonal with antidiagonal elements all zero, and A′ denotes A transposed across the antidiagonal. In either case we may take a Cartan subalgbra consisting of the diagonal matrices in 𝔤, and simple root vectors

x_{α_{i}} = \begin{cases}E_{i,i+1} + E_{N−i,N+1−i} & if i < l; \\ E_{l+1,l+2} & if i = l in type B_{l}; \\ E_{l−1,l+1} + E_{l,l+2} & if i = l in type D_{l}. \end{cases}

Assume N = 2l + 1. Arguing similarly to the symplectic case, one finds that x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} with 2r − 1 < l lies in the orbit O(2_{1}^{2r}, 1_{1}^{N−4r}), while x_{α_{l}} + x_{α_{l−2}} + ⋯ + x_{α_{l−2r}} with r ≥ 0 has q_{1}(v) = q(xv) = v_{l+2}^{2} ≠ 0 so it is in the orbit O(2_{2}^{2r+1}, 1_{1}^{N−4r−2}).

Assume N = 2l. Using the same techniques as in types B_{l} and C_{l}, it is straightforward to check that x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{2r−1}} with 2r − 1 < l lies in the orbit O(2_{1}^{2r}, 1_{1}^{N−4r}). Similarly when l is even, x_{α_{1}} +x_{α_{3}} +⋯+x_{α_{l−3}} +x_{α_{l}} is in the orbit O(2_{1}^{l}). Consider x = x_{α_{l−1}} +x_{α_{l}} = E_{l−1,l} + E_{l−1,l+1} + E_{l,l+2} + E_{l+1,l+2}. Note that x: e_{l+2} ↦ e_{l} + e_{l+1} ↦ 0, e_{l} ↦ e_{l−1} ↦ 0 (where {e_{i}: 1 ≤ i ≤ N} is the standard basis of V); this shows that x has Jordan block sizes (2^{2}, 1^{N−4}). And q_{1}(v) = q(xv) = v_{l+2}^{2} ≠ 0 so x is in the orbit O(2_{2}^{2}, 1_{1}^{N−4}).

To complete the proof it remains only to verify the final claim of the proposition. Assume N = 2l with l even. We have shown that x = x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{l−3}} + x_{α_{l−1}} and y = x_{α_{1}} + x_{α_{3}} + ⋯ + x_{α_{l−3}} + x_{α_{l}} both lie in the O(N)-orbit O(2_{1}^{l}). By [He], this orbit splits into two distinct SO(N)-orbits. Also by [He], the group O(N) is generated by the reflections r_{w}, for w ∈ V with q(w) ≠ 0, given by

r_{w}(v) = v − β(v, w)q(w)^{−1}w,

while SO(N) consists of the products of an even number of such reflections. Let w = e_{l}+e_{l+1}, with q(w) = 1. Then r_{w} interchanges e_{l} and e_{l+1} while fixing all other e_{i}. Thus, conjugation by r_{w} interchanges x_{α_{l−1}} and x_{α_{l}}, and fixes all other x_{α_{i}}. In particular, r_{w} is an element of O(N) − SO(N) which takes x to y. Since SO(N) has index 2 in O(N), it follows that x and y must belong to the two different SO(N) orbits in the O(N) orbit O(2_{1}^{l}). _{□}

3.2. Generic W -orbits in X_{1}(T). Let Φ be of type X_{l}, where X = B, C, or D, and let p = 2. In view of the strategy in Section 2.8, we consider the action of W on X(T)/pX(T), which we identify with X_{1}(T) by taking p-restricted parts of weights. In this subsection and the next, we will classify the W -orbits in X_{1}(T). For convenience in stating the results, we work here with the ordinary (not the dot) action of W; the ρ shifts will be inserted later. And for simplicity of notation, we write equalities when we really mean congruences modulo pX(T).

Set ω_{0} = 0 and S_{k} = W (ω_{k}) for 0 ≤ k ≤ l. We plan to show that X_{1}(T) = ∪_{k=0}^{l}S_{k}. To do this, we first identify S_{1} explicitly as a set of l elements in each type. We then show that for k ≤ m (a certain integer depending on the type, defined below), S_{k} consists of sums of k distinct elements of S_{1}. We determine when an element of S_{k} can be written in more than one way as such a sum, and also when S_{k} = S_{j}. In the next subsection we will identify ∪_{k=m+1}^{l}S_{k}. These data allow us to count | ∪_{k=0}^{l} S_{k}| = 2^{l} = |X_{1}(T)|.

We fix the integer m as follows:

m = \begin{cases}l − 1, & X = B, \\ l, & X = C, \\ l − 2, & X = D. \end{cases}

We will repeatedly use formulas (modulo pX(T)) for s_{i}(ω_{j}) = ω_{j} − δ_{ij}α_{i}. The following formulas are easily checked by taking the inner product of each side with α̌_{k} for all k:

(3.2.1)  s_{i}(ω_{i}) = \begin{cases}ω_{l−2} + ω_{l−1}, & i = l − 1, type B_{l}; \\ ω_{l}, & i = l, type C_{l}; \\ ω_{l−3} + ω_{l−2} + ω_{l−1} + ω_{l}, & i = l − 2, type D_{l}; \\ ω_{l−2} + ω_{l−1}, & i = l − 1, type D_{l}; \\ ω_{l−2} + ω_{l}, & i = l, type D_{l}; \\ ω_{i−1} + ω_{i} + ω_{i+1}, & otherwise \end{cases}

with the convention that ω_{0} = ω_{l+1} = 0. The following formula follows immediately:

(3.2.2)  s_{k−1}s_{k−2} … s_{1}(ω_{1}) = ω_{k−1} + ω_{k} for 1 ≤ k ≤ m.

The remaining elements of S_{1} are determined case-by-case, again using (3.2.1). If X = C then s_{l}s_{l−1} … s_{1}(ω_{1}) = ω_{l−1} + ω_{l}, which is already accounted for in S_{1}, so we get no new elements. If X = B then s_{l−1}s_{l−2} … s_{1}(ω_{1}) = ω_{l−1}, from which we can produce no additional new elements. Finally if X = D then s_{l−2}s_{l−3} … s_{1}(ω_{1}) = ω_{l−2} + ω_{l−1} + ω_{l}, and s_{l−1}s_{l−2} … s_{1}(ω_{1}) = s_{l}s_{l−2} … s_{1}(ω_{1}) = ω_{l−1} + ω_{l}, from which we can produce no additional new elements. We summarize these results.

Proposition. Let Φ be of type X_{l} where X = B, C, or D, and let p = 2. The W -orbit of ω_{1} in X_{1}(T) consists of l distinct elements, as follows:

S_{1} = {ω_{1}, s_{1}(ω_{1}), s_{2}s_{1}(ω_{1}), …, s_{l−1}s_{l−2} … s_{1}(ω_{1})}

= \begin{cases}\{ω_{1}, ω_{1} + ω_{2}, …, ω_{l−2} + ω_{l−1}, ω_{l−1}\}, & X = B; \\ \{ω_{1}, ω_{1} + ω_{2}, …, ω_{l−2} + ω_{l−1}, ω_{l−1} + ω_{l}\}, & X = C; \\ \{ω_{1}, ω_{1} + ω_{2}, …, ω_{l−2} + ω_{l−1} + ω_{l}, ω_{l−1} + ω_{l}\}, & X = D. \end{cases}

Moreover, for i ≤ l, s_{i−1}s_{i−2} … s_{1}(ω_{1}) does not involve any ω_{j} with j < i − 1.

Next we have a technical lemma which allows us to write sums of k elements of S_{1} as elements of S_{k}.

Lemma. Let k ≤ m and 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ l be positive integers. Then

s_{i_{1}−1}s_{i_{1}−2} … s_{1}(ω_{1}) + s_{i_{2}−1}s_{i_{2}−2} … s_{1}(ω_{1}) + ⋯ + s_{i_{k}−1}s_{i_{k}−2} … s_{1}(ω_{1}) = (s_{i_{1}−1}s_{i_{1}−2} … s_{1})(s_{i_{2}−1}s_{i_{2}−2} … s_{2}) … (s_{i_{k}−1}s_{i_{k}−2} … s_{k})(ω_{k}).

Proof. We use induction on k. If k = 1 the result is trivial. In general, substitute ω_{k} = ω_{k−1} +s_{k−1}s_{k−2} … s_{1}(ω_{1}) from (3.2.1) (recall we are working mod 2) on the right hand side. Observe that s_{r}s_{r−1} … s_{j} fixes s_{i−1}s_{i−2} … s_{1}(ω_{1}) for r < i − 1 < l (by the last statement of the Proposition), so the right hand side of the above equation is equal to

(s_{i_{1}−1}s_{i_{1}−2} … s_{1})(s_{i_{2}−1}s_{i_{2}−2} … s_{2}) … (s_{i_{k}−1}s_{i_{k}−2} … s_{k})(ω_{k−1}) + s_{i_{k}−1}s_{i_{k}−2} … s_{1}(ω_{1}).

Since s_{r} fixes ω_{k−1} for r > k − 1, this is equal to

(s_{i_{1}−1}s_{i_{1}−2} … s_{1})(s_{i_{2}−1}s_{i_{2}−2} … s_{2}) … (s_{i_{k−1}−1}s_{i_{k−1}−2} … s_{k−1})(ω_{k−1}) + s_{i_{k}−1}s_{i_{k}−2} … s_{1}(ω_{1}).

The result now follows by induction. _{□}

Theorem. For 1 ≤ k ≤ m, S_{k} consists of all sums of k distinct elements of S_{1}. The expression for an element of S_{k} as such a sum is unique, except when X = B or D and k = l/2, when every element of S_{k} can be written in precisely two ways as a sum of k distinct elements of S_{1}. The orbits S_{k} are distinct, except that S_{k} = S_{l−k} for 1 ≤ k ≤ l − 1 in type B, and for 2 ≤ k ≤ l − 2 in type D.

Proof. Let S̃_{k} denote the set of sums of k distinct elements of S_{1}. According to the proposition, every μ ∈ S̃_{k} can be expressed as on the left side of the equation in the lemma. Then μ ∈ S_{k}, by the right side of that equation. To prove the reverse inclusion, notice that ω_{k} is the sum of the first k elements of S_{1}, in the order listed in the proposition. Hence any W -translate of ω_{k} is the sum of the W -translates of these k elements, which are in turn distinct elements of S_{1} = W (ω_{1}).

A similar argument shows that the number of ways of writing ω_{k} as a sum of distinct elements of S_{1} is the same as the number of such ways of writing every element of S_{k}. In type C it is clear from the explicit description of S_{1} that ω_{1}+(ω_{1}+ω_{2})+⋯+(ω_{k−1}+ω_{k}) is the only way of so writing ω_{k}. In type B there is another way: (ω_{k}+ω_{k+1})+(ω_{k+1}+ω_{k+2})+⋯+ω_{l−1}, a sum of l − k elements. In type D there is also a second way of writing ω_{k}, namely (ω_{k} + ω_{k+1}) + (ω_{k+1} + ω_{k+2}) + ⋯ + (ω_{l−2} + ω_{l−1} + ω_{l}) + (ω_{l−1} + ω_{l}), also a sum of l − k elements. This shows that S̃_{k} = S̃_{l−k} in types B and D, and hence that S_{k} = S_{l−k} (provided, in type D, that k ≥ 2, since m = l − 2). These second expressions for ω_{k} are in S̃_{k} (and hence every element of S_{k} has two realizations as a sum of k elements of S_{1}) if and only if k = l/2. _{□}

3.3. Special W -orbits in X_{1}(T). Here we use a different technique to identify ∪_{k=m+1}^{l}S_{k}. In type C_{l} there is nothing to do since m = l. In type B_{l} we need only identify S_{l}.

Proposition (A). In type B_{l}, ∪_{k=0}^{l−1}S_{k} = {λ ∈ X_{1}(T) | ⟨λ, α̌_{l}⟩ = 0} and S_{l} = {λ ∈ X_{1}(T) | ⟨λ, α̌_{l}⟩ = 1}.

Proof. Since, by (3.2.1), the coefficients of ω_{l} in λ and s_{i}(λ) are the same for every simple reflection s_{i}, it is clear that L_{0} := {λ ∈ X_{1}(T) | ⟨λ, α̌_{l}⟩ = 0} and L_{1} := {λ ∈ X_{1}(T) | ⟨λ, α̌_{l}⟩ = 1} are W -invariant. In particular, S_{l} ⊂ L_{1} and S_{k} ⊂ L_{0} for 0 ≤ k ≤ l − 1. In fact, by Theorem 3.2 we have

| ⋃_{k=0}^{l−1} S_{k}| = 1 + | ⋃_{k=1}^{l−1} S_{k}| = 1 + \frac{1}{2} ∑_{k=1}^{l−1} \binom{l}{k} = 1 + \frac{1}{2}(2^{l} − 2) = 2^{l−1} = |L_{0}|,

so that

(3.3.1)  L_{0} = ⋃_{k=0}^{l−1} S_{k}.

To prove that L_{1} ⊂ S_{l}, we first claim that ω_{l−i} + ω_{l} ∈ S_{l} for 1 ≤ i ≤ l − 1. We prove this by induction on i. Since s_{l}(ω_{l}) = ω_{l−1} + ω_{l} and s_{l}s_{l−1}s_{l}(ω_{l}) = ω_{l−2} + ω_{l} by (3.2.1), the claim is true for i = 1 and 2. Assume it is true for some i ≥ 2. Use (3.2.1) to check that

s_{l−2}s_{l−3} … s_{l−i}(ω_{l−i} + ω_{l}) = ω_{l−(i+1)} + ω_{l−2} + ω_{l−1} + ω_{l}.

Apply s_{l−1} and finally s_{l} to obtain ω_{l−(i+1)} + ω_{l}, as required to complete the induction.

Now let λ ∈ L_{1}. Then λ = μ + ω_{l} where μ ∈ L_{0}. Then by (3.3.1), μ ∈ S_{k} for some 0 ≤ k < l. So we may write μ = w(ω_{k}) for some w ∈ W. By Lemma 3.2, we may in fact assume that w ∈ ⟨s_{1}, …, s_{l−1}⟩. Then w(ω_{l}) = ω_{l} so λ = μ + ω_{l} = w(ω_{k}) + ω_{l} = w(ω_{k} + ω_{l}) ∈ S_{l}, by the claim. _{□}

Finally in type D_{l} we need to identify S_{l−1} ∪ S_{l}.

Proposition (B). In type D_{l}, ∪_{k=0}^{l−2}S_{k} = {λ ∈ X_{1}(T) | ⟨λ, α̌_{l−1}⟩ = ⟨λ, α̌_{l}⟩} and S_{l−1} ∪ S_{l} = {λ ∈ X_{1}(T) | ⟨λ, α̌_{l−1}⟩ ≠ ⟨λ, α̌_{l}⟩}. When l is odd, S_{l−1} = S_{l}.

Proof. Since, by (3.2.1), no simple reflection changes the coefficient of exactly one of ω_{l−1}, ω_{l} in a decomposition of λ ∈ X_{1}(T) as a linear combination of fundamental weights, the sets L_{0} := {λ ∈ X_{1}(T) | ⟨λ, α̌_{l−1}⟩ = ⟨λ, α̌_{l}⟩} and L_{1} := {λ ∈ X_{1}(T) | ⟨λ, α̌_{l−1}⟩ ≠ ⟨λ, α̌_{l}⟩} are W -invariant. In particular, S_{l−1} ∪ S_{l} ⊂ L_{1} and S_{k} ⊂ L_{0} for 0 ≤ k ≤ l − 2. A counting argument similar to the one in the previous proposition shows that

(3.3.2)  L_{0} = ⋃_{k=0}^{l−2} S_{k}.

To prove that L_{1} ⊂ S_{l−1} ∪ S_{l}, we first claim that ω_{l−i} + ω_{l} ∈ S_{l} for i even, 2 ≤ i < l, and ω_{l−i} + ω_{l−1} ∈ S_{l} for i odd, 3 ≤ i < l. (The same statements, with ω_{l} and ω_{l−1} interchanged, hold for S_{l−1}.) Begin the induction on i with the computations s_{l}(ω_{l}) = ω_{l−2} +ω_{l} and s_{l−1}s_{l−2}s_{l}(ω_{l}) = ω_{l−3} +ω_{l−1}. The inductive step follows from the computation s_{l−1}s_{l−2}s_{l−3} … s_{l−i}(ω_{l−i} + ω_{l}) = ω_{l−(i+1)} + ω_{l−1} (and similarly interchanging ω_{l} and ω_{l−1}, and using s_{l} in place of s_{l−1}).

Now let λ ∈ L_{1}. Then without loss of generality λ = μ + ω_{l} where μ ∈ L_{0}. (The other possibility is λ = μ − ω_{l}, but these are congruent modulo 2ω_{l} and we are working in X(T)/2X(T).) Then by (3.3.2), μ ∈ S_{k} for some 0 ≤ k ≤ m. So μ = w(ω_{k}) for some w ∈ W, but in fact by Lemma 3.2, we may assume w ∈ ⟨s_{1}, …, s_{l−1}⟩. Then w(ω_{l}) = ω_{l} so λ = μ + ω_{l} = w(ω_{k}) + ω_{l} = w(ω_{k} + ω_{l}) ∈ S_{l} ∪ S_{l−1}, by the claim.

Finally we prove the last statement of the proposition. When l is odd, we have l − 1 even, and so by the claim of the second paragraph with i = l − 1, ω_{1} + ω_{l} ∈ S_{l}. But s_{l−1}s_{l−2} … s_{1}(ω_{1} + ω_{l}) = ω_{l−1}, so ω_{l−1} ∈ S_{l} and thus S_{l−1} ⊂ S_{l}. The reverse inclusion follows by symmetry, so we have S_{l−1} = S_{l} as desired. _{□}

Theorem. Let Φ be of type X_{l} where X = B, C, or D, and p = 2. Then X_{1}(T) = ∪_{k=0}^{l}S_{k}.

Proof. First suppose X = C. Then by Theorem 3.2, the sets S_{k} are disjoint subsets of X_{1}(T) for 0 ≤ k ≤ l, and |S_{k}| = \binom{l}{k}. Thus | ∪_{k=0}^{l} S_{k}| = 2^{l} = |X_{1}(T)|. The theorem follows.

If X = B or D the theorem follows from the two previous propositions, since X_{1}(T) = L_{0} ∪ L_{1}. _{□}

3.4. Computation of |Φ| − |Φ_{λ}|. Define r as follows:

r = \begin{cases}l − 2k, & Φ = B_{l}, C_{l}, D_{l}, 1 ≤ k ≤ \frac{l}{2}, \\ −l + 2k, & Φ = B_{l}, D_{l}, \frac{l}{2} < k ≤ l − 1, \\ −l + 2k − 1, & Φ = C_{l}, \frac{l}{2} < k ≤ l. \end{cases}

When r is defined in this way and k is in the defined range, one has 0 ≤ l − r ≤ l. This will make our labelling of partitions make sense later in Section 3.6. We now calculate |Φ|−|Φ_{λ}| for classical Lie algebras. Recall the results of the previous two subsections for the values of k which index the W -orbits in X_{1}(T).

Proposition. Let Φ be of type X_{l} where X = B, C, or D. Let λ = ω_{k} − ρ, and define r as above. Then

|Φ| − |Φ_{λ}| = \begin{cases}l^{2} − r^{2}, & X = B_{l}, 1 ≤ k ≤ \frac{l}{2}, \\ l^{2} + l, & X = B_{l}, k = l, \\ l^{2} + l − r^{2} − r, & X = C_{l}, 1 ≤ k ≤ l, \\ l^{2} − r^{2}, & X = D_{l}, 1 ≤ k ≤ \frac{l}{2}, \\ l^{2} − l, & X = D_{l}, k = l − 1, l. \end{cases}

Proof. Set λ = ω_{k} − ρ. Let us first assume that for types B_{l} and D_{l}, we have 1 ≤ k ≤ \frac{l}{2} and for type C_{l}, 1 ≤ k ≤ l. Then Φ_{λ} = {α | ⟨ω_{k}, α̌⟩ ∈ 2ℤ}. Express α̌ = n_{1}α̌_{1} +n_{2}α̌_{2} +⋯+n_{l}α̌_{l}. If α ∈ Φ_{λ} then n_{k} = 0, ±2. If n_{k} = 0 then α is in the root subsystem A_{k−1}×X_{l−k}. Moreover, the number of roots such that n_{k} = ±2 is k(k + 1) (resp. k(k − 1), k(k − 1)) when X = B (resp. C, D). This gives |Φ| − |Φ_{λ}| = 4lk − 4k^{2} when X = B or D and 4lk − 4k^{2} + 2k when X = C. Substituting r as defined before the proposition yields the desired results.

On the other hand, for type B_{l} (resp. D_{l}) when k = l (resp. k = l −1, l), Φ_{λ} is isomorphic to A_{l−1}. Therefore, |Φ| − |Φ_{λ}| = 2l^{2} − l(l − 1) = l^{2} + l (resp. 2(l^{2} − l) − l(l − 1) = l^{2} − l). _{□}

3.5. Dimensions of Orbits. In [UGA2], the restricted nullcone N_{1}(𝔤) was calculated. The Hasse diagram for N_{1}(𝔤) was deduced from work of Großer [Gr] and Spaltenstein [Sp1] or can be obtained by using [UGA2, Thm. 2.3]. We provide this information in Section 6. The dimensions of the orbits in N_{1}(𝔤) can be computed from formulas given by Hesselink [He, Thm. 4.4]. This data will be essential for our computations so we have recorded it below.

Φ  Orbit  Dimension
B_{l} O(2_{2}^{l−r}, 1_{1}^{2r+1}) l^{2} + l − r^{2} − r
O(2_{1}^{l−r}, 1_{1}^{2r+1})  l^{2} − r^{2}
C_{l} O(2_{1}^{l−r}, 1_{0}^{2r}) l^{2} + l − r^{2} − r
O(2_{0}^{l−r}, 1_{0}^{2r})  l^{2} − r^{2}
D_{l} O(2_{2}^{l−r}, 1_{1}^{2r})  l^{2} − r^{2}
O(2_{1}^{l−r}, 1_{1}^{2r}) l^{2} − l − r^{2} + r

3.6. Supports of induced modules. The following theorem presents the computation of the supports of the induced modules H^{0}(w · λ), w · λ ∈ X(T)_{+} when G is O(N) or Sp(N) and char k = 2. By Theorem 3.3 and the remarks in Section 2.8, it suffices to consider λ of the form ω_{k} − ρ.

Theorem. Let G = O(2l + 1), Sp(2l), or O(2l) be a classical group with root system Φ of type B_{l}, C_{l} or D_{l} and assume that char k = 2. Let r be as defined in Section 3.4. Let λ = ω_{k} − ρ. The support varieties of the induced modules H^{0}(w · λ) where w · λ ∈ X(T)_{+}, are given in the following table.

Φ  k  Constrictors  |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
B_{l}  1 ≤ k ≤ \frac{l}{2} O(2_{1}^{l−r+2}, 1_{1}^{2r−3}),  l^{2} − r^{2}  \overline{O(2_{1}^{l−r}, 1_{1}^{2r+1})}
O(2_{2}, 1_{1}^{2l−1})
B_{l}  l  l^{2} + l  \overline{O(2_{2}^{l}, 1_{1}^{1})}
C_{l}  1 ≤ k ≤ \frac{l}{2} O(2_{0}^{l−(r−2)}, 1_{0}^{2(r−2)}), l^{2} + l − r^{2} − r \overline{O(2_{1}^{l−r}, 1_{0}^{2r})}
O(2_{1}^{l−(r−1)}, 1_{0}^{2(r−1)})
C_{l}  \frac{l}{2} < k ≤ l O(2_{0}^{l−(r−1)}, 1_{0}^{2(r−1)}) l^{2} + l − r^{2} − r \overline{O(2_{1}^{l−r}, 1_{0}^{2r})}
D_{l}  1 ≤ k ≤ \frac{l}{2} O(2_{1}^{l−(r−2)}, 1_{1}^{2(r−2)})  l^{2} − r^{2}  \overline{O(2_{2}^{l−r}, 1_{1}^{2r})}
D_{l} (l even) l − 1, l  O(2_{2}^{2}, 1_{1}^{2(l−2)})  l^{2} − l  \overline{O(2_{1}^{l})}
D_{l} (l odd) l − 1, l  O(2_{1}^{2}, 1_{1}^{2(l−2)})  l^{2} − l  \overline{O(2_{1}^{l−1}, 1_{1}^{2})}

Proof. In Proposition 3.4, we calculated |Φ|−|Φ_{λ}| for an element λ = ω_{k} −ρ in a typical W - orbit on X_{1}(T). According to Section 3.5, there exists an orbit in N_{1}(𝔤) of dimension equal to |Φ| − |Φ_{λ}|. The orbits considered are given by O := O(2_{1}^{l−r}, 1_{1}^{2r+1}) (resp. O(2_{1}^{l−r}, 1_{0}^{2r}), O(2_{2}^{l−r}, 1_{1}^{2r})) when Φ is of type B_{l} (resp. C_{l}, D_{l}); we postpone the cases k = l in B_{l}, k = l − 1, l in D_{l}. From the Hasse diagrams given in Section 6, one can easily determine the constrictors for these orbits (which are given in the statement of the theorem).

From Proposition 3.1, it follows that the constrictors are all of the form G · x_{J} where J ⊆ Δ. The results of this theorem will now follow from Theorem 2.7, if we verify condition (iii), which says that w(Φ_{λ}) ∩ Φ_{J} ≠ ∅ for all w ∈ W. Note that w^{−1}(Φ_{λ}) = Φ_{w·λ}; indeed,

⟨w · λ + ρ, α̌⟩ = ⟨w(ω_{k}), α̌⟩.

Since Φ_{J} ≅ A_{1} × A_{1} × ⋯ × A_{1}, our task is equivalent to proving that for every w ∈ W, 2|⟨w(ω_{k}), α̌⟩ for some α ∈ J.

We will proceed case by case. For type B_{l}, we have N = 2l + 1. Assume 1 ≤ k ≤ l/2, so r = l − 2k. Then O(2_{1}^{l−r+2}, 1_{1}^{2r−3}) = O(2^{2(k+1)}, 1_{1}^{N−4(k+1)}). Recall that S_{1} = {ω_{1}, ω_{1} + ω_{2}, …, ω_{l−2} + ω_{l−1}, ω_{l−1}}. An element in the W orbit of ω_{k} must be the sum of k distinct elements of S_{1}. Consider such an element of the form

w(ω_{k}) = n_{1}ω_{1} + n_{2}(ω_{1} + ω_{2}) + ⋯ + n_{l−1}(ω_{l−2} + ω_{l−1}) + n_{l}ω_{l−1}
= (n_{1} + n_{2})ω_{1} + (n_{2} + n_{3})ω_{2} + … (n_{l−1} + n_{l})ω_{l−1}

where 0 ≤ n_{j} ≤ 1 for all j. Observe that J = {α_{1}, α_{3}, …, α_{2(k+1)−1}} so |J| = k + 1.

Suppose that ⟨w(ω_{k}), α̌_{2j−1}⟩ ≡ 1 (mod 2) for j = 1, 2, …, k + 1. Then the k + 1 terms n_{1} + n_{2}, n_{3} + n_{4}, n_{5} + n_{6}, n_{2(k+1)−1} + n_{2k} must all be equal to 1. But, this implies that there are k +1 distinct elements of S_{1} in the expression of w(ω_{k}), which is a contradiction. Hence, 2|⟨w(ω_{k}), α̌_{j}⟩ for some j = 1, 2, …, 2(k + 1) − 1. For the other constrictor O(2_{2}, 1_{1}^{2l−1}), we have J = {α_{l}} according to the table in Proposition 3.1. Since w(ω_{k}) never contains ω_{l} as a summand (because of our description of S_{k}), it follows that ⟨w(ω_{k}), α̌_{l}⟩ = 0 for all w ∈ W. The case when k = l can be deduced immediately because the restricted nullcone is the closure of O(2_{2}^{l}, 1_{1}^{1}) (an irreducible variety). The variety has dimension l^{2} + l which is also a lower bound for the dimension of the support variety (i.e., |Φ| − |Φ_{λ}|).

For type C_{l} (resp. D_{l}) when r = l − 2k (i.e., 0 ≤ k ≤ \frac{l}{2}) one has N = 2l and O(2_{0}^{l−r+2}, 1_{0}^{2r−4}) = O(2_{0}^{2(k+1)}, 1_{0}^{N−4(k+1)}), O(2_{1}^{l−r+1}, 1_{0}^{2r−2}) = O(2_{1}^{2k+1}, 1_{0}^{N−4k−2}), (resp. O(2_{1}^{l−r+2}, 1_{1}^{2r−4}) = O(2_{1}^{2(k+1)}, 1_{1}^{N−4(k+1)})). The same argument as for type B_{l} can be used to verify that for w ∈ W, 2|⟨w(ω_{k}), α̌⟩ for some α ∈ J.

For type C_{l} when r = −l + 2k − 1 (i.e., \frac{l}{2} < k ≤ l) we have O(2_{0}^{l−(r−1)}, 1_{0}^{2(r−1)}) = O(2_{0}^{2(l−k+1)}, 1_{0}^{N−4(l−k+1)}). From the tables in Proposition 3.1, we have |J| = l − k + 1. Since \frac{l}{2} < k, we have |J| = l − k + 1 ≤ k with equality holding if and only if k = \frac{l+1}{2}. If |J| < k then we can use the same argument as in the type B_{l} case to deduce that condition (iii) of Theorem 2.7 holds. On the other hand, if k = \frac{l+1}{2} then l must be odd and l − (r − 1) = 2(l − k + 1) = l + 1. But in the case when l is odd the only allowable orbits O(2_{0}^{l−s}, 1_{0}^{2s}) occur when l − s < l. That is, there are no constrictors in this case.

The remaining cases for D_{l} when k = l − 1, l can be easily verified using arguments similar to those given in the preceding paragraphs. _{□}

4. Exceptional Lie algebras

4.1. In this section we compute the support varieties for the induced modules for exceptional Lie algebras. The strategy follows the one outlined in Section 2.8. Let us illustrate this with the following example.

Example. Let Φ = G_{2} and p = 3. The positive roots with their corresponding coroots are given in the following table. Note that Δ = {α_{1}, α_{2}} where α_{1} is the short root.

root  coroot
α_{1}  α̌_{1}
α_{2}  α̌_{2}
α_{1} + α_{2}  α̌_{1} + 3α̌_{2}
2α_{1} + α_{2} 2α̌_{1} + 3α̌_{2}
3α_{1} + α_{2}  α̌_{1} + α̌_{2}
3α_{1} + 2α_{2} α̌_{1} + 2α̌_{2}

Recall that Φ_{λ} = {α ∈ Φ | ⟨λ + ρ, α̌⟩ ∈ pℤ}. The restricted region X_{1}(T) contains nine weights and their stablizers are computed below.

\begin{matrix}λ & λ + ρ & Φ_{λ} \\ (0, 0) & (1, 1) & \{±(3α_{1} + 2α_{2})\} \\ (1, 0) & (2, 1) & \{±(3α_{1} + α_{2})\} \\ (2, 0) & (3, 1) & \{±α_{1}, ±(α_{1} + α_{2}), ±(2α_{1} + α_{2})\} \\ (0, 1) & (1, 2) & \{±(3α_{1} + α_{2})\} \\ (1, 1) & (2, 2) & \{±(3α_{1} + 2α_{2})\} \\ (2, 1) & (3, 2) & \{±α_{1}, ±(α_{1} + α_{2}), ±(2α_{1} + α_{2})\} \\ (0, 2) & (1, 3) & \{±α_{2}\} \\ (1, 2) & (2, 3) & \{±α_{2}\} \\ (2, 2) & (3, 3) & Φ \end{matrix}

One can verify that there are 3 W -orbits in X_{1}(T) under the dot action of the extended affine Weyl group, namely, {(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2)}, {(2, 0), (2, 1)}, and {(2, 2)}. The classes have stabilizer Φ_{λ} isomorphic to A_{1}, A_{2} and Φ respectively.

First assume that Φ_{λ} is isomorphic to A_{1}. Then |Φ| − |Φ_{λ}| = 10. The restricted nullcone N_{1}(𝔤) = \overline{O(G_{2}(a_{1}))} by Theorem 2.1(B)(v), an irreducible 10-dimensional variety. Hence, V_{G_{1}}(H^{0}(λ)) = \overline{O(G_{2}(a_{1}))} by Corollary 2.5.

Next assume that Φ_{λ} is isomorphic to A_{2}. In this case |Φ| − |Φ_{λ}| = 6. There are two 6-dimensional G-orbits in N_{1}(𝔤). Consider O = O(A_{1}) and its constrictor O_{1} = O(Ã_{1}) as in Theorem 2.7. Condition (i) is satisfied because dim O = 6. Moreover, condition (ii) holds because O(Ã_{1}) = G · x_{α_{1}}. Finally, condition (iii) is satisfied by inspecting the table above for the weights (2, 0) and (2, 1). Hence, V_{G_{1}}(H^{0}(λ)) = \overline{O(A_{1})} by Theorem 2.7.

When Φ_{λ} = Φ then λ = (2, 2) and H^{0}((2, 2)) is the Steinberg module for G_{1}. This module is projective so V_{G_{1}}(H^{0}(λ)) = {0}.

4.2. For the exceptional groups the computation of V_{G_{1}}(H^{0}(λ)) follows the paradigm given in the preceding example. With the help of GAP [Sch], we are able to compute the dot action of W on X_{1}(T) (identified with X(T)/pX(T)). This allows us first to determine the W -orbits on X_{1}(T). Next for a given orbit representative λ, we identify a potential G-orbit of dimension |Φ| − |Φ_{λ}| with constrictors {O_{1}, O_{2}, …, O_{s}} having the form G · x_{J_{i}} for i = 1, 2, …, s.

Since

w(Φ_{λ}) = Φ_{w·λ} = {α ∈ Φ : ⟨w · λ + ρ, α̌⟩ ∈ pℤ} = {α ∈ Φ : ⟨w(λ + ρ), α̌⟩ ∈ pℤ},

we can again use GAP, to compute the ordinary W action on λ + ρ to determine w(Φ_{λ}). Finally, this allows us to check that w(Φ_{λ})∩Φ_{J_{i}} ≠ ∅ for i = 1, 2, …, s. Our GAP programs are available on the web at http://www.math.uga.edu/∼nakano/vigre/vigre.html.

The following tables record our results. In the leftmost column, we give an orbit representative in X_{1}(T) of X(T) under the extended affine Weyl group. Note our results show that V_{G_{1}}(H^{0}(λ)) = V_{G_{1}}(H^{0}(w · λ + pν)) for all λ, w · λ + pν ∈ X(T)_{+} where w ∈ W and ν ∈ X(T).

E_{6}, p = 2

λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(1, 0, 1, 1, 1, 1) 36  40  \overline{O(3A_{1})}
(0, 1, 1, 1, 1, 1) 27  O(3A_{1})  32  \overline{O(2A_{1})}
(1, 1, 1, 1, 1, 1)  1  O(A_{1})  0  {0}

E_{6}, p = 3

λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(2, 2, 0, 2, 2, 2) 80  54  \overline{O(2A_{2} + A_{1})}
(2, 2, 2, 2, 0, 2) 216  O(2A_{2})  50  \overline{O(A_{2} + 2A_{1})}
(2, 2, 0, 2, 2, 2) 216  O(2A_{2})  50  \overline{O(A_{2} + 2A_{1})}
(2, 2, 2, 2, 0, 2) 90 O(A_{2} + 2A_{1})  48  \overline{O(2A_{2})}
(2, 0, 2, 2, 2, 2) 72  O(A_{2} + A_{1})  42  \overline{O(A_{2})}
(2, 2, 2, 2, 2, 0) 27  O(3A_{1})  32  \overline{O(2A_{1})}
(0, 2, 2, 2, 2, 2) 27  O(3A_{1})  32  \overline{O(2A_{1})}
(2, 2, 2, 2, 2, 2)  1  O(A_{1})  0  {0}

E_{7}, p = 2

λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(1, 0, 1, 1, 1, 1, 1) 36  70  \overline{O(4A_{1})}
(0, 1, 1, 1, 1, 1, 1) 63  O((3A_{1})′′)  64  \overline{O((3A_{1})′)}
(1, 1, 1, 1, 1, 1, 0) 28  O((3A_{1})′)  54  \overline{O((3A_{1})′′)}
(1, 1, 1, 1, 1, 1, 1)  1  O(A_{1})  0  {0}

E_{7}, p = 3

λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0, 0, 0, 0, 0, 0) 672  90  \overline{O(2A_{2} + A_{1})}
(0, 2, 0, 0, 0, 0, 0) 756 O(A_{2} + 3A_{1})  84  \overline{O(2A_{2})}
(0, 0, 0, 1, 0, 1, 0) 576  O(2A_{2})  84  \overline{O(A_{2} + 3A_{1})}
(0, 2, 0, 2, 2, 2, 2) 126  O((3A_{1})′′)  66  \overline{O(A_{2})}
(0, 2, 1, 2, 0, 1, 2) 56  O((3A_{1})′)  54  \overline{O((3A_{1})′′)}
(2, 2, 2, 2, 2, 2, 2)  1  O(A_{1})  0  {0}

E_{8}, p = 2
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 1, 1, 1, 1, 1, 1, 1) 135  128  \overline{O(4A_{1})}
(1, 1, 0, 1, 1, 1, 1, 1) 120  O(4A_{1})  112  \overline{O(3A_{1})}
(1, 1, 1, 1, 1, 1, 1, 1)  1  O(A_{1})  0  {0}

E_{8}, p = 3
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0, 0, 0, 0, 0, 0, 0) 1920  168  \overline{O(2A_{2} + 2A_{1})}
(0, 0, 1, 0, 0, 0, 0, 0) 2240 O(2A_{2} + 2A_{1})  162  \overline{O(2A_{2} + A_{1})}
(0, 0, 0, 0, 0, 2, 1, 0) 2160 O(2A_{2} + A_{1})  156  \overline{O(2A_{2})}
(1, 1, 0, 2, 2, 2, 0, 1) 240  O(4A_{1})  114  \overline{O(A_{2})}
(2, 2, 2, 2, 2, 2, 2, 2)  1  O(A_{1})  0  {0}

E_{8}, p = 5
λ  |W · λ|  Constrictors  |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0, 0, 0, 0, 0, 0, 0) 48, 484  200  \overline{O(A_{4} + A_{3})}
(1, 0, 0, 0, 0, 0, 0, 0) 69, 120  O(A_{4} + A_{3})  196  \overline{O(A_{4} + A_{2} + A_{1})}
(3, 1, 1, 1, 1, 1, 1, 1) 69, 120  O(A_{4} + A_{3})  196  \overline{O(A_{4} + A_{2} + A_{1})}
(1, 0, 0, 0, 1, 0, 0, 0) 60, 480 O(A_{4} + A_{2} + A_{1})  194  \overline{O(A_{4} + A_{2})}
(3, 1, 1, 1, 3, 1, 1, 1) 60, 480 O(A_{4} + A_{2} + A_{1})  194  \overline{O(A_{4} + A_{2})}
(0, 4, 0, 4, 4, 4, 4, 4) 17, 280  O(A_{4}), O(2A_{3})  184  \overline{O(D_{4}(a_{1}) + A_{2})}
(1, 4, 1, 4, 4, 4, 4, 4) 17, 280  O(A_{4}), O(2A_{3})  184  \overline{O(D_{4}(a_{1}) + A_{2})}
(0, 2, 0, 0, 0, 0, 4, 4) 30, 240 O(A_{3} + A_{2} + A_{1})  180  \overline{O(A_{4})}
(0, 4, 0, 0, 1, 2, 1, 4) 6720  O(2A_{2} + 2A_{1})  166  \overline{O(D_{4}(a_{1}))}
(1, 4, 1, 1, 3, 0, 3, 4) 6720  O(2A_{2} + 2A_{1})  166  \overline{O(D_{4}(a_{1}))}
(0, 4, 1, 1, 0, 1, 4, 4) 2160 O(2A_{2} + A_{1}), O(A_{3})  156  \overline{O(2A_{2})}
(1, 4, 3, 3, 1, 3, 4, 4) 2160 O(2A_{2} + A_{1}), O(A_{3})  156  \overline{O(2A_{2})}
(0, 4, 0, 3, 4, 4, 4, 4) 240  O(4A_{1})  114  \overline{O(A_{2})}
(1, 4, 1, 2, 4, 4, 4, 4) 240  O(4A_{1})  114  \overline{O(A_{2})}
(4, 4, 4, 4, 4, 4, 4, 4)  1  O(A_{1})  0  {0}

F_{4}, p = 2
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0, 0, 0)  12  28  \overline{O(A_{1} + Ã_{1})}
(1, 0, 1, 1)  3  O(Ã_{1})  3  \overline{O(A_{1})}
(1, 1, 1, 1)  1  O(A_{1}), O(Ã_{1})  0  {0}

F_{4}, p = 3
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0, 0, 0)  32  36  \overline{O(A_{1} + Ã_{2})}
(2, 2, 2, 0)  24  O(A_{2})  30  \overline{O(Ã_{2})}
(0, 2, 2, 2)  24  O(Ã_{2})  30  \overline{O(A_{2})}
(2, 2, 2, 2)  1  O(A_{1})  0  {0}

G_{2}, p = 2
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0)  3  8  \overline{O(Ã_{1})}
(1, 1)  1  O(A_{1})  0  {0}

G_{2}, p = 3
λ  |W · λ| Constrictors |Φ| − |Φ_{λ}| V_{G_{1}}(H^{0}(w · λ))
(0, 0)  6  10  \overline{O(G_{2}(a_{1}))}
(2, 0)  2  O(Ã_{1})  6  \overline{O(A_{1})}
(2, 2)  1  O(A_{1}), O(Ã_{1})  0  {0}

5. Applications

In this section we prove several consequences of our support variety computations.

5.1. Dimension equality. Here we prove the dimension equality conjecture stated earlier as (1.4.1).

Theorem. Let G be a simple algebraic group defined over an algebraically closed field of characteristic p > 0. Let λ ∈ X(T)_{+}. Then dim V_{G_{1}}(H^{0}(λ)) = |Φ| − |Φ_{λ}|.

Proof. For p good this is Theorem 1.4(b). For p bad it follows by observation from the tables in Sections 3 and 4. _{□}

5.2. Irreducibility. For induced modules we can now show that their supports are always irreducible varieties.

Theorem. Let G be a simple algebraic group defined over an algebraically closed field of characteristic p > 0. Let λ ∈ X(T)_{+}. Then V_{G_{1}}(H^{0}(λ)) is an irreducible variety.

Proof. For p good this follows from Theorem 1.4(a). It is also true for p bad since, from the tables in Sections 3 and 4, V_{G_{1}}(H^{0}(λ)) is always the closure of a single G-orbit. _{□}

5.3. Realization of orbit closures. An open problem from [FP] is: For a simple algebraic group G, which G-stable, closed, conical subvarieties of N_{1}(𝔤) can be realized as V_{G_{1}}(M) for some G-module M? Because the support of a direct sum is the union of the supports, it is enough to determine which G-orbit closures can be realized. In [Jan1, Jan2], Jantzen showed that all orbit closures can be realized in type A and in type B_{2} (when p ≠ 2). It follows from [CLNP] that when p is good, the closure of every Richardson orbit in N_{1}(𝔤) can be realized as the support variety of an induced module. Recently, Nakano and Tanisaki [NT] have proved that when p is good, with the possible exception of a few non-Richardson orbits in type E, all other orbit closures can be realized.

We will employ the ideas used in [NT] with some modifications. Assume that ρ̂: G → GL(V) is a representation such that the differential ρ : 𝔤 → 𝔤𝔩(V) satisfies the property that Ker ρ ∩ N (𝔤) = {0} (here we are also viewing ρ : G_{1} → GL(V) via the equivalence between restricted representations of 𝔤 and representations of G_{1}). For any finite-dimensional ρ(G_{1})- module M, we have

V_{GL(V)_{1}}(M) ∩ ρ(𝔤) = V_{ρ(G_{1})}(M).

Now M becomes a G_{1}-module by composing with ρ. Furthermore, ρ induces a natural map ρ^{∗} : V_{G_{1}}(M) → V_{ρ(G_{1})}(M) which is surjective. This map is injective because ker ρ ∩ N (𝔤) = {0}. Hence,

(5.3.1)  V_{GL(V)_{1}}(M) ∩ ρ(𝔤) ≅ V_{G_{1}}(M).

This isomorphism will be used to realize orbit closures as support varieties of G-modules.

Theorem. Assume that p is a bad prime for a simple algebraic group G. Then the closure of every G-orbit in N_{1}(𝔤) is realized as V_{G_{1}}(M) for some G-module M, with the following possible exceptions.

(1) Type C_{l}: O(2_{0}^{k}1_{0}^{2(n−k)}) for k ∈ ℕ even (p = 2).

(2) Type F_{4}: O(Ã_{1}) (p = 2).

(3) Type G_{2}: O(Ã_{1}) (p = 3).

Proof. We use the following principle [UGA1, Prop. 2.4]. Assume ρ: G → GL(V) is a representation satisfying Ker ρ ∩ N (𝔤) = {0}. Given a G-orbit O ⊂ N (𝔤), we identify O with ρ(O) (and similarly for \overline{O}), and let λ(O) be the Jordan block partition associated with ρ(O). Let O_{λ(O)} denote the corresponding GL(V) orbit in 𝔤𝔩(V). Then

(5.3.2)  \overline{O_{λ(O)}} ∩ ρ(𝔤) = ⋃ \overline{O′},

where the union is over all G-orbits O′ ⊆ N_{1}(𝔤) which are maximal with respect to ⪯ (the inclusion of orbit closures) and satisfy λ(O′) ⊴ λ(O) (where ⊴ is the dominance ordering on partitions). In particular, if

(5.3.3)  O′ ⪯̸ O ⟹ λ(O′) ⋬ λ(O),

then \overline{O_{λ(O)}} ∩ ρ(𝔤) = \overline{O}. In this case, by [NT] and (5.3.1) there is a GL(V)-module M such that

\overline{O} = \overline{O_{λ(O)}} ∩ ρ(𝔤) = V_{GL(V)_{1}}(M) ∩ ρ(𝔤) = V_{G_{1}}(M|_{G_{1}}),

whence \overline{O} is realized as a support variety of a G-module. In any case there is at most one orbit O′ besides O on the right side of (5.3.2).

Begin with the exceptional groups and let ρ be the adjoint representation. Using the tables of adjoint partitions associated to the G-orbits in N_{1}(𝔤) from [UGA2, §6] along with the Hasse diagrams giving the order ⪯ (see Section 6), we find that there are four cases where (5.3.3) does not hold (and O is not already realized as the support of an induced module). These are tabulated below.

\begin{matrix}Type & p & O & O′ & λ(O) & λ(O′) \\ F_{4} & 2 & O(Ã_{1}) & O(A_{1}) & (2^{16}, 1^{20}) & (2^{16}, 1^{20}) \\ F_{4} & 3 & O(A_{2} + Ã_{1}) & O(A_{1} + Ã_{2}) & (3^{16}, 2^{2}) & (3^{16}, 2^{2}) \\ G_{2} & 2 & O(A_{1}) & O(Ã_{1}) & (2^{6}, 1^{2}) & (2^{6}, 1^{2}) \\ G_{2} & 3 & O(Ã_{1}) & O(A_{1}) & (3^{3}, 1^{5}) & (3, 2^{4}, 1^{3}) \end{matrix}

But we can investigate these same orbits O under the minimal representation ρ, using the orbit representatives in [UGA2, §5] and MAGMA [BC, BCP]. The Jordan block partitions are given in the next table. (Note that in F_{4}, p = 3 we have replaced O′ = A_{1} + Ã_{2} by the smaller orbit Ã_{2}.)

\begin{matrix}Type & p & O & O′ & λ(O) & λ(O′) \\ F_{4} & 2 & O(Ã_{1}) & O(A_{1}) & (2^{10}, 1^{6}) & (2^{6}, 1^{14}) \\ F_{4} & 3 & O(A_{2} + Ã_{1}) & O(Ã_{2}) & (3^{7}, 2^{2}, 1) & (3^{8}, 2) \\ G_{2} & 2 & O(A_{1}) & O(Ã_{1}) & (2^{2}, 1^{3}) & (2^{3}, 1) \\ G_{2} & 3 & O(Ã_{1}) & O(A_{1}) & (3, 2^{2}) & (2^{2}, 1^{3}) \end{matrix}

In the two middle rows of the table, λ(O′) ⋬ λ(O), so by the discussion at the beginning of the proof, \overline{O} is realized as a support variety in those two cases, leaving Ã_{1} in F_{4}, p = 2 and in G_{2}, p = 3 as the only exceptional orbits possibly not realized as support varieties of G-modules in bad characteristic.

For the classical groups, let ρ be the standard representation V with dim V = N. Assume first that we are in type B_{l}, where N = 2l + 1. The orbit closures not realized as support varieties of induced modules are parametrized by symbols μ_{χ} = (2_{2}^{k}, 1_{1}^{N−2k}), k ≥ 1. Recall that the associated Jordan block partition for the minimal representation is obtained from μ_{χ} by deleting the subscripts χ; i.e., it is simply μ. Observe from the Hasse diagram that condition (5.3.3) is satisfied for these orbits. Thus they are all realized as supports of restrictions to G of GL(N)-modules.

Next consider type D_{l}, where N = 2l. The orbit closures not realized as support varieties of induced modules are parametrized by symbols μ_{χ} = (2_{1}^{k}, 1_{1}^{N−2k}), 2 ≤ k ≤ l − 2, k even. Here condition (5.3.3) is not satisfied, but we can use another trick. Assume for definiteness that l is even and G = O(2l). There are G-modules M and N whose support varieties are \overline{O(2_{1}^{l})} and \overline{O(2_{2}^{l−2}, 1_{1}^{4})}, respectively. Now by a basic property of support varieties,

V_{G_{1}}(M ⊗ N) = V_{G_{1}}(M) ∩ V_{G_{1}}(N) = \overline{O(2_{1}^{l−2}, 1_{1}^{4})}.

Iterating this process, we realize all the remaining orbit closures in type D. The argument is identical, with a shift in indices, if l is odd. If G = SO(2l) and l is even then the O(2l)-orbit O(2_{1}^{l}) splits into a union of two G-orbits, both of whose closures are support varieties of induced G-modules, and both of which contain O(2_{1}^{l−2}, 1_{1}^{4}), so the same argument works. _{□}

Remark. In type C the argument used for type B does not apply because each “unrealized” orbit is dominated (in the closure ordering) by an orbit having the same partition. And the argument used for type D does not work because we do not know that \overline{O(2_{0}^{l−ε}, 1_{0}^{2ε})} is realized (where ε = 0 if l is even and ε = 1 if l is odd). However, if that orbit were realized, then all the remaining ones in type C would be, too, by the same tensor product construction used in type D.

5.4. Richardson orbits. When p is a good prime, it follows from the solution to the Jantzen conjecture on support varieties [NPV] that V_{G_{1}}(H^{0}(λ)) is the closure of a Richardson orbit, and every Richardson orbit closure in N_{1}(𝔤) is realized in this fashion. We now consider to what extent these statements are true for bad primes.

Theorem. Let G be a simple algebraic group defined over an algebraically closed field of bad characteristic p. Let O be a Richardson orbit in N_{1}(𝔤). Then there exists λ ∈ X(T)_{+} such that V_{G_{1}}(H^{0}(λ)) = \overline{O}, except when O = O(2_{2}^{2}, 1_{1}^{2l−3}) in type B_{l}.

Proof. Recall that O is Richardson if and only if \overline{O} = G · 𝔲_{J} for some subset J ⊂ Δ. Also dim G · 𝔲_{J} = 2 dim 𝔲_{J} = |Φ| − |Φ_{J}|. The zero orbit is trivially Richardson (corresponding to J = ∅), and is also the support variety of the Steinberg module, so we assume henceforth that O is not the zero orbit and J ≠ ∅.

Assume first that G is classical, so p = 2. The formula (x + y)^{[2]} = x^{[2]} + [x, y] + y^{[2]} (cf. [NPV, Section 6.3]) implies that 𝔲_{J} ⊂ N_{1}(𝔤) if and only if 𝔲_{J} is abelian. One checks that 𝔲_{J} is abelian (in characteristic 2) if and only if Δ − J is a single simple root at an “end” of the Dynkin diagram (i.e., a simple root α such that ⟨α, β⟩ ≠ 0 for exactly one other simple root β). Comparing |Φ| − |Φ_{J}| (computed using, for instance, [Hum1, Table 12.2.1]) for these J with the dimensions of the orbits from Hesselink (see Section 3), there is always a unique orbit of the correct dimension. The orbit parameters for the nonzero Richardson orbits are given in the next table.

Type  Richardson Orbits
B_{l}  O(2_{2}^{2}, 1_{1}^{2l−3}), O(2_{2}^{l}, 1_{1}^{1})
C_{l}  O(2_{1}^{2}, 1_{0}^{2l−4}), O(2_{1}^{l})
D_{l}, l even  O(2_{2}^{2}, 1_{1}^{2l−4}), O(2_{1}^{l})
D_{l}, l odd O(2_{2}^{2}, 1_{1}^{2l−4}), O(2_{1}^{l−1}, 1_{1}^{2})

(Recall that for l even, the O(2l) orbit O(2_{1}^{l}) splits into two SO(2l) orbits; these are both Richardson.) The theorem now follows in the classical cases by comparison with the circled orbits in Figures 1 and 2.

Now assume G is exceptional. The Richardson orbits in characteristic zero are given by Hirai in [Hi]. None of these orbits (in N_{1}(𝔤)) “split” in bad characteristic; therefore the Richardson orbits in characteristic p are given by the same parameters. The list of nontrivial Richardson orbits in N_{1}(𝔤) is given in the following table.

Type  Richardson Orbits
E_{6}  O(2A_{1}), O(A_{2}), O(A_{2} + 2A_{1}), O(2A_{2})
E_{7}  O((3A_{1})′′), O(A_{2}), O(A_{2} + 3A_{1}), O(2A_{2})
E_{8} O(A_{2}), O(2A_{2}), O(D_{4}(a_{1})), O(A_{4}), O(D_{4}(a_{1}) + A_{2}), O(A_{4} + A_{2}), O(A_{4} + A_{2} + A_{1})
F_{4}  O(A_{2}), O(Ã_{2})
G_{2}  O(G_{2}(a_{1}))

Comparison with Figures 3–7 shows that the closures of all these orbits arise as support varieties of induced modules. _{□}

Remark. It is clear from the data that, in contrast to the situation when p is good, many non-Richardson orbit closures arise as support varieties of induced modules when p is bad.

6. Hasse diagrams

In Figures 1–7, we provide the Hasse diagrams for N_{1}(𝔤) when k is a field of bad characteristic for Φ. The orbits whose closures are support varieties of induced/Weyl modules are circled (cf. Sections 3.6 and 4.2).

1_{1}^{N} 2_{2} 1_{1}^{N−2} 2_{1}^{2} 1_{1}^{N−4}  2_{2}^{2} 1_{1}^{N−4} 2_{2}^{3} 1_{1}^{N−6} 2_{1}^{4} 1_{1}^{N−8}  2_{2}^{4} 1_{1}^{N−8} 2_{1}^{l−4} 1_{1}^{9}  2_{2}^{l−4} 1_{1}^{9} 2_{2}^{l−3} 1_{1}^{7} 2_{1}^{l−2} 1_{1}^{5}  2_{2}^{l−2} 1_{1}^{5} 2_{2}^{l−1} 1_{1}^{3} 2_{1}^{l} 1_{1}^{1}  2_{2}^{l} 1_{1}^{1}  1_{0}^{N} 2_{1} 1_{0}^{N−2} 2_{0}^{2} 1_{0}^{N−4}  2_{1}^{2} 1_{0}^{N−4} 2_{1}^{3} 1_{0}^{N−6} 2_{0}^{4} 1_{0}^{N−8}  2_{1}^{4} 1_{0}^{N−8} 2_{0}^{l−4} 1_{0}^{8}  2_{1}^{l−4} 1_{0}^{8} 2_{1}^{l−3} 1_{0}^{6} 2_{0}^{l−2} 1_{0}^{4}  2_{1}^{l−2} 1_{0}^{4} 2_{1}^{l−1} 1_{0}^{2} 2_{0}^{l}  2_{1}^{l}  1_{1}^{N} 2_{1}^{2} 1_{1}^{N−4}  2_{2}^{2} 1_{1}^{N−4} 2_{1}^{4} 1_{1}^{N−8}  2_{2}^{4} 1_{1}^{N−8} 2_{1}^{l−4} 1_{1}^{8}  2_{2}^{l−4} 1_{1}^{8} 2_{1}^{l−2} 1_{1}^{4}  2_{2}^{l−2} 1_{1}^{4} 2_{1}^{l}  2_{2}^{l}

Figure 1. O(2l + 1), Sp(2l), O(2l) for l even (N = 2l + 1, 2l, 2l, respectively)

1_{1}^{N} 2_{2} 1_{1}^{N−2} 2_{1}^{2} 1_{1}^{N−4}  2_{2}^{2} 1_{1}^{N−4} 2_{2}^{3} 1_{1}^{N−6} 2_{1}^{4} 1_{1}^{N−8}  2_{2}^{4} 1_{1}^{N−8} 2_{1}^{l−5} 1_{1}^{11}  2_{2}^{l−5} 1_{1}^{11} 2_{2}^{l−4} 1_{1}^{9} 2_{1}^{l−3} 1_{1}^{7}  2_{2}^{l−3} 1_{1}^{7} 2_{2}^{l−2} 1_{1}^{5} 2_{1}^{l−1} 1_{1}^{3}  2_{2}^{l−1} 1_{1}^{3} 2_{2}^{l} 1_{1}^{1}  1_{0}^{N} 2_{1} 1_{0}^{N−2} 2_{0}^{2} 1_{0}^{N−4}  2_{1}^{2} 1_{0}^{N−4} 2_{1}^{3} 1_{0}^{N−6} 2_{0}^{4} 1_{0}^{N−8}  2_{1}^{4} 1_{0}^{N−8} 2_{0}^{l−5} 1_{0}^{10}  2_{1}^{l−5} 1_{0}^{10} 2_{1}^{l−4} 1_{0}^{8} 2_{0}^{l−3} 1_{0}^{6}  2_{1}^{l−3} 1_{0}^{6} 2_{1}^{l−2} 1_{0}^{4} 2_{0}^{l−1} 1_{0}^{2}  2_{1}^{l−1} 1_{0}^{2} 2_{1}^{l}  1_{1}^{N} 2_{1}^{2} 1_{1}^{N−4}  2_{2}^{2} 1_{1}^{N−4} 2_{1}^{4} 1_{1}^{N−8}  2_{2}^{4} 1_{1}^{N−8} 2_{1}^{l−5} 1_{1}^{10}  2_{2}^{l−5} 1_{1}^{10} 2_{1}^{l−3} 1_{1}^{6}  2_{2}^{l−3} 1_{1}^{6} 2_{1}^{l−1} 1_{1}^{2}  2_{2}^{l−1} 1_{1}^{2}

Figure 2. O(2l + 1), Sp(2l), O(2l) for l odd (N = 2l + 1, 2l, 2l, respectively)

∅ A_{1} 2A_{1} 3A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} A_{2} + A_{1} 2A_{2} A_{2} + 2A_{1} 2A_{2} + A_{1}

Figure 3. E_{6} for p = 2 (left) and p = 3 (right)

∅ A_{1} 2A_{1} (3A_{1})′ (3A_{1})′′ 4A_{1}  ∅ A_{1} 2A_{1} (3A_{1})′ (3A_{1})′′  A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1}2A_{2} A_{2} + 3A_{1} 2A_{2} + A_{1}

Figure 4. E_{7} for p = 2 (left) and p = 3 (right)

∅ A_{1} 2A_{1} 3A_{1} 4A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1} A_{2} + 3A_{1} 2A_{2} 2A_{2} + A_{1} 2A_{2} + 2A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1} A_{2} + 3A_{1} 2A_{2} 2A_{2} + A_{1} A_{3}  2A_{2} + 2A_{1} A_{3} + A_{1} A_{3} + 2A_{1} D_{4}(a_{1}) D_{4}(a_{1}) + A_{1} A_{3} + A_{2} A_{3} + A_{2} + A_{1} D_{4}(a_{1}) + A_{2} A_{4} A_{4} + A_{1}  2A_{3} A_{4} + 2A_{1} A_{4} + A_{2} A_{4} + A_{2} + A_{1} A_{4} + A_{3}

Figure 5. E_{8} for p = 2 (left), p = 3, (center), and p = 5 (right)

∅ A_{1} Ã_{1}^{(2)} Ã_{1} A_{1} + Ã_{1}  ∅ A_{1} Ã_{1} A_{1} + Ã_{1} A_{2} Ã_{2}  A_{2} + Ã_{1} A_{1} + Ã_{2}

Figure 6. F_{4} for p = 2 (left) and p = 3 (right)

∅ A_{1} Ã_{1}  ∅ A_{1} Ã_{1}^{(3)} Ã_{1} G_{2}(a_{1})

Figure 7. G_{2} for p = 2 (left) and p = 3 (right)

7. VIGRE Algebra Group at the University of Georgia

This project was initiated during Spring Semester 2004 under the Vertical Integration of Research and Education (VIGRE) Program sponsored by the National Science Foundation (NSF) at the Department of Mathematics at the University of Georgia (UGA). We would like to acknowledge the NSF grant DMS-0089927 for its financial support of this project. The VIGRE Algebra Group consists of 4 faculty members, 3 postdoctoral fellows and 5 graduate students. The group was led in Spring 2004 by David J. Benson, Brian D. Boe and Daniel K. Nakano. During Summer 2004, Leonard Chastkofsky directed a group of the students (Bergonio, Platt, and Wright) who completed the computations for this project. The email addresses of the members of the group are given below.

Faculty:

David J. Benson  bensondj@maths.abdn.ac.uk
Brian D. Boe  brian@math.uga.edu
Leonard Chastkofsky lenny@math.uga.edu
Daniel K. Nakano  nakano@math.uga.edu

Postdoctoral Fellows:

Jo Jang Hyun  jhjo@sogang.ac.kr
Jonathan Kujawa  kujawa@math.uga.edu
Nadia Mazza  n.mazza@maths.abdn.ac.uk

Graduate Students:

Philip Bergonio  pbergonio@math.uga.edu
Bobbe Cooper  bcooper@math.uga.edu
Jeremiah Hower  jhower@math.uga.edu
Kenyon J. Platt  platt@math.uga.edu
Caroline Wright  cwright@math.uga.edu

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