SUPPORT VARIETIES FOR MODULES OVER CHEVALLEY GROUPS AND CLASSICAL LIE ALGEBRAS

JON F. CARLSON, ZONGZHU LIN, AND DANIEL K. NAKANO

1. Introduction

1.1. Let G be a connected reductive algebraic group scheme defined over the finite field of 𝔽_{p} of p-elements. For a fixed r ≥ 1, let G_{r} be the scheme theoretic kernel of the rth iteration of the Frobenius map. When r = 1 the category of G_{1}-modules is equivalent to the category of restricted modules for the restricted Lie algebra 𝔤 = Lie G. On the other hand, for r ≥ 1 and q = p^{r}, let G(𝔽_{q}) be the finite Chevalley group obtained by taking the 𝔽_{q} rational points of G. Given a G-module one can restrict the action to either G_{r} or to G(𝔽_{q}) (see Figure 1). Much progress has been made in relating the representation and cohomology theory for the finite Chevalley group and the Frobenius kernel by using the ambient algebraic group. Even though the category of modules for the finite group and the Frobenius kernels share many intrinsic features, difficulties in making comparisons arise because there is no direct functorial relationship between these two categories.

[figure: mod(G); mod(G(𝔽_{q})); mod(G_{r}); res; res]

Figure 1

In 1987, B. Parshall in an article in the Arcata Conference Proceedings entitled Cohomology of Algebraic Groups stated two open problems where the first problem is a precursor to the second:

(1.1.1) If M ∈ mod(G) such that M is projective as G_{1}-module, is M projective over G(𝔽_{p})?

(1.1.2) Let M ∈ mod(G). Can one relate the support variety of M for G_{1}, |G_{1}|_{M}, with the support variety of M for the finite group |G(𝔽_{p})|_{M}?

Date: October 2000.

1991 Mathematics Subject Classification. Primary 17B55, 20G; Secondary 17B50.

Research of the first author was supported in part by NSF grant DMS-9870035.

Research of the second author was supported in part by NSF grant DMS-9970603.

Research of the third author was supported in part by NSF grant DMS-9800960.

In [LN], the second and third authors showed that for M ∈ mod(G),

dim |G(𝔽_{p})|_{M} ≤ \frac{1}{2} dim |G_{1}|_{M}.

This inequality was proved by developing a new approach to relate the cohomology theories of G(𝔽_{p}) and G_{1}. It follows that (1.1.1) (known as the “Parshall conjecture”) is an immediate corollary of this result. Furthermore, in that paper it is shown that if M ∈ mod(G) and the G-composition factors are p-bounded then the converse to (1.1.1) also holds.

This paper is devoted to providing an affirmative answer to (1.1.2). We show that there are (up to an isogeny) finite maps ϕ and ψ making the diagram in Figure 2 commute. This allows us to identify the support variety |G(𝔽_{p})|_{M} as a subvariety inside the algebraic group G factored out by the action of G(𝔽_{p}). Furthermore, we can also identify |G(𝔽_{p})|_{M} as a subvariety of |G_{1}|_{M}/G(𝔽_{p}). In general the algebraic group G does not act on |G(𝔽_{p})|_{M}. Nevertheless, the existence of these maps allows us to use the rich geometric theory of nilpotent orbits in 𝔤 to obtain results for finite groups. Applications will be presented in the last section of the paper.

[figure: G/G(𝔽_{p}); |G(𝔽_{p})|_{M}; |G_{1}|_{M}/G(𝔽_{p}); ψ; ϕ; log]

Figure 2

1.2. The paper is organized as follows. In Section 2 we define the notion of 𝔽_{p}-expressible nilpotent elements and 𝔽_{p}-rational unipotent elements via elementary abelian subgroups. These varieties behave well with respect to the logarithm map between the unipotent and nilpotent varieties. In Section 3, we show how to map an elementary abelian subgroup in G(𝔽_{p}) into the unipotent variety using results by Testerman. This construction will allow us to define a finite map of the support variety of a module for G(𝔽_{p}) into the algebraic group G factored out by the action of the finite group G(𝔽_{p}). The quotient space G/G(𝔽_{p}) can be thus viewed as “model” for the theory of supports for G(𝔽_{p}). The logarithm map can also be used to construct a finite map from the support variety of a module for G(𝔽_{p}) into 𝔤/G(𝔽_{p}) where 𝔤 = Lie G. The images of these maps are identified at the end of the section by using our definition of 𝔽_{p}-expressibility. In the final section, applications are provided which are both conceptual as well as computational. We indicate how earlier results of Alperin and Mason, and Janiszczak and Jantzen for finite Chevalley groups can be further extended by using the theory of nilpotent orbits in 𝔤. Furthermore, we demonstrate how one can compute the dimension of the support varieties for modules over finite Chevalley groups via earlier computations involving support varieties for infinitesimal group schemes. In particular we compute the complexity of the simple modules for G(𝔽_{p}) for all rank two simple groups. Also, we determine the complexity for all induced modules H^{0}(λ) over GL_{n}(𝔽_{p}) for n ≤ 5 when the highest weights are not large.

1.3. Notation: Throughout this paper let k be an algebraically closed field of characteristic p > 0 which contains the finite field of p elements 𝔽_{p}. Let G be a reductive algebraic kgroup defined and split over the prime field 𝔽_{p} having root system Φ with respect to a maximal split torus T. Let Π be the set of simple roots and B ⊃ T be the Borel subgroup defined by −Φ^{+}. Furthermore, let X(T) be the integral weight lattice obtained from Φ. If α^{∨} = 2α/⟨α, α⟩ is the coroot corresponding to α ∈ Φ, then the set of dominant integral weights is defined by

X(T)_{+} = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ for all α ∈ Π}.

The Weyl group W is the group generated by reflections associated to the root system Φ and let W_{p} = W ⋉ pℤΦ be the affine Weyl group. Furthermore, the set of p-restricted weights is

X_{1}(T) = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ < p for all α ∈ Π}.

The Weyl group W (affine Weyl group W_{p}) acts on X(T) via the “dot action” given by w · λ = w(λ + ρ) − ρ where w ∈ W (w ∈ W_{p}), λ ∈ X(T), and ρ is the half sum of positive roots. For the reductive algebraic group G with root system Φ, the simple modules will be denoted by L(λ) and the induced modules H^{0}(λ) = ind_{B}^{G}λ for λ ∈ X(T)_{+}.

Let H be an affine algebraic group scheme over k and let H_{1} = ker(F r). Here F r : H → H^{(1)} is the Frobenius map. There is a categorical equivalence between restricted Lie(H)-modules and H_{1}-modules.

For a finite k-group scheme K, the cohomology ring R = H^{2•}(K, k) is a commutative, finitely generated k-algebra [FS]. If M ∈ mod(K), the support variety of M, |K|_{M}, is the maximal ideal spectrum of R/J_{M} where J_{M} is the annihilator of R on Ext_{K}^{•}(M, M). The dimension of |K|_{M} can be interpreted as the complexity of the module M [Ben, §5.7]. In this paper we will be interested in the cases when K = G(𝔽_{p}) and K = G_{1}.

2. 𝔽_{p}-expressibility

2.1. Let 𝔤_{ℂ} be a complex semisimple Lie algebra. Fix a Chevalley basis and let 𝔤_{ℤ} be the ℤ-span of the Chevalley basis. Set 𝔤 := k ⊗_{ℤ}𝔤_{ℤ} and 𝔤_{𝔽_{p}} := 𝔽_{p} ⊗_{ℤ}𝔤_{ℤ}. Let G be the algebraic group over k with a split 𝔽_{p}-form G_{𝔽_{p}} such that 𝔤_{𝔽_{p}} is the Lie algebra Lie(G_{𝔽_{p}}) over 𝔽_{p} in 𝔤 = Lie(G).

Definition . An element x ∈ 𝔤 is 𝔽_{p}-expressible if x = ∑_{i=1}^{n} c_{i}x_{i} where c_{i} ∈ k, x_{i} ∈ 𝔤_{𝔽_{p}}, x_{i}^{[p]} = 0, and [x_{i}, x_{j}] = 0 for 1 ≤ i, j ≤ n.

Let N = {x ∈ 𝔤 : x^{[p]} = 0} and N^{𝔽_{p}} be the set of 𝔽_{p}-expressible elements of 𝔤. From the definition given above we have N^{𝔽_{p}} ⊆ N. The algebraic group G acts on the variety N via the adjoint representation on 𝔤. This induces an action of G(𝔽_{p}) on the variety N^{𝔽_{p}}.

2.2. From this point on we assume that p ≥ 2h − 1 (h is the Coxeter number of Φ). By taking a quotient over a central toral subgroup, we can assume that G is semisimple and of adjoint type. Let U = {x ∈ G : x^{p} = 1}. Under the assumption that p ≥ 2h − 1, U is precisely the variety of unipotent elements of G and N is the set of nilpotent elements in 𝔤. For x ∈ N, let

exp(x) = 1 + (ad x) + \frac{(ad x)^{2}}{2!} + \frac{(ad x)^{3}}{3!} + ⋯ + \frac{(ad x)^{p−1}}{(p − 1)!} .  (2.2.1)

Observe that for p ≥ 2h − 1, exp : N → G is well defined because (ad x)^{p} = ad x^{[p]} = 0 by [C, Prop. 5.5.2]. For each 0 ≠ x ∈ N, the map x̂ : 𝔸 → G, defined by x̂(t) = exp(tx), is a homomorphism of algebraic groups. Note that we are naturally identifying 𝔸 with the additive group G_{a}. The result below shows that two elements x and y in N define the same elementary abelian subgroup if and only if x and y are 𝔽_{p}-linearly dependent.

Proposition . Let p ≥ 2h − 1, x, y ∈ N − {0}. Then x̂(𝔽_{p}) = ŷ(𝔽_{p}) as subsets of G, if and only if x = ay for some nonzero a ∈ 𝔽_{p}.

Proof. First suppose that x = ay for some nonzero a ∈ 𝔽_{p}. Then x̂(t) = ây(t) = ŷ(ta) for all t ∈ 𝔽_{p}, thus x̂(𝔽_{p}) = ŷ(𝔽_{p}). Conversely, the equality of sets x̂(𝔽_{p}) = ŷ(𝔽_{p}) implies that x̂(1) = ŷ(a) for some non-zero a ∈ 𝔽_{p}. Therefore, x̂(t) = ŷ(ta) for t ∈ 𝔽_{p}, or equivalently, exp(tx) = exp(tay) for t ∈ 𝔽_{p}. For any ϕ ∈ 𝔤^{∗} and v ∈ 𝔤 fixed, f(t) = ϕ(exp(tx)v − exp(tay)v) is a polynomial in t of degree at most p − 1. The above observation shows that f(𝔽_{p}) = 0. Thus f(t) = 0 for all t ∈ k, otherwise, f would have p distinct roots in 𝔽_{p}.

2.3. We begin by defining an analogous notion of 𝔽_{p}-expressibility for elements in G.

Definition . An element g ∈ G is 𝔽_{p}-expressible if g = exp(t_{1}x_{1}) exp(t_{2}x_{2}) … exp(t_{n}x_{n}) where t_{i} ∈ k, x_{i} ∈ 𝔤_{𝔽_{p}}, x_{i}^{[p]} = 0, and [x_{i}, x_{j}] = 0 for 1 ≤ i, j ≤ n.

Let U^{𝔽_{p}} be the variety of 𝔽_{p}-expressible elements of G. If g ∈ U^{𝔽_{p}} then g^{p} = 1, so U^{𝔽_{p}} ⊆ U. The two notions 𝔽_{p}-expressibility are related using the exponential map. This is shown in the following result.

Proposition . Let p ≥ 2h − 1 and x_{1}, x_{2}, …, x_{n} ∈ N such that [x_{i}, x_{j}] = 0 for all i, j = 1, 2, …, n.

(a) For any t_{1}, t_{2}, …, t_{n} ∈ k,

exp(∑_{i=1}^{n} t_{i}x_{i}) = ∏_{i=1}^{n} exp(t_{i}x_{i}).  (2.3.1)

(b) An element g of G is 𝔽_{p}-expressible if and only if g = exp(x) for some x ∈ N^{𝔽_{p}}. Moreover, the map exp maps N^{𝔽_{p}} onto U^{𝔽_{p}}.

Proof. The statements in (b) follow directly from (a). Since [x_{i}, x_{j}] = 0 for all i, j, then x = t_{1}x_{1} + t_{2}x_{2} + ⋯ + t_{n}x_{n} ∈ N. By using induction n, it suffices to prove the case when n = 2, that is, exp(x + y) = exp(x) exp(y) for x, y ∈ N with [x, y] = 0. By computing out exp(x) exp(y) , it is enough to show that

∑_{i=0}^{n} \frac{(ad x)^{i}}{i!} \frac{(ad y)^{n−i}}{(n − i)!} = 0  (2.3.2)

for all n ≥ p. Since p ≥ 2h − 1, (ad x)^{n} = 0 for all n ≥ p. Therefore, whenever i or n − i is at least p, the corresponding term is zero in (2.3.2). We can assume that both x and y are in 𝔫^{+} and that both x and y are linear combinations of the root vectors of positive roots by conjugating x and y by an element of G. Consider the direct sum decomposition of 𝔤 = ⊕_{m∈ℤ}𝔤_{m} with 𝔤_{m} = ∑_{α∈Φ∪{0},ht(α)=m} 𝔤_{α}. Here 𝔤_{α} is the root space of α ∈ Φ and 𝔤_{0} = 𝔥. Therefore, (ad x)^{i}(ad y)^{n−i}(𝔤_{m}) ⊆ ⊕_{l≥m+n}𝔤_{l}. Since the highest root has height h − 1 and n ≥ p ≥ 2h − 1, it follows that n + m ≥ h. Hence, (ad x)^{i}(ad y)^{n−i}(𝔤_{m}) = 0.

2.4. In this subsection we discuss the inverse of exp. For our constructions, we will use a result of Testerman [T, Thm.4.1].

(2.4.1) For any u ∈ G(𝔽_{p}) such that u^{p} = 1, there is a subgroup K of G defined over 𝔽_{p} such that u ∈ K(𝔽_{p}) and K is isomorphic to either SL_{2} or PSL_{2}, i.e., there is an algebraic group homomorphism ρ : SL_{2} → G which is defined over 𝔽_{p} and with nonzero differential.

Theorem . Assume that p ≥ 2h − 1.

(a) For each u ∈ G(𝔽_{p}) with u^{p} = 1, there exists x ∈ 𝔤_{𝔽_{p}} such that u = exp(x).

(b) The map exp : N → U is a G-equivariant isomorphism of varieties. Moreover, exp : N^{𝔽_{p}} → U^{𝔽_{p}} is a G(𝔽_{p})-equivariant isomorphism of varieties.

(c) For any x, y ∈ N, [x, y] = 1 if and only if exp(x) exp(y) = exp(y) exp(x).

Proof. (a) Let u ∈ G(𝔽_{p}) with u^{p} = 1 and let σ : SL_{2} → G be a homomorphism of algebraic groups with image K (as in (2.4.1)). Let I be the identity matrix and E_{ij} be the matrix with 1 in the (i, j)th-entry and zero everywhere else. Since any unipotent matrix in SL_{2}(𝔽_{p}) is conjugate to I + E_{12} in SL_{2}(𝔽_{p}) and u is in the image of ϕ without loss of generality we may assume u = σ(I + E_{12}).

The homomorphism σ : SL_{2} → G is defined over 𝔽_{p}, so the differential dσ : Lie(K) → Lie(G) sends elements of (𝔰𝔩_{2})_{𝔽_{p}} to elements in 𝔤_{𝔽_{p}}. In fact the differential is the restriction of the homomorphism of the distribution algebras dσ : Dist(SL_{2}) → Dist(G) which restricts to a homomorphism of 𝔽_{p}-forms Dist_{𝔽_{p}}(SL_{2}) → Dist_{𝔽_{p}}(G) (see [Jan1, I, Ch.7]). For a rational SL_{2}-module V, the action of I + E_{12} on V coincides with the action of ∑_{n≥0} E_{12}^{(n)},

where E_{12}^{(n)} is the divided power of E_{12} in Dist_{𝔽_{p}}(SL_{2}).

Set x^{(n)} = dσ(E_{12}^{(n)}) ∈ Dist_{𝔽_{p}}(G). Then x = x^{(1)} is the image of E_{12} ∈ (𝔰𝔩_{2})_{𝔽_{p}} in 𝔤_{𝔽_{p}} under the differential dσ. Furthermore, x is a nilpotent element and exp(tx) = ∑_{i=0}^{p−1} t^{i}(ad x)^{i}/i! ∈ G has the property that exp(tx) ∈ G(𝔽_{p}) for all t ∈ 𝔽_{p}. Now G acts on Lie(G) via conjugation so the rational SL_{2}-module Lie(G) acts through the homomorphism σ. Thus the action of u on Lie(G) is given by the action of ∑_{n≥0} ad x^{(n)}. One can also see from Testerman’s construction that the map σ is actually defined over ℤ_{(p)} (the localization of ℤ at p). Consequently, x is also defined over ℤ_{(p)}. Therefore, x^{(n)} is well-defined and ad(x^{(n)}) = 0 for all n ≥ 2h − 1.

(b) The group G acts on both U and N and the map exp : N → U is an G-equivariant morphism of algebraic varieties. From (a) this map is surjective, and the injectivity holds by Proposition 2.2 (i.e. if u = exp(y) then x = y). From Proposition 2.3, the map exp : N^{𝔽_{p}} → U^{𝔽_{p}} is surjective, and thus a G(𝔽_{p})-equivariant isomorphism of varieties (see [Hum4, 6.20].

(c). One direction follows from the proof of Proposition 2.3. Assume that exp(x) exp(y) = exp(y) exp(x). For any v ∈ 𝔤 = Lie(G) and any linear function f on 𝔤, we consider the function g(t) = f(exp(x) exp(ty) exp(−x)v − exp(ty)v), which is a polynomial in t of degree at most p − 1 for all t ∈ k. Since g(t) = 0 for all t ∈ 𝔽_{p}, we must have g(t) = 0 as polynomial. Hence exp(x) ad(y) = ad(y) exp(x). Similarly, for the polynomial h(t) = f(exp(tx) ad(y)v − ad(y) exp(tx)) in t of degree at most p − 1, we have h(t) = 0 for all t ∈ 𝔽_{p}. Hence, h(t) = 0 and ad(x) ad(y) = ad(y) ad(x). Thus, [x, y] = 0 since 𝔤 is semisimple.

3. Finite morphisms of varieties

3.1. Assume throughout this section that p ≥ 2h − 1. Let E be an elementary abelian subgroup of G(𝔽_{p}) of rank n. We may write E = ⟨y_{1}⟩ × ⟨y_{2}⟩ × ⋯ × ⟨y_{n}⟩ with independent generators y_{1}, y_{2}, …, y_{n}. From Theorem 2.4(a), there exists nilpotent elements of N^{𝔽_{p}}, namely x_{1}, x_{2}, …, x_{n} such that x̂_{i}(1) = y_{i} for i = 1, 2, …, n. We can now define a map σ : 𝔸^{n} → G of algebraic varieties by σ(t_{1}, …, t_{n}) = x̂_{1}(t_{1})x̂_{2}(t_{2}) ⋯ x̂_{n}(t_{n}). Let Ê = σ(𝔸^{n}). The following proposition demonstrates that σ is a homomorphism of algebraic groups so σ(𝔸^{n}) is an abelian algebraic subgroup of G.

Proposition . The map σ is a homomorphism of algebraic groups with the following properties:

(a) σ(𝔸^{n}) is an abelian algebraic subgroup of G.

(b) Im(σ) ⊆ U^{𝔽_{p}} and Im(dσ) ⊆ N^{𝔽_{p}};

(c) σ is a finite map on 𝔸^{n};

Proof. (a) It follows from Theorem 2.4(c) that [x_{i}, x_{j}] = 0. Then Proposition 2.3 shows that σ is a homomorphism of algebraic group and σ(𝔸^{n}) is abelian.

(b) Since Im(σ) is an abelian subgroup of G and x_{1}, x_{2}, …, x_{n} ∈ N^{𝔽_{p}}, it follows that Im(σ) ⊆ U^{𝔽_{p}}. Now consider the differential dσ : Lie(G_{a}^{n}) → Lie(G) with image being an abelian Lie subalgebra which is the k-linear span of the set x_{1}, x_{2}, …, x_{n}. Thus, Im(dσ) ⊆ N^{𝔽_{p}}.

(c) Suppose that Ker(σ) has positive dimension. Then x_{1}, …, x_{n} would be linearly dependent over k. Since x_{i} ∈ 𝔤_{𝔽_{p}} for 1 ≤ i ≤ n, x_{1}, …, x_{n} would have to be linearly dependent over 𝔽_{p}. Without a loss of generality we may assume that x_{1} is an 𝔽_{p}-linear combination of x_{2}, …, x_{n}. This shows that y_{1} = y_{2}^{a_{2}} ⋯ y_{n}^{a_{n}} for some integers a_{2}, …, a_{n}. This contradicts the fact that y_{1}, … y_{n} are independent generators of E.

3.2. Let E = ⟨y_{1}⟩ × ⟨y_{2}⟩ × ⋯ × ⟨y_{n}⟩ be an elementary abelian subgroup of rank n. For M ∈ mod(E), let

|E|_{M} = {(t_{1}, t_{2}, …, t_{n}) : M is not free over ⟨1 + ∑_{i=1}^{n} t_{i}(y_{i} − 1)⟩.}  (3.2.1)

For each g ∈ G(𝔽_{p}), gEg^{−1} = ⟨gy_{1}g^{−1}⟩ × ⟨gy_{2}g^{−1}⟩ × ⋯ × ⟨gy_{n}g^{−1}⟩ is an elementary abelian subgroup of rank n.

Let E′ = ⟨u_{1}⟩ × ⟨u_{2}⟩ × ⋯ × ⟨u_{m}⟩ be another elementary abelian subgroup of G(𝔽_{p}) such that gE′g^{−1} ⊆ E for some g ∈ G(𝔽_{p}). We write gu_{i}g^{−1} = ∏_{j=1}^{n} y_{j}^{r_{ij}}. Then the matrix (r_{ij}) has full rank over the field 𝔽_{p}. Let J = Rad(kE) be the ideal generated by elements y − 1 for all y ∈ E in kE. For each (t_{1}, …, t_{n}) ∈ |E|_{k}, one has

gug^{−1} = 1 + ∑_{i=1}^{m} t_{i}(gu_{i}g^{−1} − 1) = 1 + ∑_{j=1}^{n}(∑_{i=1}^{m} t_{i}r_{ij})(y_{j} − 1) + r_{2}  (3.2.2)

where r_{2} is an element in J^{2}. To prove the relation above one uses the identity (yz − 1) = (y − 1) + (z − 1) + (y − 1)(z − 1). This shows there is a map |E′|_{k} → |E|_{k} defined by the matrix (r_{ij}). However this map depends on the choice of the basis. But when |E|_{k} is identified with J/J^{2}, then the induced map S(g) : |E′|_{k} → |E|_{k} will be independent of the choice of the basis. In particular change of basis of E gives map on |E|_{k} defined by the invertible matrix (r_{ij}) over 𝔽_{p}, which gives the identity map on J/J^{2}.

Let E be the category whose objects are elementary abelian p-subgroups E of the finite group G(𝔽_{p}) and the morphism sets are (E, E′) = {g ∈ G(𝔽_{p}) : gEg^{−1} ⊆ E′}. Let S : E → V be the functor that assigns to each E the cohomological varieties of E where V is the category of all algebraic k-varieties. The morphism g : E → E′ induces a map S(g) : |E|_{k} → |E′|_{k}. Since the category E has only finitely many objects, the direct limit of the functor S exists.

Let { \atop |G(𝔽_{p})|}_{k} = lim_{E} |E|_{k}. This direct limit has the following universal property: for any algebraic variety V and any morphisms ϕ_{E} : |E|_{k} → V of algebraic varieties satisfying the following commutative diagram,

|E|_{k} ⟶^{S(g)} |E′|_{k}
ϕ_{E} ↓  ^{ϕ_{E′}} ↓
V ⟶^{Id} V

for any morphism g : E → E′ in E, there is a unique morphism of algebraic varieties ψ : lim_{E} |E|_{k} → V making the following diagram commute

\begin{matrix}|E|_{k} & ⟶ & lim_{E} |E|_{k} \\ ^{ϕ_{E}} ↓ & & ^{ψ}↓ \\ V & ⟶^{Id} & V \end{matrix}.

Given E = ⟨y_{1}⟩ × ⟨y_{2}⟩ × ⋯ × ⟨y_{n}⟩ and x_{1}, x_{2}, …, x_{n} ∈ N^{𝔽_{p}} such that exp(x_{i}) = y_{i} for i = 1, 2, …, n, define ζ̂_{E} : |E|_{k} → G by

ζ̂_{E}(t_{1}, t_{2}, …, t_{n}) = ∏_{i=1}^{n} exp(t_{i}x_{i}).  (3.2.3)

Now let ζ_{E} : |E|_{k} → G/G(𝔽_{p}) be the map defined by ζ̂_{E} composed with the quotient map π : G → G/G(𝔽_{p}). We will now show that these maps can be extended to { \atop |G(𝔽_{p})|}_{k} = lim_{E} |E|_{k}.

Theorem . Assume p ≥ 2h−1. There are finite surjective maps of varieties ϕ : { \atop |G(𝔽_{p})|}_{k} →

U^{𝔽_{p}}/G(𝔽_{p}) and ψ : { \atop |G(𝔽_{p})|}_{k} → N^{𝔽_{p}}/G(𝔽_{p}) such that ψ = log ∘ϕ.

Proof. The finite group G(𝔽_{p}) acts on the algebraic group G by conjugation. Let E = ⟨y_{1}⟩ × ⟨y_{2}⟩ × ⋯ × ⟨y_{n}⟩ and E′ = ⟨y′_{1}⟩ × ⟨y′_{2}⟩ × ⋯ × ⟨y′_{m}⟩ be elementary abelian subgroups of G(𝔽_{p}) with gE′g^{−1} ⊆ E′ for some g ∈ G(𝔽_{p}). Moreover, let y_{i} = exp(x_{i}) for i = 1, 2, …, n and y′_{j} = exp(x′_{j}) for j = 1, 2, …, m where x_{i}, x′_{j} ∈ N^{𝔽_{p}} for all i, j. Under the adjoint action of g on G, Ad(g)(x′_{i}) = ∑_{j=1}^{n} r_{ij}x_{j}. Then for each (t_{1}, t_{2}, …, t_{m}) ∈ |E′|_{k},

\begin{matrix}π(ζ̂_{E′}(t_{1}, …, t_{m})) = π(g · ζ̂_{E′}(t_{1}, …, t_{m})) = π(g(∏_{i=1}^{m} exp(t_{i}x_{i}))g^{−1}) \\ = π(∏_{i=1}^{n} exp(∑_{i=1}^{n} t_{i}r_{ij}x′_{j})) = ζ_{E}(S(g)(t_{1}, t_{2}, …, t_{n})). \end{matrix}  (3.2.4)

The above identity proves that the diagram

|E′|_{k}  ⟶^{S(g)} |E|_{k}
ζ_{E′} ↓  ^{ζ_{E}} ↓
G/G(𝔽_{p}) ⟶^{id} G/G(𝔽_{p})

commutes for all g ∈ G(𝔽_{p}). The horizontal map on the bottom of the diagram is induced by the conjugation of g and thus is the identity map on G/G(𝔽_{p}).

Hence, by the universal property of the direct limit, there exists a map ϕ : lim_{E} |E|_{k} → G/G(𝔽_{p}). The image of each ζ_{E} is in U^{𝔽_{p}}, thus ϕ : lim_{E} |E|_{k} → U^{𝔽_{p}}/G(𝔽_{p}). The map ϕ is finite because there are only finitely many elementary abelian subgroups and each ζ_{E} is finite. The map ϕ is onto since for each u ∈ U^{𝔽_{p}}, there exists an elementary abelian subgroup E such that ζ̂_{E}(|E|_{k}) contains u. Since log is an G(𝔽_{p})-isomorphism of U^{𝔽_{p}} with N^{𝔽_{p}}, the map log induces an isomorphism of between the quotient spaces U^{𝔽_{p}}/G(𝔽_{p}) and N^{𝔽_{p}}/G(𝔽_{p}). One can define ψ = log ∘ ϕ with ψ : { \atop |G(𝔽_{p})|}_{k} → N^{𝔽_{p}}/G(𝔽_{p}). The finiteness of ψ follows from the finiteness of ϕ.

We remark that, by Quillen’s Stratification Theorem [Ben, Cor 5.6.4], |G(𝔽_{p})|_{k} is homeomorphic to { \atop |G(𝔽_{p})|}_{k} = lim_{E} |E|_{k}. For the sake of convenience, we will define our maps ϕ and ψ explicitly on { \atop |G(𝔽_{p})|}_{k} rather than on |G(𝔽_{p})|_{k}.

3.3. For M ∈ mod(G(𝔽_{p})), let { \atop |G(𝔽_{p})|}_{M} = lim_{E} |E|_{M}. Observe that { \atop |G(𝔽_{p})|}_{M} is a closed subvariety of { \atop |G(𝔽_{p})|}_{k}. By a well-known result of Avrunin and Scott [AS], the restriction maps induce an inseparable isogeny δ : { \atop |G(𝔽_{p})|}_{M} → |G(𝔽_{p})|_{M}. A fundamental question is to describe the image of { \atop |G(𝔽_{p})|}_{M} under the maps ϕ and ψ. We begin this investigation by proving a result which compares restrictions of nilpotent (and unipotent) elements on a rational G-module M.

Proposition . Let M be a finite k-dimensional vector space and x, y ∈ End_{k}(M) be two commuting elements such that x ≠ 0 and x^{p} = y^{p−1} = 0. Then 1 + x and 1 + x + y both are invertible and that M is free upon restriction to the algebra k⟨1 + x⟩ implies that M is free upon restriction to the algebra k⟨1 + x + y⟩.

Proof. Since x^{p} = 0 and 1 + x ≠ 0, the element 1 + x is invertible of order p. There exists an isomorphism of algebras k⟨1 + x⟩ ≅ k[T]/⟨T^{p}⟩ where x ↦ T. Since M is free as a module over k⟨1 + x⟩, M is free over k[T]/⟨T^{p}⟩ with basis {v_{1}, …, v_{r}}. Therefore, {v_{i}, T v_{i}, …, T^{p−1}v_{i} | i = 1, …, r} is a k-basis of M. Under this basis, the rp × rp matrix representing 1 + x consists of r Jordan blocks of size p × p of the form

(\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 1 & 1 & 0 & ⋯ & 0 \\ 0 & 1 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \end{matrix}) .

The matrix above can be viewed as the regular unipotent class in GL_{p}(k). Therefore, the centralizer of x in GL_{rp}(k) has the following block form

(\begin{matrix}A_{11} & A_{12} & A_{13} & ⋯ & A_{1r} \\ A_{21} & A_{22} & A_{23} & ⋯ & A_{2r} \\ A_{31} & A_{23} & A_{33} & ⋯ & A_{3r} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ A_{r1} & A_{r2} & A_{r3} & ⋯ & A_{rr} \end{matrix}) with A_{ij} = (\begin{matrix}λ_{1} & 0 & 0 & ⋯ & 0 \\ λ_{2} & λ_{1} & 0 & ⋯ & 0 \\ λ_{3} & λ_{2} & λ_{1} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ λ_{p} & λ_{p−1} & λ_{p−2} & ⋯ & λ_{1} \end{matrix}) .

Since 1 + y commutes with 1 + x, by taking a conjugate we can assume that y is lower triangular and blocks A_{ij} of y have the properties the A_{ij} = 0 for i < j and the diagonal of A_{ii} are all zero. Since y^{p−1} = 0, we have A_{ii}^{p−1} = 0. This forces the subdiagonal of A_{ii} (entries right below the diagonal) to be zero. Therefore, y is a lower triangular matrix with the diagonal and subdiagonal being zero. One now compute the matrix x + y = (B_{ij}) with triangular p × p-block matrices B_{ij} such that B_{ij} = 0 for i ≤ j and B_{ii} has zeros on the diagonal and 1 on the lower subdiagonal. By matrix computation one sees that (x + y)^{p−1} = (C_{ij}) is a triangular matrix having C_{ij} being the p × p-block matrices with the following properties.

C_{ii} = (\begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & 0 & 0 & ⋯ & 0 \end{matrix}) , C_{(i+1),i} = (\begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ ∗ & 0 & 0 & ⋯ & 0 \end{matrix}) .

This shows that dim_{k}((x + y)^{p−1}M) ≥ r = dim_{k}(M)/p. But, (x + y)^{p} = 0 so dim_{k}(x + y)^{p−1}M ≤ dim_{k} M/p = r. Therefore, equality holds and M is free as a module for the algebra k⟨1 + x + y⟩.

3.4. Let σ : SL_{2} → G be a homomorphism of algebraic group schemes. The differential dσ : Dist(SL_{2}) → Dist(G) is homomorphism of associative algebras. Note that the distribution algebra of the unipotent subgroup 1 + kE_{21} of SL_{2} has a basis E_{21}^{(n)} satisfying E_{21}^{(n)}E_{21}^{(m)} = \binom{n+m}{n}E_{21}^{(n+m)}. Let x ∈ 𝔤 such that x = dτ(E_{12}). We set x^{(n)} = dτ(E_{12}^{(n)}) ∈ Dist(G).

For any rational G-module M, the action of exp(tx) is the same as ∑_{n≥0} t^{n}x^{(n)}. Note that x^{(n)}x^{(m)} = \binom{n+m}{n}x^{(n+m)}. For n = ∑_{i} r_{i}p^{i} with 0 ≤ r_{i} < p, we have x^{(n)} = ∏_{i}(x^{(p^{i})})^{r_{i}}/r_{i}!.

Let β_{s} = ∑_{i=0}^{s} r_{i}p^{i}. By considering the images in End_{k}(M), we have

exp(tx) = ∑_{s≥0} ∑_{0≤r_{0},...,r_{s}<p} x^{r_{0}}(x^{(p)})^{r_{1}} · (x^{(p^{s})})^{r_{s}}t^{β_{s}}  (3.4.1)

3.5. We now apply Proposition 3.3 to compare the restriction of M ∈ mod(G) to ⟨x⟩, ⟨u⟩ and ⟨y⟩. This allows us to describe the image of { \atop |G(𝔽_{p})|}_{M} in U^{𝔽_{p}} (resp. N^{𝔽_{p}}) under ϕ (resp. ψ).

Theorem . Let G be a connected reductive algebraic group with p ≥ 2h − 1 and let M ∈ mod(G).

(a) {u ∈ U^{𝔽_{p}} : u^{p} = 1, M|_{⟨x⟩} is not free} ⊆ exp({x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free});

(b) ψ({ \atop |G(𝔽_{p})|}_{M}) ⊆ \{x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free\}/G(𝔽_{p}) ∪ \{0\}.

Proof. (a) Let x ∈ N^{𝔽_{p}} with M|_{⟨x⟩} projective. We have x = ∑_{i=1}^{n} t_{i}x_{i} with x_{1}, x_{2}, …, x_{n} in 𝔤_{𝔽_{p}} and [x_{i}, x_{j}] = 0. Set u = exp(x) = ∏_{i=1}^{n} exp(t_{i}x_{i}) ∈ U^{𝔽_{p}}. It suffices to show that M|_{⟨u⟩} is free.

Seitz [Se] showed there is a homomorphism of algebraic groups σ : SL_{2} → G such that the differential dσ : 𝔰𝔩_{2} → 𝔤 is nonzero. In fact dσ(E_{21}) = x and σ(I + E_{21}) = u. Let 𝔫′ = kE_{21} be the Lie algebra of the unipotent subgroup of SL_{2} corresponding to the upper triangular matrices. Since M is free when restricted to 𝔫′, and M is an SL_{2}-module, thus it is projective as u(𝔰𝔩_{2})-module by [FP2, (1.2) Thm.]. Therefore, M is projective as an SL_{2}(𝔽_{p})-module by [LN, Cor. 3.5]. Note that σ is an embedding of the cyclic group ⟨1 + E_{21}⟩ to G(𝔽_{p}) with image ⟨u⟩. Hence, M|_{⟨u⟩} is also projective.

(b) Let x = ∑_{i=1}^{n} t_{i}x_{i} with x_{i} ∈ 𝔤_{𝔽_{p}} being linearly independent and x_{i}^{[p]} = 0, [x_{i}, x_{j}] = 0. Set y_{i} = exp(x_{i}) ∈ G(𝔽_{p}). We show that the projectivity of M|_{⟨x⟩} implies the projectivity of M|_{⟨y⟩} for y = 1+∑_{i} t_{i}(y_{i}−1) ∈ kG(𝔽_{p}). We start by using the homomorphism σ : SL_{2} → G defined by x such that dσ(E_{21}) = x. Then M becomes a rational SL_{2}-module. Moreover, M|_{⟨x⟩} is projective implies that M projective as kE_{21}-module. Thus, M is projective for the first Frobenius kernel of the Borel subgroup of lower triangular matrices in 𝔰𝔩_{2}. Thus the module M has a filtration M ⊂ M^{i} ⊂ M^{i+1} ⊂ ⋯ with subquotients M^{i}/M^{i+1} isomorphic to the “baby” Verma modules Z_{1}(λ_{i}) for (SL_{2})_{1}T′ [Jan1, II, 9.4]. Here T′ is the maximal torus of SL_{2}. We can assume that λ_{i} ≥ λ_{i+1}. There exists a basis of M from the filtrations consisting of T′-weight vectors. The matrix of x is exactly its Jordan canonical form. Each x^{(j)} for j > 1 commutes with x and has lower triangular matrix with zeros on the lower subdiagonal entries of all principal p × p-blocks (determined by Z_{1}(λ)). Thus acting on the vector space M with respect to the aforementioned basis, the matrix of

exp(tx) − 1 − tx = ∑_{j>1} t^{j}x^{(j)}

(for all t ∈ k) has all zeros on the lower subdiagonal of each principal p × p-blocks.

We now claim that with respect to the aforementioned basis of M, the matrix of each x_{i}^{(p^{r})} corresponding to the action on M, has all zeros on the lower subdiagonal of each principal p × p-block matrix. Let us fix a principal p × p-block and let λ_{i,r} be the lower subdiagonal entry of x_{i}^{(p^{r})}. Consider the image of exp(tx) in End_{k}(M) for all 0 ≠ t ∈ k.

Then the lower subdiagonal entry of the matrix exp(tx) − 1 − tx is

∑_{i=1}^{n} ∑_{r≥1} t^{p^{r}}t_{i}^{p^{r}}λ_{i,r} = ∑_{r≥1}(∑_{i=1}^{n} t_{i}^{p^{r}}λ_{i,r})t^{p^{r}} = 0

for all t ∈ k (k is infinite). Thus ∑_{i=1}^{n} t_{i}^{p^{r}}λ_{i,r} = 0. Let 𝔢 be the abelian Lie subalgebra of 𝔤 spanned by x_{1}, x_{2}, …, x_{n}. The support variety |𝔢|_{M} of M as 𝔢-module is a closed subvariety of 𝔢. We have shown that the coordinates t_{i} of each x = ∑_{i} t_{i}x_{i} in the open set 𝔢 \ |𝔢|_{M}, satisfy the equations ∑_{i=1}^{n} t_{i}^{p^{r}}λ_{i,r} = 0. But 𝔢 is irreducible and therefore all coefficients λ_{i,r} have to be zero. This proves the claim.

In End_{k}(M) we write

y = 1 + x + ∑_{i=1}^{n} t_{i} ∑_{j>1} x_{i}^{(j)}.  (3.5.1)

In the second summation, except for the terms x_{i}^{(p^{r})} with r ≥ 1, all other terms are product of at least two triangular nilpotent matrices, and therefore have lower subdiagonal entries being zeros. Now the claim also implies the exceptional term also have all zeros on lower subdiagonals. Hence, the matrix determined by the summation in (3.5.1) has zero subdiagonal and therefore M|_{⟨y⟩} is projective by Proposition 3.3. We have proven that for each elementary abelian p-subgroup E, ψ(|E|_{M}) ⊆ {x ∈ N^{𝔽_{p}} : M|_{⟨x⟩}is not free}, thus (b) holds.

We remark that when p ≥ 2h−1 part (b) of the theorem also implies Parshall’s conjecture (i.e. a G-module that is projective for the restricted Lie algebra 𝔤 is also projective for the finite group G(𝔽_{p})). We should also add that the inclusions given in Theorem 3.5(b) can be proper. If M = St^{(1)}, the Frobenius twist of the Steinberg module, one has { \atop |G(𝔽_{p})|}_{M} = \{0\}, but M|_{⟨x⟩} is not free for all x ∈ N.

3.6. Let x ∈ N. There is a homomorphism of σ : SL_{2} → G such that x = dσ(E_{21}). As in Section 3.4, x^{(n)} ∈ Dist(G) is defined with fixed σ. The following proposition will provide some motivation for the results in the following two subsections.

Proposition . Let M ∈ mod(G). Suppose x satisfies (x^{(p^{i})})^{p−1} = 0 in End_{k}(M) for i ≥ 1. Then for each fixed t ∈ k^{∗}, M|_{<exp(tx)>} is free if and only if M|_{⟨x⟩} is free.

Proof. Note that in End_{k}(M) we can write exp(tx) = 1 + tx + y such that tx and y satisfies the condition of Proposition 3.3. Since t ∈ k^{∗}, M|_{⟨x⟩} is free implies that M|_{<1+tx>} is free. Therefore, by Proposition 3.3, M|_{<exp(tx)>} is free. Conversely, one can substitute tx + y for x and −y for y in Proposition 3.3. Then the same argument shows that if M|_{<exp(tx)>} is free then M|_{⟨x⟩} is free.

For G = SL_{2} and x is a positive root vector, the condition of the proposition above is satisfied for any G-module with all weights strictly less than p(p − 1). Therefore, under these conditions M is projective over the Sylow p-subgroup of SL_{2}(𝔽_{p}) if and only if it is projective over the nilpotent radical of the Borel subalgebra. Consequently, M is projective as SL_{2}(𝔽_{p})-module if and only if M is projective for the Lie algebra 𝔰𝔩_{2}. This result was proved in [LN, Thm. 4.3] for SL_{2} by classifying all such indecomposable modules of G in the bounded category defined by weight condition then computing their dimensions. The aforementioned proposition provides a much more natural (and enlightening) proof of this result.

3.7. We still assume that p ≥ 2h−1. For the sake of convenience, let us fix some notations for the rest of this section. Let x = t_{1}x_{1} +t_{2}x_{2} +⋯+t_{n}x_{n} ∈ N^{𝔽_{p}} such that x_{1}, …, x_{n} ∈ 𝔤_{𝔽_{p}} satisfy [x_{i}, x_{j}] = 0 and x_{i}^{[p]} = 0. Set u = ∏_{i=1}^{n} exp(t_{i}x_{i}) ∈ U^{𝔽_{p}} and y = 1 + ∑_{i=1}^{n} t_{i}(y_{i} − 1) where y_{i} = exp(x_{i}) for i = 1, 2, …, n. We can now prove that under certain nilpotency conditions for the module M, the projectivity of M when restricted to ⟨x⟩, ⟨u⟩, and ⟨y⟩ are equivalent.

Theorem . Let M ∈ mod(G) such that (x_{i}^{(p^{s})})^{p−1} = 0 in End_{k} M for all s ≥ 1. Then the following three conditions are equivalent.

(a) M|_{⟨x⟩} is not free where x ∈ N^{𝔽_{p}}.

(b) M|_{⟨u⟩} is not free where u ∈ U^{𝔽_{p}}.

(c) M|_{⟨y⟩} is not free where y ∈ kG(𝔽_{p}).

Proof. (a) ⇔ (c) First observe that in End_{k}(M)

y − 1 = ∑_{i=1}^{n} t_{i}x_{i} + ∑_{i=1}^{n} ∑_{j=2}^{p−1} t_{i}x_{i}^{(j)} = x + ∑_{i=1}^{n} ∑_{j=2}^{p−1} t_{i}x_{i}^{(j)}.  (3.7.1)

Since x_{i}^{(j)} are all commuting elements in End_{k}(M), the assumption shows that (x_{i}^{(j)})^{p−1} = 0 for all j ≥ 2. Therefore, by applying Proposition 3.3 repeatedly by adding the terms t_{i}x_{i}^{(j)} it follows that if M|_{⟨x⟩} is free, then M|_{⟨y⟩} is free. Conversely, assume that M|_{⟨y⟩} is free, then (y − 1)^{p} = 0 in End_{k}(M) and by adding −t_{i}x_{i}^{(j)} for j ≥ 2 it follows by Proposition 3.3 that M|_{⟨x⟩} is free.

(a) ⇔ (b) An expansion of u in End_{k}(M) gives

u = 1 + ∑_{i=1}^{n} t_{i}x_{i} + ∑_{j_{1},j_{2},...,j_{n}} ∏_{i=1}^{n} t_{i}^{j_{i}} ∏_{i=1}^{n} x_{i}^{(j_{i})} = x + ∑_{j_{1},j_{2},...,j_{n}} ∏_{i=1}^{n} t_{i}^{j_{i}} ∏_{i=1}^{n} x_{i}^{(j_{i})}. (3.7.2)

The summation in the second part is over all (j_{1}, j_{2}, …, j_{n}) ∈ ℕ^{n} such that at least one component is at least 2. Since all factors in (3.7.2) commute in End_{k}(M), we have

(∏_{i=1}^{n} t_{i}^{j_{i}} ∏_{i=1}^{n} x_{i}^{(j_{i})})^{p−1} = 0  (3.7.3)

by the assumption. Then the same argument as above shows that M|_{⟨x⟩} is free if and only if M|_{⟨u⟩} is free.

3.8. Let G be a simple algebraic group and let C_{p} be the full subcategory of G-modules with all weights λ satisfying the condition

∑_{i=1}^{l} ∑_{j=1}^{l}⟨λ, α_{i}^{∨}⟩b_{ij} < p(p − 1)/2  (3.8.1)

where the matrix (b_{ij}) is the inverse matrix of the Cartan matrix (⟨α_{i}, α_{j}^{∨}⟩). For simple algebraic groups let c be the value given in the table below.

\begin{matrix}type & A_{l} & B_{l} & C_{l} & D_{l} & E_{6} & E_{7} & E_{8} & F_{4} & G_{2} \\ c & ((l + 1)/2)^{2} & l(l + 1)/2 & l^{2}/2 & l(l − 1)/2 & 16 & 27 & 46 & 11 & 4 \end{matrix}

The condition ⟨λ, α_{0}^{∨}⟩ < p(p−1)/c, where α_{0} is the highest short root of the indecomposable root system, will insure that λ satisfies the condition (3.8.1). The computation of c is provided in the Appendix in §5. For M ∈ C_{p}, we can identify the images of ϕ and ψ.

Theorem . Let G be a connected reductive algebraic group with p ≥ 2h − 1 and M ∈ C_{p}.

(a) {u ∈ U^{𝔽_{p}} : u^{p} = 1, M|_{⟨u⟩} is not free} = exp({x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free});

(b) ϕ({ \atop |G(𝔽_{p})|}_{M}) = \{u ∈ U^{𝔽_{p}} : u^{p} = 1, M|_{⟨u⟩} is not free\}/G(𝔽_{p}) ∪ \{0\};

(c) ψ({ \atop |G(𝔽_{p})|}_{M}) = \{x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free\}/G(𝔽_{p}) ∪ \{0\}.

Proof. Let M be a G-module in the category C_{p}. According to Theorem 3.5, it suffices to prove that

{u ∈ U^{𝔽_{p}} : u^{p} = 1, M|_{⟨u⟩} is not free} ⊇ exp({x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free}); (3.8.2)

ϕ({ \atop |G(𝔽_{p})|}_{M}) ⊇ \{u ∈ U^{𝔽_{p}} : u^{p} = 1, M|_{⟨u⟩} is not free\}/G(𝔽_{p}) ∪ \{0\};  (3.8.3)

ψ({ \atop |G(𝔽_{p})|}_{M}) ⊇ \{x ∈ N^{𝔽_{p}} : x^{[p]} = 0, M|_{⟨x⟩} is not free\}/G(𝔽_{p}) ∪ \{0\}. (3.8.4)

Observe that (3.8.2) follows once that statements (3.8.1) and (3.8.3) are established. Let x ∈ N^{𝔽_{p}} such that M|_{⟨x⟩} is not free. We have x = t_{1}x_{1} + t_{2}x_{2} + ⋯ + t_{n}x_{n} where x_{i} ∈ 𝔤_{𝔽_{p}}, and [x_{i}, x_{j}] = 0. Set y_{i} = exp(x_{i}) ∈ U^{𝔽_{p}} for 1 ≤ i ≤ n and y = 1 + ∑_{i=1}^{n} t_{i}(y_{i} − 1) with E = ⟨y_{1}⟩ × ⟨y_{2}⟩ × ⋯ × ⟨y_{n}⟩.

Let 1 ≤ i ≤ n. According to [Se], there exists a homomorphism of algebraic groups σ : SL_{2} → G of algebraic groups such that dσ(E_{12}) = x_{i}. Let T′ be the maximal torus of SL_{2} and T be the maximal torus in G. Since M is a G-module, we can conjugate x_{i} and assume that x_{i} ∈ 𝔫^{+} and the image of T′ is contained in T. Since Hom(T′, T) = Hom(G_{m}, T) = Hom_{ℤ}(X(T), ℤ), the homomorphism σ : T′ → T determines an element t^{∨} ∈ Hom_{ℤ}(X(T), ℤ). On any G-module, T′ acts on the weight space of T -weight λ by the T′-weight ⟨λ, t^{∨}⟩ = t^{∨}(λ).

Let Π = {α_{1}, α_{2}, …, α_{l}} be the simple roots. By Dynkin’s theorem [C, Prop 5.6.6], the possible values for a_{i} = t^{∨}(α_{i}) are 0, 1, 2. For any weight λ ∈ X(T), write λ = ∑_{i=1}^{l} λ_{i}ω_{i} where ω_{1}, ω_{2}, …, ω_{l} are the fundamental weights. If one expresses ω_{i} = ∑_{j=1}^{l} b_{ij}α_{j}, then the matrix (b_{ij}) is the inverse of the Cartan matrix of the root system. Consequently,

t^{∨}(λ) = ∑_{i=1}^{l} λ_{i}t^{∨}(ω_{i}).  (3.8.5)

Now let λ be a weight of the G-module M. We can assume that λ is dominant via conjugation by an element in the Weyl group. Then λ_{i} ≥ 0 are integers. By a direct computation, one will get

t^{∨}(λ) ≤ ∑_{i,j=1}^{l}⟨λ, α_{i}^{∨}⟩2b_{ij} < p(p − 1).  (3.8.6)

Thus for any M ∈ C_{p}, any dominant weight λ of M satisfies t^{∨}(λ) < p(p − 1). Note that each x_{i}^{(p^{s})} has T′-weight 2p^{s}. If v ∈ M has T -weight λ then (x_{i}^{(p^{s})})^{p−1}v will have T′-weight 2(p − 1)p^{s} + t^{∨}(λ) ≥ p(p − 1), which is impossible. Thus, (x_{i}^{(p^{s})})^{p−1}v = 0 for each weight vector v of dominant weight λ. Using conjugation by elements in the Weyl group, we have (x_{i}^{(p^{r})})^{p−1}.M = 0. Hence, by Theorem 3.7, we have M|_{⟨y⟩} is not free so (3.8.3) holds. Furthermore, M|_{⟨u⟩} is not free also by Theorem 3.7, so (3.8.1) is verified.

3.9. The following corollary provides a precise relationship between |G(𝔽_{p})|_{M} and |G_{1}|_{M} when M is a module in C_{p}.

Corollary . Let G be a connected reductive algebraic group with p ≥ 2h − 1. If M is in C_{p}, then ψ({ \atop |G(𝔽_{p})|}_{M}) = (N^{𝔽_{p}} ∩ |G_{1}|_{M})/G(𝔽_{p}). Furthermore,

dim |G(𝔽_{p})|_{M} = dim(N^{𝔽_{p}} ∩ |G_{1}|_{M}).

Proof. The first statement follows from Theorem 3.8(c) and the equality of dimensions holds because ψ is a finite map.

For M in C_{p} with p ≥ 2h − 1, the preceding result provides a complete answer to the question (1.1.2) raised by Parshall. This relationship will be used in Section 4 to compute the complexity of modules for the finite group G(𝔽_{p}). We should also add that for M ∈ mod(G), |G_{1}|_{M} is a closed G-invariant subvariety of N and thus the union of (finitely many) nilpotent orbits. An interesting and useful problem would be to determine the dimension of the intersection of a nilpotent orbit with N^{𝔽_{p}}.

4. Applications

4.1. Alperin and Mason [AM2] proved that for a finite Chevalley group G(𝔽_{q}) with underlying root system of type A, D or E, a simple module S is the Steinberg module if and only if S upon restriction to a root subgroup X_{α}(t) is projective. Our first application extends this result to all root systems in the case when q = p and p > c(h − 1) where c is defined in Section 3.8.

Theorem . Let G be a reductive algebraic group with p > c(h − 1). Moreover, let S be a simple module for G(𝔽_{p}). Then S is the Steinberg module if and only if S_{X_{α}(t)} is projective for some long root α.

Proof. The assumption on p insures that all simple G(𝔽_{p})-modules are in the category C_{p}. If S is the Steinberg module then S is projective over G(𝔽_{p}) and thus projective over X_{α}(t) for any long root α. On the other hand, suppose that S is not the Steinberg module. Then S is not projective over G_{1}. Since |G_{1}|_{S} is G-invariant and non-zero, it follows that G·x_{β} ⊆ |G_{1}|_{S} for β the highest long root. More generally we have x_{α} ∈ |G_{1}|_{S} ∩ N^{𝔽_{p}} ⊆ ψ(|G(𝔽_{p})|_{S}) for all α which are long roots. Hence, by Theorem 3.8, we have that S cannot be projective over X_{α}(t).

We should remark that the assumption on p in Theorem 4.1 can be substantially improved if one uses [LN, Thm. 4.5]. This result shows that if ⟨(p−1)ρ, α_{0}^{∨}⟩ < p(p−1) for the highest short root α_{0}, then an simple module is projective if and only if is projective for a root subgroup. The restriction on the prime arises from the result in 3.8 and 3.9.

4.2. Janiszczak and Jantzen [JJ] proved that there are no periodic non-projective simple modules for a Chevalley group G(𝔽_{q}) when rank(G) ≥ 2. The following result demonstrates how the map ψ and the geometry of nilpotent orbits can be used to extend this result in several different directions.

Theorem . Let G be a simple algebraic group with p ≥ 2h − 1 and let M ∈ C_{p} be nonprojective as a G(𝔽_{p}) module.

(a) If M is indecomposable in mod(G(𝔽_{p})) and rank(G) ≥ 2, then dim |G(𝔽_{p})|_{M} ≥ 2.

(b) If G has underlying root system of type A_{n} then dim |G(𝔽_{p})|_{M} ≥ n.

Proof. (a) Let M be a non-projective indecomposable module over G(𝔽_{p}). Since M is non-projective over G(𝔽_{p}), it follows that M is non-projective over G_{1} by [LN, Cor. 3.5]. Therefore, G · x_{β} ⊆ |G_{1}|_{M} for β the highest long root vector, and x_{α} ∈ |G_{1}|_{M} ∩ N^{𝔽_{p}} ⊆ ψ(|G(𝔽_{p})|_{M}) for all α long. There are at least two such long root vectors because rank G ≥ 2. It follows that |G(𝔽_{p})|_{M} contains two lines. This cannot happen when M is indecomposable and dim |G(𝔽_{p})|_{M} ≤ 1 because the projective variety of |G(𝔽_{p})|_{M} is connected.

(b) As in part (a) we have G·x_{β} ⊆ |G_{1}|_{M} for β the highest long root vector. However, for type A_{n} the minimal orbit is Richardson. More precisely, we have G · x_{β} = G · 𝔲_{J} where 𝔲_{J} is the nilpotent radical of the parabolic subalgebra obtained by taking the Levi subalgebra of type A_{n−1} inside of A_{n}. The unipotent Lie algebra u_{J} is abelian of dimension equal to n. Furthermore, 𝔲_{J} ⊆ |G_{1}|_{M} ∩ N^{𝔽_{p}} ⊆ ψ(|G(𝔽_{p})|_{M}). Hence, dim |G(𝔽_{p})|_{M} ≥ n.

It is interesting to note that for reductive groups with root systems other than of type A_{n}, the minimal orbit is rigid [CM, Lem. 7.1.5], and thus cannot be Richardson.

4.3. Rank 2 groups: For λ ∈ X(T), let Φ_{λ} = {α ∈ Φ^{+} : ⟨λ + ρ, α^{∨}⟩ ∈ pℤ}. When p is good, there exists a subset J of simple roots and w ∈ W such that w(Φ_{λ}) = Φ_{J}. Let 𝔲_{J} be the nilpotent radical of the parabolic algebra with Levi subalgebra corresponding to J. In [NPV, (6.6.1) Cor.] the support varieties of simple modules for G_{1} were calculated for rank 2 groups. The next result shows how to use this information to calculate the dimensions of the support varieties for simple modules over G(𝔽_{p}) when G has rank two.

Theorem . Let G be a simple algebraic group of rank 2 with underlying root system A_{2} (resp. B_{2} or G_{2}) with p > 3 (resp. p > 7, or p > 19).

(a) If λ = (p − 1)ρ then dim |G(𝔽_{p})|_{L(λ)} = 0.

(b) If λ ≠ (p − 1)ρ then dim |G(𝔽_{p})|_{L(λ)} = \begin{cases}2 & for A_{2} \\ 3 & for B_{2} and G_{2}. \end{cases}

Proof. (a) Observe that λ = (p − 1)ρ if and only if Φ_{λ} = Φ. Furthermore, L((p − 1)ρ) is the Steinberg module which is projective over G(𝔽_{p}), thus |G(𝔽_{p})|_{L(λ)} = {0}.

(b) We may now assume that Φ_{λ} is a subroot system of Φ which is empty or of type A_{1}. Let J be the set consisting of one simple root. Then by [NPV, (6.6.1) Cor.], 𝔲_{J} is abelian and in both cases 𝔲_{J} ⊆ |G_{1}|_{L(λ)} ∩ N^{𝔽_{p}} ⊆ ψ(|G(𝔽_{p})|_{L(λ)}). But, dim 𝔲_{J} is equal to the p-rank of G(𝔽_{p}) in all cases. This is 2 for G of type A_{2} and 3 for G of types B_{2} and G_{2}.

4.4. Induced modules: In [NPV, (6.2.1) Thm.], it was shown that for p good and λ ∈ X(T)_{+}, if J ⊂ Π with w(Φ_{λ}) = Φ_{J}, then |G_{1}|_{H^{0}(λ)} = G · 𝔲_{J}. The theorem below allows us to use this computation to determine the dimension of support varieties for H^{0}(λ) over G(𝔽_{p}).

Theorem . Let G be a connected reductive algebraic group and p ≥ 2h − 1. Suppose λ ∈ X(T)_{+} such that ⟨λ, α_{0}^{∨}⟩ < p(p − 1)/c.

(a) If J′ is a subset of the simple roots such that 𝔲_{J′} is abelian and 𝔲_{J′} ⊆ |G_{1}|_{H^{0}(λ)} then

dim_{k} 𝔲_{J′} ≤ dim |G(𝔽_{p})|_{H^{0}(λ)} ≤ min\{p -rank G(𝔽_{p}), \frac{1}{2} dim |G_{1}|_{H^{0}(λ)}\}.

(b) If w(Φ_{λ}) = Φ_{J} for some w ∈ W and 𝔲_{J} is abelian then dim_{k} |G(𝔽_{p})|_{H^{0}(λ)} = dim_{k} 𝔲_{J}.

Proof. (a) Suppose that 𝔲_{J′} ⊆ |G_{1}|_{H^{0}(λ)} where 𝔲_{J′} is abelian. Then 𝔲_{J′} ⊆ N^{𝔽_{p}}∩|G_{1}|_{H^{0}(λ)} ⊆ ψ(|G(𝔽_{p})|_{H^{0}(λ)}). Hence, dim_{k} 𝔲_{J′} ≤ dim |G(𝔽_{p})|_{H^{0}(λ)}. The other inequality follows by [LN, Thm. 3.4(b)] and the fact that the dimension of the support variety of a module is always bounded by the dimension of the support variety of the trivial module which is precisely the p-rank of the group.

(b) Under the hypotheses we have |G_{1}|_{H^{0}(λ)} = G · 𝔲_{J} where 𝔲_{J} is abelian. Therefore by part (a), dim_{k} 𝔲_{J} ≤ dim |G(𝔽_{p})|_{H^{0}(λ)}. On the other hand, by [LN, Thm. 3.4(b)], we have

dim |G(𝔽_{p})|_{H^{0}(λ)} ≤ \frac{1}{2} dim |G_{1}|_{H^{0}(λ)} = dim_{k} 𝔲_{J}.

The last equality uses the dimension formula (see [C, 5.10.2] and [Hum4, 6.7]).

4.5. Computations for GL_{n+1}. We now concentrate on the group G = GL_{n+1} with underlying root system of type A_{n}.The p-rank of GL_{n+1}(𝔽_{p}) is given for n ≤ 6 in the table below.

group  p-rank
GL_{2}(𝔽_{p}) 1
GL_{3}(𝔽_{p}) 2
GL_{4}(𝔽_{p}) 4
GL_{5}(𝔽_{p}) 6
GL_{6}(𝔽_{p}) 9
GL_{7}(𝔽_{p}) 12

By using Theorem 4.4, one can compute the dimension of the support varieties for the induced modules H^{0}(λ) with ⟨λ, α_{0}^{∨}⟩ < p(p − 1)/c for n ≤ 4 completely and for n = 5 and n = 6 in all but two cases. These computation are given in the tables below. In these two exceptional cases the unipotent radical corresponding to the stabilizer Φ_{λ} is not abelian and an elementary abelian of largest dimension is not contained in the unipotent radical.

\begin{matrix}GL_{2}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 2 \\ A_{1} & 0 \end{matrix}  \begin{matrix}GL_{3}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 2 \\ A_{1} & 2 \\ A_{2} & 0 \end{matrix}

\begin{matrix}GL_{4}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 4 \\ A_{1} & 4 \\ A_{1} × A_{1} & 4 \\ A_{2} & 3 \\ A_{3} & 0 \end{matrix}  \begin{matrix}GL_{5}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 6 \\ A_{1} & 6 \\ A_{1} × A_{1} & 6 \\ A_{2} & 6 \\ A_{2} × A_{1} & 6 \\ A_{3} & 4 \\ A_{4} & 0 \end{matrix}

\begin{matrix}GL_{6}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 9 \\ A_{1} & 9 \\ A_{1} × A_{1} & 9 \\ A_{2} & 9 \\ A_{1} × A_{1} × A_{1} & ? \\ A_{2} × A_{1} & 9 \\ A_{3} & ? \\ A_{2} × A_{2} & 9 \\ A_{3} × A_{1} & 8 \\ A_{4} & 5 \\ A_{5} & 0 \end{matrix}  \begin{matrix}GL_{7}(𝔽_{p}) \\ Φ_{λ} & dim |G(𝔽_{p})|_{H^{0}(λ)} \\ ∅ & 12 \\ A_{1} & 12 \\ A_{1} × A_{1} & 12 \\ A_{2} & 12 \\ A_{1} × A_{1} × A_{1} & 12 \\ A_{2} × A_{1} & 12 \\ A_{3} & 12 \\ A_{2} × A_{1} × A_{1} & 12 \\ A_{2} × A_{2} & 12 \\ A_{3} × A_{1} & 12 \\ A_{4} & ? \\ A_{3} × A_{2} & 12 \\ A_{4} × A_{1} & 10 \\ A_{5} & 6 \\ A_{6} & 0 \end{matrix}

5. Appendix

5.1. For a dominant integral weight λ ∈ X(T)_{+}, express λ = ∑_{i=1}^{l} λ_{i}ω_{i} in terms of the fundamental weights {ω_{1}, ω_{2}, …, ω_{l}}. We will give an estimate for t^{∨}(ω_{i}) for each root system using the table in [Hum1, 13.2] and the fact that 0 ≤ t^{∨}(α_{i}) ≤ 2 (see 3.8).

In this appendix we list all a_{i} = t^{∨}(ω_{i}) = ∑_{j=1}^{l} 2b_{ij} for t^{∨}(α_{i}) = 2. Hence, a_{i} is twice the sum of the coefficients of ω_{i} when expressed in terms of simple roots. Set n_{i} = ⟨ω_{i}, α_{0}^{∨}⟩ and b_{i} = a_{i}/n_{i}. Then ⟨λ, α_{0}^{∨}⟩ = ∑_{i=1}^{l} n_{i}λ_{i} and t^{∨}(λ) ≤ ∑_{i=1}^{l} b_{i}n_{i}λ_{i} ≤ c⟨λ, α_{0}^{∨}⟩, where c = max_{i}{b_{i}}. The Coxeter number h is given by the formula h = 1 + ∑_{i=1}^{l} n_{i}.

\begin{matrix}Type & (n_{1}, …, n_{l}) & h & c \\ A_{l} & (1, 1, …, 1) & l + 1 & ((l + 1)/2)^{2} \\ B_{l} & (2, 2, …, 2, 1) & 2l & l(l + 1)/2 \\ C_{l} & (1, 2, …, 2) & 2l & l^{2}/2 \\ D_{l} & (1, 2, …, 2, 1, 1) & 2l − 2 & l(l − 1)/2 \\ E_{6} & (1, 2, 2, 3, 2, 1) & 12 & 16 \\ E_{7} & (2, 2, 3, 4, 3, 2, 1) & 18 & 27 \\ E_{8} & (2, 3, 4, 6, 5, 4, 3, 2) & 30 & 46 \\ F_{4} & (2, 4, 3, 2) & 12 & 11 \\ G_{2} & (2, 3) & 6 & 4 \end{matrix}

5.2. The quantity a_{i} is given below for all indecomposable root systems. Type A. For i = 1, 2, …, l,

a_{i} = \frac{2}{l + 1}[(l − i + 1) ∑_{j=1}^{i} j + i ∑_{j=i+1}^{l}(l − j + 1)] = i(l − i + 1)  (5.2.1)
c = (l + 1)^{2}/4.  (5.2.2)

Type B. For i = 1, 2, …, l − 1,

a_{i} = ∑_{j=1}^{i} 2j + i ∑_{j=i+1}^{l} 2 = i(2l − i + 1) ≤ l(l + 1),  (5.2.3)
a_{l} = \frac{1}{2} ∑_{j=1}^{l} 2j = l(l + 1)/2,  (5.2.4)
c = l(l + 1)/2.  (5.2.5)

Type C. For i = 1, 2, …, l − 1,

a_{i} = ∑_{j=1}^{i−1} 2j + i(2 + ⋯ + 2 + 1) = i(2l − i),  (5.2.6)
c = \begin{cases}l^{2}/2 & if (l > 3) \\ 5 & if i = 3 \end{cases} .  (5.2.7)

Type D. For i = 1, 2, …, l − 2,

a_{i} = ∑_{j=1}^{i} 2j + i(2 + ⋯ + 2 + 1 + 1) = i(2l − 3 − i),  (5.2.8)
a_{l−1} = a_{l} ≤ (1 + 2 + ⋯ + (l − 2)) + 2l − 2 = l(l − 1)/2  (5.2.9)
c = l(l − 1)/2 (l ≥ 4).  (5.2.10)

Exceptional groups.

Index Type E_{6} Type E_{7} Type E_{8} Type F_{4} Type G_{2}
i  a_{i} b_{i} a_{i} b_{i} a_{i} b_{i} a_{i} b_{i} a_{i} b_{i}
1 16 16 34 17 92 46 22 11 6 3
2 22 11 49 25 136 46 42 11 10 4
3 30 15 66 22 182 26 30 10
4 42 12 96 24 270 45 16 8
5 30 15 75 25 220 44
6 16 16 52 26 168 42
7  27 27 114 38
8  58 29

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

   E-mail address: jfc@sloth.math.uga.edu

Department of Mathematics, Kansas State University, Manhattan, KS 66506

   E-mail address: zlin@math.ksu.edu

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322

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