Proceedings of the Summer School on “Representation theory of Algebras, Finite and Reductive Groups” “Babeş-Bolyai” University Cluj, 1998; pp. 101-126.

IV. FINITE REDUCTIVE GROUPS

MEINOLF GECK

1. Introduction

The aim of this chapter is to give an introduction to the theory of finite groups of Lie type (or, “finite reductive groups”) and their ordinary representations. This is best done in the framework of reductive algebraic groups, and this is also the way we begin to develop the theory here. Our intention is to explain some of the basic background material from algebraic geometry and the theory of algebraic groups, and then to show how this can be used to obtain information about representations. We also discuss the theory of Hecke algebras in the simplest possible case.

From this description, it should be clear that we cannot go really far into the representation theory of finite groups of Lie type and that, usually, proofs have to be omitted. But it is hoped that these notes may serve as a kind of first “appetizer” before one goes on to consult more comprehensive treatments like those mentioned in our guide to further reading in Section 7.

Before we begin to develop the theory, let us describe the place of groups of Lie type in finite group theory and discuss their original construction.

1.A. Finite simple groups. The classification of finite simple groups (see Gorenstein et al. [13]) states that if G is a finite group which has no non-trivial normal subgroup then G is isomorphic to one of the following groups.

• a cyclic group of prime order,

• an alternating group 𝔄_{n} for n ≥ 5,

• a simple group of Lie type,

• one of 26 sporadic simple groups. Among the simple groups of Lie type we find, for example, “classical” groups like PSL_{n}(q), PSp_{2n}(q), PSO_{2n+1}(q) etc., and “exceptional” groups like G_{2}(q). Here, q is a prime power and the group has a “natural” representation over a finite field with q elements. Among the sporadic simple groups we find, for example, the 5

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Mathieu groups (discovered in the last century) and groups like Janko’s group J_{1} (the first sporadic simple group discovered after the Mathieu groups in 1966) and M, the Fischer–Griess “Monster”.

The above classification result was completed around 1981. One of the important problems in the theory of finite groups is the determination of the irreducible representations of the finite simple groups. We naturally have two major subcases of this problem.

Ordinary representations. This is the case where we study representations over a field of characteristic 0. One of the aims here is to find the character tables of all finite simple groups. (A basic reference about the character theory of finite groups is Isaacs [15]). For example, the Cambridge ATLAS [7] contains the individual character tables of PSL_{2}(q), where q = 4, 7, 9, … (all these tables are available, in electronic form, in the library of GAP [24]). What we would like is to find a uniform description of these tables, where q occurs a parameter. The result in this case can be found in [9, (15.9)] (for example), but let us mention right away that the character tables for PSL_{n}(q) (any n, q) and for most other types of groups are not yet completely determined. The tables which are known are electronically accessible through the CHEVIE system [12].

Modular representations. This is the case where we study representations over a field of characteristic ℓ > 0, where ℓ divides the order of the group G. Here, there are a number of different branches, concentrating on general properties of blocks and decomposition numbers, on character formulas for simple modules or on relations with the ℓ-local structure of G. It should be pointed out that the state of knowledge here is much less complete than in the case of ordinary representations. In these talks, we will not go into any more detail but give references for further reading in Section 7.

1.B. Chevalley’s construction. It was Chevalley who found, in 1955, a uniform construction of all groups of Lie type (finite or infinite); see Carter [4] and Steinberg [28]. The starting point is the classification of finite-dimensional simple Lie algebras over ℂ. This classification, due to Cartan and Killing, shows that every such Lie algebra is uniquely determined by one of the diagrams in Table 1. For example, the simple Lie algebra corresponding to the diagram of type A_{n−1} is 𝔤 = 𝔰𝔩_{n}(ℂ) = {A ∈ M_{n}(ℂ) | trace(A) = 0}, with Lie product given by [A, B] = AB − BA.

Let 𝔤 be a finite-dimensional simple Lie algebra over ℂ. Then 𝔤 contains a Cartan subalgebra 𝔥 ⊆ 𝔤, which is a commutative subalgebra (i.e., we have [x, y] = 0 for all x, y ∈ 𝔥) and it is maximal with this property. For any λ ∈ Hom(𝔥, ℂ), we define

Table 1. Dynkin diagrams of simple Lie algebras

[figure: E_{7}; 1; 3; 4; 2; 5; 6; 7; E_{8}; 1; 3; 4; 2; 5; 6; 7; 8; G_{2}; 1; >; 2; F_{4}; 1; 2; >; 3; 4; E_{6}; 1; 3; 4; 2; 5; 6; D_{n}; n≥4; 1; 2; 3; 4; n; C_{n}; n≥3; 1; >; 2; 3; n; A_{n}; n≥1; 1; 2; 3; n; B_{n}; n≥2; 1; <; 2; 3; n]

𝔤_{λ} = {x ∈ 𝔤 | [h, x] = λ(h)x for all h ∈ 𝔥}. Then we have a Cartan decomposition

𝔤 = 𝔥 ⊕ ⨁_{α∈Φ} 𝔤_{α}

where Φ := {0 ≠ λ ∈ Hom(𝔥, ℂ) | 𝔤_{λ} ≠ 0} is the set of “roots” of 𝔤. Moreover, we have 𝔥 = 𝔤_{0} and dim 𝔤_{α} = 1 for all α ∈ Φ.

Chevalley has shown that 𝔤 has a basis B = {h_{1}, …, h_{n}; e_{α}, α ∈ Φ}, where n is the number of nodes in the corresponding diagram of Table 1, h_{1}, …, h_{n} form a basis of 𝔥 and 0 ≠ e_{α} ∈ 𝔤_{α}, such that the structure constants of 𝔤 with respect to B lie in ℤ, i.e., if we write [b, b′] = ∑_{b′′∈B} z_{b,b′,b′′}b′′ (for b, b′ ∈ B) then z_{b,b′,b′′} ∈ ℤ for all b, b′, b′′. Hence,

𝔤_{ℤ} := ⟨h_{1}, …, h_{n}; e_{α}, α ∈ Φ⟩_{ℤ}

is a Lie algebra over ℤ. So, if we take any field k, then 𝔤_{k} := k ⊗_{ℤ} 𝔤_{ℤ} is a Lie algebra over k (not necessarily simple any more, but this doesn’t matter).

A corresponding group of Lie type is constructed using certain automorphisms of 𝔤. For any x ∈ 𝔤, we have a ℂ-linear map ad(x): 𝔤 → 𝔤, y ↦ [x, y]. If ad(x) is a nilpotent endomorphism, then

exp(ad(x)) = id_{𝔤} + ad(x) + \frac{1}{2} ad(x)^{2} + \frac{1}{6} ad(x)^{3} + ⋯

is a well-defined automorphism of 𝔤. One can show that if we take x = te_{α} for some α ∈ Φ and t ∈ ℂ, then t ad(e_{α}) is nilpotent and, moreover, the entries of the matrix of exp(t ad(e_{α})) with respect to the basis B can be written as polynomials in t with coefficients in ℤ. It is this property which makes it possible to define, for any α ∈ Φ and t ∈ k (where k is any field), a k-linear automorphism x_{α}(t) of 𝔤_{k}, by transferring the above exponential construction to k. Then we define

G(k) := ⟨x_{α}(t) | α ∈ Φ, t ∈ k⟩ ⊆ Aut_{k}(𝔤),

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and call this the Chevalley group over k associated with 𝔤. The group G(k) always has a trivial center, and G(k) is simple except possibly if k is a finite field of very small order. For example, for type A_{n−1}, the above construction yields the group PSL_{n}(k). In order to obtain related groups like SL_{n}(k), one has to work with more general types of 𝔤-modules and construct linear automorphisms of them; for more details see Steinberg’s lectures [28]. Exercise 1.1 Let 𝔤 = 𝔰𝔩_{2}(ℂ). Then a Chevalley basis is given by

h = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] , e = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , f = [ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ];

note that dim 𝔤 = 3. We have 𝔥 = ⟨h⟩ and 𝔤 = 𝔥⊕⟨e⟩⊕⟨f⟩ is the Cartan decomposition of 𝔤; check that [h, e] = 2e and [h, f] = −2f. Let k be a field. By the above description, G(k) is realized as a subgroup of Aut_{k}(𝔤) and, hence, by taking matrices, as a subgroup of GL_{3}(k). First let k = ℂ and construct, for any t ∈ k, explicitly the matrices of exp(t ad(e)) and exp(t ad(f)) with respect to the basis {h, e, f} of 𝔤. Then find similar matrices for any field k. Check that we have G(k) ⊆ SL_{3}(k). Show that G(k) ≅ 𝔖_{3} if k = 𝔽_{2}.

1.C. Algebraic groups and Frobenius maps. Let k be a field and consider the group G = SL_{n}(k). As already mentioned in (1.B), this group can be obtained by a variation of the above construction from a Lie algebra of type A_{n−1}. The point that we wish to make here is that G is defined by a polynomial equation, namely by det(A) = ∑_{σ∈𝔖_{n}} sign(σ)a_{1,σ(1)} ⋯ a_{n,σ(n)} = 1 where A = (a_{ij}).

Thus, if k is algebraically closed, then G is both a group and an affine variety, and we can use results from algebraic geometry to study properties of G. But how do we get back our finite groups? For this purpose, we work over \overline{𝔽}_{q}, an algebraic closure of the finite field with q elements. Then we can define a map F: G → G by (a_{ij}) ↦ (a_{ij}^{q}); it is called a Frobenius map. Its fixed point set G^{F} = {g ∈ G | F (g) = g} is still a group; furthermore, we have F ((a_{ij})) = (a_{ij}^{q}) = (a_{ij}) if and only if a_{ij} ∈ 𝔽_{q}, i.e., we have in fact G^{F} = SL_{n}(q).

This is the point of view under which we will study our finite groups of Lie type: we begin with algebraic groups over an algebraic closure of 𝔽_{q} and then take fixed point sets under a Frobenius maps F. (This point of view was worked out systematically by Steinberg [29].) Exercise 1.2 Let k = \overline{𝔽}_{q}, where q is odd. Let G(k) ⊆ SL_{3}(k) be as in Exercise 1.1 and consider the Frobenius map F on SL_{3}(k) given by (a_{ij}) ↦ (a_{ij}^{q}). Show that G(k) is invariant under F and that the order of G(k)^{F} is q(q^{2} −1); moreover, G(k) is simple but G(k)^{F} is not.

2. Zariski topology

In this section, we explain the most basic concepts from the theory of affine varieties. (For further reading see, for example, Mumford’s “red book” [23].) Throughout, k denotes an algebraically closed field.

2.A. The operators V and I. Let k[X_{1}, …, X_{n}] be the polynomial ring over k in n variables. Given f ∈ k[X_{1}, …, X_{n}] and v = (v_{1}, …, v_{n}) ∈ k^{n}, we simply write f(v) = f(v_{1}, …, v_{n}). For any subset I ⊆ k[X_{1}, …, X_{n}], we set

V(I) := {v ∈ k^{n} | f(v) = 0 for all f ∈ I}.

Then we have ∅ = V({1}), k^{n} = V({0}), ⋂_{i∈I} V(I_{i}) = V(⋃_{i∈I} I_{i}) (for any index set I) and V(I_{1}) ∪ ⋯ ∪ V(I_{m}) = V({f_{1} ⋯ f_{m} | f_{i} ∈ I_{i}}). In other words, the sets

{V(I) | I ⊆ k[X_{1}, …, X_{n}]}

form the closed sets of a topology on k^{n}, which is called the Zariski topology.

Let ⟨I⟩ be the ideal spanned by I ⊆ k[X_{1}, …, X_{n}]. Then we clearly have V(I) = V(⟨I⟩). Now Hilbert’s Basissatz states that every ideal of k[X_{1}, …, X_{n}] is generated by finitely many elements. It follows that every closed subset of k^{n} is defined by the vanishing of finitely many polynomials.

Conversely, we can also start with a subset V ⊆ k^{n} and set

I(V) := {f ∈ k[X_{1}, …, X_{n}] | f(v) = 0 for all v ∈ V }.

Then I(V) is an ideal in k[X_{1}, …, X_{n}], and we have V(I(V)) = \overline{V}, where we denote by \overline{V} the Zariski closure of V.

Thus, V and I define correspondences between closed subsets of k^{n} (“geometric picture”) and ideals in k[X_{1}, …, X_{n}] (“algebraic picture”). The basic result that controls these correspondences is:

2.B. Hilbert’s Nullstellensatz. In its weak form, it says that if I ⊆ k[X_{1}, …, X_{n}] spans a proper ideal of k[X_{1}, …, X_{n}] then we have V(I) ≠ ∅, i.e., there exists some v ∈ k^{n} such that f(v) = 0 for all f ∈ I. Now the weak forms implies the strong form, which states that, for any ideal I ⊆ k[X_{1}, …, X_{n}], we have

I(V(I)) = \sqrt{I} := \{f ∈ k[X_{1}, …, X_{n}] | f^{m} ∈ I for some m ≥ 1\}.

Any ideal I such that I = \sqrt{I} is called a radical ideal. It is easy to check that I(V) is a radical ideal, for any subset V ⊆ k^{n}. Thus, we see that V and I define bijective correspondences between closed subsets of k^{n} and radical ideals of k[X_{1}, …, X_{n}]. This also means that every statement about the topology of k^{n} (under the Zariski topology)

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should have a counterpart in a statement about ideals of k[X_{1}, …, X_{n}], and vice versa. Indeed, the map

v ∈ k^{n} ↦ m_{v} := {f ∈ k[X_{1}, …, X_{n}] | f(v) = 0} ⊆ k[X_{1}, …, X_{n}]

defines a bijection between k^{n} and the set of maximal ideals of k[X_{1}, …, X_{n}]. Under this bijection, a closed subset V ⊆ k^{n} corresponds to the collection of maximal ideals of k[X_{1}, …, X_{n}] which contain I(V). Thus, the Zariski topology on k^{n} is completely determined in terms of k[X_{1}, …, X_{n}].

2.C. Affine varieties. Let V ⊆ k^{n} be a closed subset of k^{n}. Then V itself is a topological space, with the induced topology. The k-algebra k[V] := k[X_{1}, …, X_{n}]/I(V) is called the coordinate ring of V. It is a finitely generated algebra which is reduced (i.e., 0 is the only nilpotent element; this holds since I(V) is a radical ideal). We may view the elements of k[V] as functions on V. Then the pair (V, k[V]) has the following formal properties (see [30, (1.1)]).

(1) V is a set and k[V] is a finitely generated k-algebra of functions V → k.

(2) For v ∈ V, define the “evaluation map” λ_{v}: k[V] → k by λ_{v}(f) = f(v). Then v ↦ λ_{v} is a bijection between V and the set of k-algebra homomorphisms k[V] → k.

We can see as before that the set V together with its topology is completely determined by k[V]. Any pair (V, A) consisting of a set V and a finitely generated k-algebra A of functions V → k satisfying (1) and (2) will be called an affine variety (over k). We have just seen that if V ⊆ k^{n} is a closed subset then (V, k[V]) is an affine variety in this sense. The point of this “formal” definition is that now we are independent of any embedding of V as a closed subset of some affine space k^{n}.

The advantage of this formal definition can also be seen in the following constructions. Let (V, A) and (W, B) be two affine varieties.

A map φ: V → W is called a morphism if f ∘ φ ∈ A for all f ∈ B. In this case, φ^{∗}: B → A, f ↦ f ∘ φ, is a k-algebra homomorphism. Conversely, assume that ψ: B → A is any k-algebra homomorphism. Then there is a unique morphism φ: V → W such that φ^{∗} = ψ. (Indeed, given v ∈ V, the map λ_{v} ∘ ψ is a k-algebra homomorphism B → k, hence it is equal to λ_{w} for a unique w ∈ W. We set φ(v) = w.)

The pair (V × W, A ⊗_{k} B) is an affine variety, called the product of (V, A) and (W, B). Note that the elements of A ⊗_{k} B can be naturally regarded as functions on V × W, via (f ⊗ g)(v, w) = f(v)g(w). Exercise 2.1 (a) Let (V, A) be an affine variety. Show that there exists some n ≥ 1 and a k-algebra homomorphism ψ: k[X_{1}, …, X_{n}] → A such that the following holds. Let I be the kernel of ψ and V′ = V(I) ⊆ k^{n}. Then (V, A) is isomorphic to the affine

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variety (V′, k[V′]), and the corresponding k-algebra homomorphism between k[V′] and A is the canonical map induced by ψ.

(b) Let V ⊆ k^{n} and W ⊆ k^{m} be closed subsets. Show that a map φ: V → W is a morphism if and only if there exist polynomials φ_{1}, …, φ_{m} ∈ k[X_{1}, …, X_{n}] such that φ(v) = (φ_{1}(v), …, φ_{m}(v)) for all v ∈ V.

Exercise 2.2 Let (V, A) be an affine variety and 0 ≠ f ∈ A. Consider the open subset V_{f} := {v ∈ V | f(v) ≠ 0} ⊆ V. Show that (V_{f}, A_{f}) is an affine variety, where A_{f} denotes the localization of A in f. (Hint: The map v ↦ (v, 1/f(v)) defines a bijection between V_{f} and the closed set Ṽ_{f} := {(v, t) ∈ V × k | f(v)t = 1}; determine the coordinate ring of Ṽ_{f}.)

2.D. Linear algebraic groups. An affine algebraic group is an affine variety (G, A) which is also a group, in such a way that these two structures are compatible, i.e, if μ: G × G → G and ι: G → G denote the maps defining the multiplication and inversion in G, respectively, then we require that μ, ι are morphisms of affine varieties. Example 2.3 (a) Let SL_{n}(k) = {A ∈ M_{n}(k) | det(A) = 1} ⊆ M_{n}(k). This is clearly a closed subset of M_{n}(k). The multiplication and inversion are given in terms of polynomial equations, hence Exercise 2.1(b) shows that SL_{n}(k) is an affine algebraic group. Can you determine its coordinate ring?

(b) Let G ⊆ SL_{n}(k) be a subgroup which is closed. Then G also is an affine algebraic group. In fact, it is known that every affine algebraic group is isomorphic to a closed subgroup of SL_{n}(k) for some n (see [30, §1.9]). Therefore, affine algebraic groups are also called linear algebraic groups.

(c) Consider k as an additive group. Then 𝔾_{a} = (k, k[X]) (where X is an indeterminate) is an affine algebraic group. We can find it as a closed subgroup of SL_{2}(k), as follows:

k ≅ \{[ \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ] | t ∈ k\} ⊆ SL_{2}(k).

(d) Consider k^{×} as a multiplicative group. Then 𝔾_{m} = (k^{×}, k[X, X^{−1}]) (where X is an indeterminate) is an affine algebraic group. We can find it as a closed subgroup of SL_{2}(k), as follows:

k^{×} ≅ \{[ \begin{matrix}x & 0 \\ 0 & y \end{matrix} ] | x, y ∈ k, xy = 1\} ⊆ SL_{2}(k).

(e) The pair (GL_{n}(k), k[X_{ij} | 1 ≤ i, j ≤ n]_{det}) is a linear algebraic group. (This follows using Exercise 2.2.)

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Finally, let 𝔤 be a simple Lie algebra as in (1.B) and G(k) ⊆ Aut_{k}(𝔤_{k}) be the corresponding Chevalley group over k. We have seen that G(k) is generated by elements x_{α}(t) for α ∈ Φ and t ∈ k. Now, from the construction, it is not difficult to check that {x_{α}(t) | t ∈ k} (for fixed α ∈ Φ) is a closed subgroup of Aut_{k}(𝔤_{k}) isomorphic to 𝔾_{a}. Thus, G(k) is generated by closed subgroups of Aut_{k}(𝔤_{k}). By a general result, this implies that G(k) itself is a closed subgroup of Aut_{k}(𝔤_{k}) and, hence, a linear algebraic group. Chevalley (Séminaire sur la classification des groupes de Lie algébriques, 1957– 1958) has shown that every linear algebraic group which is simple must be isomorphic to some G(k) as above.

3. Some general properties of algebraic groups

Throughout this section, we let G be a linear algebraic group. (We omit the coordinate ring from the notation, if there is no danger of confusion.) We now discuss some general properties of G. Proofs are usually omitted; they can be found in any text book on linear algebraic groups, for example Steinberg [30] or Springer [26].

3.A. Connected components. As a topological space, G is the disjoint union of its connected components. We shall usually denote by G^{∘} the connected component which contains the identity element of G. Then G^{∘} is a closed normal subgroup of finite index in G, and the cosets of G^{∘} are the connected components of G. Here are some simple criteria which make sure that G = G^{∘}:

(1) G has no proper closed normal subgroup of finite index, or

(2) k[G] is an integral domain, or

(3) as a group, G is generated by closed connected subgroups. Using these criteria, we can see that all the groups in Example 2.3 are connected. Indeed, every proper normal subgroup of SL_{n}(k) is contained in the center and, hence, is finite. So SL_{n}(k) cannot have any normal subgroup of finite index, and we can apply (1). The groups 𝔾_{a} and 𝔾_{m} are connected since their coordinate rings are integral domains, so we can apply (2). The same argument also works for GL_{n}(k). The Chevalley groups G(k) ⊆ Aut_{k}(𝔤_{k}) are connected since they are generated by closed connected subgroups (see the discussion at the end of (2.D)), and so (3) applies.

Note also that any finite group is a linear algebraic group (why?), but it is never connected unless it is the trivial group. Exercise 3.1 Assume that G is connected. Show that every normal subgroup of G which is finite must be contained in the center of G.

3.B. The Jordan decomposition of elements. By Example 2.3(b), we can embed G as a closed subgroup into SL_{n}(k) for some n. An element g ∈ G is called unipotent,

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if all eigenvalues of g (as an element of SL_{n}(k)) are 1. We say that g is semisimple if g is diagonalizable. Then every g ∈ G can be written uniquely in the form g = su = us where s ∈ G is semisimple and u ∈ G is unipotent. This is called the Jordan decomposition of g. It is a non-trivial fact that these notions do not depend on the chosen embedding G ⊆ SL_{n}(k). Exercise 3.2 Assume that k = \overline{𝔽}_{p} for some prime power p. Show that every element g ∈ G has finite order; moreover, g is unipotent or semisimple according to whether the order of g is a power of p or coprime to p. Finally, the Jordan decomposition of g is just the factorization of g into its p-part and its p′-part.

A linear algebraic group H is called diagonalizable if H is abelian and consists of semisimple elements. Or, equivalently, there exists an embedding of H as a closed subgroup of some SL_{n}(k) such that H consists of diagonal matrices. A torus is a connected, diagonalizable group; it is isomorphic to a direct product of a finite number of copies of 𝔾_{m}. A diagonalizable group is always a direct product of a torus and a finite group. The main result about diagonalizable groups is the Rigidity Theorem, which states that if H is a closed diagonalizable subgroup of a linear algebraic group G then

N_{G}(H)^{∘} = C_{G}(H)^{∘} and [N_{G}(H) : C_{G}(H)] < ∞.

3.C. Borel subgroups. From now on, we assume that G is a connected linear algebraic group.

A Borel subgroup of G is a maximal closed connected solvable subgroup of G. A typical example is the subgroup of all upper (or all lower) triangular matrices in G = GL_{n}(k). The Borel subgroups and their properties are of central importance in the structure theory of linear algebraic groups. Here are some of their basic properties.

(1) All Borel subgroups of G are conjugate in G and any Borel subgroup is equal to its normalizer.

(2) Every element of G lies in a Borel subgroup.

(3) Any two Borel subgroups have a maximal torus of G in commun. Let B ⊆ G be a fixed Borel subgroup. The group theoretical structure of B is described as follows. We have a semidirect product decomposition B = U ⋊ T, where U is a connected unipotent group and T is a torus, which is also a maximal torus in G and we have C_{G}(T) ⊆ B. Moreover, all maximal tori of B are conjugate in B. As a consequence of these results, we have the following double coset decomposition of G,

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called the Bruhat decomposition: let T ⊆ B be a maximal torus; then

G = ∐_{n} BnB

where n runs over a set of representatives of the cosets of C_{G}(T) in N_{G}(T). Note that, by the Rigidity Theorem of (3.B), [N_{G}(T) : C_{G}(T)] is finite, and so there are only finitely many double cosets BnB.

3.D. Connected reductive groups. The group G contains a unique maximal closed connected normal solvable subgroup; this is denoted R(G) and called the radical of G. Similairly, G contains a unique maximal closed connected normal unipotent subgroup; this is denoted R_{u}(G) and called the uniptent radical of G. By Kolchin’s Theorem, a unipotent group is nilpotent and, hence, solvable. This implies that we always have R_{u}(G) ⊆ R(G). Since R(G) is solvable, it is its own Borel subgroup, and so the results described in (3.C) show that we have R(G) = R_{u}(G) ⋊ H, where H is a torus.

The group G is called semisimple if R(G) = {1}, and G is called reductive if R_{u}(G) = {1}. Since R_{u}(G) ⊆ R(G), we see that a semisimple group is always reductive. For example, SL_{n}(k) is semisimple and GL_{n}(k) is reductive but not semisimple. The linear algebraic groups that we are interested in are the connected reductive groups. It is this class of groups for which we have a nice general structure theory. On the other hand, this class is sufficiently large so as to contain all the algebraic groups from which we want to construct finite groups.

We now explain some properties of a connected reductive group G.

First of all, we see that R(G) is a torus. By the Rigidity Theorem of (3.B), we conclude that C_{G}(R(G))^{∘} = N_{G}(R(G))^{∘} = G, and so R(G) is contained in the center Z(G). We have in fact R(G) = Z(G)^{∘}. Furthermore, the commutator subgroup [G, G] is a closed connected normal subgroup of G, and we have

G = [G, G] Z(G)^{∘} and |[G, G] ∩ Z(G)^{∘}| < ∞.

This also shows that [G, G] is semisimple. Moreover, any semisimple group contains only a finite number of minimal closed connected normal subgroups and it is the central product of these subgroups. Thus, we have

[G, G] = G_{1} ⋯ G_{r} where [G_{i}, G_{j}] = 1 for i ≠ j and G_{i}/Z(G_{i}) is simple for all i.

The Bruhat decomposition now takes a simpler form. Indeed, for any maximal torus T ⊆ G, we have C_{G}(T) = T. Then W = N_{G}(T)/T is a finite group, which is called the Weyl group of G with respect to T. Note that all maximal tori are conjugate in G, so

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W is a finite group which is uniquely determined up to isomorphism by G. Choosing T ⊆ B we have

G = ∐_{w∈W} BẇB

where ẇ denotes a representative of w in N_{G}(T). A basic result says that W is a Coxeter group, i.e., we have a presentation of the form

W = ⟨s ∈ S | s^{2} = 1 for s ∈ S and (st)^{m(s,t)} = 1 for s ≠ t⟩,

where m(s, t) denotes the order st; we have m(s, t) ∈ {2, 3, 4, 6}. (For the general theory of Coxeter groups, see Humphreys [14].)

As a standard example, one should always keep in mind the group G = GL_{n}(k). We have G = [G, G]Z(G), where [G, G] = SL_{n}(k) and Z(G) consists of all scalar matrices in G. The subgroup B of all upper triangular matrices in G is a Borel subgroup; we have B = U ⋊T, where U consists of all upper triangular matrices with 1 along the diagonal, and T is the maximal torus of all diagonal matrices in G. The subgroup N_{G}(T) consists of all monomial matrices in G and N_{G}(T)/T is canonically isomorphic to the symmetric group 𝔖_{n}. Thus, 𝔖_{n} is the Weyl group of GL_{n}(k).

4. Frobenius maps

Having introduced algebraic groups over any algebraically closed field k, we now turn to the problem of recovering finite groups via the process of taking fixed point sets under a “Frobenius map”. For this purpose, we work from now on with the field k = \overline{𝔽}_{q}, where q is prime power.

Recall that in the example where G = SL_{n}(k) we can define a “Frobenius map” by F: (a_{ij}) ↦ (a_{ij}^{q}) and then the fixed point set of G under F is the finite group SL_{n}(q). How can we generalize this to any linear algebraic group G? One way would be to embed G as a closed subset into some SL_{n}(k) and then to require that G is invariant under the map F defined above. But this would depend on the choice of an embedding. We will proceed by describing a more intrinsic definition (see, for example, Srinivasan [27, Chap. II]).

4.A. 𝔽_{q}-rational structues. Let (V, A) be any affine variety of k. We say that V is defined over 𝔽_{q} or that V has an 𝔽_{q}-rational structure if there exists a k-algebra homomorphism γ: A → A such that the following conditions hold:

(1) We have γ(A) ⊆ A^{q} and the map γ: A → A^{q} is a bijection.

(2) For each a ∈ A there exists some m ≥ 1 such that γ^{m}(a) = a^{q^{m}}.

112 MEINOLF GECK

As explained in (2.C), the algebra homomorphism γ defines a morphism F: V → V such that F^{∗} = γ. This morphism F is called the Frobenius map of V corresponding to the given 𝔽_{q}-rational structure. The fixed point set of V under F will be denoted by

V^{F} := {v ∈ V | F (v) = v}.

Let us check that this definition really yields what we wanted.

Given a homomorphism γ: A → A satisfying (1) and (2), we set A_{0} := {a ∈ A | γ(a) = a^{q}}. Then it can be checked that A_{0} is a finitely generated 𝔽_{q}-subalgebra of A such that the canonical map k ⊗_{𝔽_{q}} A_{0} → A given by multiplication is an isomorphism. This means that there exists some n ≥ 1 and a surjective homomorphism of 𝔽_{q}- algebras from 𝔽_{q}[X_{1}, …, X_{n}] onto A_{0} such that, if we let I_{0} be the kernel of this map, then the canonically extended map k[X_{1}, …, X_{n}] → A has kernel I := k ⊗_{𝔽_{q}} I_{0}. Hence, identifying V with V(I_{0}) ⊆ k^{n} as in Exercise 2.1(a), we see that V is defined by a set of polynomials with coefficients in 𝔽_{q}, and the map F is the restriction to V of the map k^{n} → k^{n}, (v_{1}, …, v_{n}) ↦ (v_{1}^{q}, …, v_{n}^{q}). This discussion shows that

V^{F} = {(v_{1}, …, v_{n}) ∈ V | v_{i}^{q} = v_{i} for all i} = V ∩ 𝔽_{q}^{n};

in particular, V^{F} is a finite set. Moreover, we see that F is a bijective morphism of affine varieties.

Conversely, let V ⊆ k^{n} be a closed subset and assume that I(V) is generated by polynomials with coefficients in 𝔽_{q}. Then we have k[V] = k ⊗_{𝔽_{q}} A_{0}, where A_{0} is a finitely generated 𝔽_{q}-algebra, and it can be checked that the map γ: k[V] → k[V], t⊗a_{0} ↦ t⊗a_{0}^{q}, satisfies the conditions (1) and (2). (For more details, see [27, Chap. II] or [9, Prop. 3.3].) Exercise 4.1 Let (V, A) be an affine variety defined over 𝔽_{q}, with corresponding Frobenius map F. Let σ: V → V be an automorphism of finite order which commutes with F. Then the morphism F ∘ σ is also a Frobenius map corresponding to some 𝔽_{q}-rational structure on V. (Hint. Let γ be the algebra homomorphism corresponding to F. Then set γ′ := γ ∘ σ^{∗} and check that γ′ also satisfies (1) and (2).)

4.B. Finite reductive groups. Now let G be a linear algebraic group. We say that G is defined over 𝔽_{q} if G has an 𝔽_{q}-rational structure with corresponding Frobenius map F and F is compatible with the group operations, i.e., we have F (gh) = F (g)F (h) and F (g^{−1}) = F (g)^{−1} for all g, h ∈ G. Thus, F is a bijective homomorphism of algebraic groups and so the fixed point set G^{F} is a finite group. If G is a connected reductive group, then G^{F} is called a finite group of Lie type or a finite reductive group.

IV. FINITE REDUCTIVE GROUPS 113

Example 4.2 The group G = GL_{n}(k) is defined over 𝔽_{q}, with Frobenius map F: (a_{ij}) ↦ (a_{ij}^{q}). This can be seen as follows. Recall from Example 2.3(e) that the coordinate ring k[G] is k[X_{ij} | 1 ≤ i, j ≤ n] localized in det. Now the map X_{ij} ↦ X_{ij}^{q} induces a k-algebra homomorphism γ: k[G] → k[G]. One checks easily that the conditions (1) and (2) in (4.A) hold. Also note that F^{∗} = γ and that F is compatible with matrix multiplication and inversion.

More generally, the groups SL_{n}(k), Sp_{2n}(k) and O_{n}(k) are defined over 𝔽_{q}, since they are closed F -stable subgroups of GL_{n}(k). In fact, the same kind of argument applies to all Chevalley groups G(k) as in (1.B). Example 4.3 Let G = GL_{n}(k) with its “standard” Frobenius map F as in Example 4.2. Then we have G^{F} = GL_{n}(q), the finite general linear group of non-singular matrices over 𝔽_{q}.

Now consider the map σ: G → G, g ↦ (g^{tr})^{−1}. This is an automorphism of order 2, which commutes with F. By Exercise 4.1, the map F_{σ} = F ∘ σ also is a Frobenius morphism. Then G^{F_{σ}} is the full unitary group over 𝔽_{q^{2}} with respect to the automorphism λ ↦ λ^{q} of 𝔽_{q^{2}}.

We remark that all finite Chevalley groups as well as the twisted groups of type 2A_{n}, ^{2}D_{n}, ^{2}E_{6}, ^{3}D_{4} arise in this way. In order to obtain the Suzuki and Ree groups one has to consider groups of the form G^{F′} where F′: G → G is a morphism such that some power of F′ is a Frobenius map (see [5, §1.19]).

4.C. Lang’s Theorem. We can now state a fundamental theorem of Lang. Assume that G is a linear algebraic group and that F: G → G is a Frobenius map. Then, if G is connected, the map G → G, g ↦ g^{−1}F (g), is surjective. (For an elegant proof, due to Steinberg, see [27, Theorem 2.4].) This result provides the tool which makes possible the passage between algebraic groups and finite groups. We illustrate this in some examples. Assume throughout that G is a connected linear algebraic group defined over 𝔽_{q}, with corresponding Frobenius map F: G → G. Example 4.4 There exist F -stable Borel subgroups and any two of them are conjugate by an element of G^{F}.

Indeed, let B ⊆ G be any Borel subgroup. Then F (B) also is a Borel subgroup (since F is a bijective homomorphism of algebraic groups) and so there exists some g ∈ G such that g^{−1}Bg = F (B) (see (3.C)(1)). By Lang’s Theorem, we can write g = h^{−1}F (h) for some h ∈ G. Let B′ := hBh^{−1}. Then we have F (B′) = F (h)F (B)F (h)^{−1} = hgF (B)(hg)^{−1} = hBh^{−1}, and so B′ is F -stable, as desired.

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Now let B_{1}, B_{2} be two F -stable Borel subgroups. Again, there exists some g ∈ G such that B_{2} = gB_{1}g^{−1}. Since B_{1}, B_{2} are F -stable, we conclude that gB_{1}g^{−1} = B_{2} = F (B_{2}) = F (g)F (B_{1})F (g)^{−1} = F (g)B_{1}F (g)^{−1} and so g^{−1}F (g) ∈ N_{G}(B_{1}) = B_{1}, where the last equality holds by (3.C)(1). Since B_{1} is connected, we can apply Lang’s Theorem to B_{1} and conclude that we can write g^{−1}F (g) = b^{−1}F (b) for some b ∈ B_{1}. Then we set h = gb^{−1} ∈ G^{F} and see that hB_{1}h^{−1} = gB_{1}g^{−1} = B_{2}, as desired.

Example 4.5 The same argument as in Example 4.4 shows that G also contains F -stable maximal tori. (Similarly, every F -stable Borel subgroup contains F -stable maximal tori.) But now it is not true that any two of them are conjugate by an element of G^{F}. More precisely, the G^{F} -conjugacy classes of F -stable maximal tori in G are parametrized by the F -conjugacy classes of W.

Here, W is the Weyl group with respect to a fixed F -stable maximal torus T_{0} ⊆ G. Since T_{0} is F -stable, F induces an automorphism of W and we say that w, w′ ∈ W are F -conjugate if w′ = x^{−1}wF (x) for some x ∈ W.

The correspondence between F -stable maximal tori in G and elements of W is defined as follows. Let T ⊆ G be such a torus. Then T = gT_{0}g^{−1} for some g ∈ G. We have gT_{0}g^{−1} = T = F (T) = F (g)F (T_{0})F (g)^{−1} = F (g)T_{0}F (g)^{−1} and so g^{−1}F (g) ∈ N_{G}(T_{0}). Let w be the image of g^{−1}F (g) in W = N_{G}(T_{0})/T_{0}. Now one has to show that this correspondence defines a bijection between G^{F} -conjugacy classes on the one hand and F -conjugacy classes on the other hand. For more details see [5, §3.3].

Exercise 4.6 Let C be an F -stable conjugacy class of G. Show by a similar argument as in Example 4.4 that there exists some g ∈ C^{F}. Now C^{F} is a union of conjugacy classes of G^{F}. Show that if C_{G}(g) is connected, then C^{F} is a single conjugacy class of G^{F}. Check that, for G = GL_{n}(k), the centralizer of any element g ∈ G is connected. Is this also true for SL_{n}(k)?

5. Hecke algebras

Let G be a connected reductive group defined over 𝔽_{q}, with corresponding Frobenius map F. We now begin to consider representations of G^{F}. The aim of this section is to show how to study those irreducible representations which occur in the permutation module of G^{F} on the cosets of B^{F}, where B ⊆ G is an F -stable Borel subgroup. (This is only the simplest possible case of a wider theory; see [5, Chap. 10].) Let W be the Weyl group of G with respect to an F -stable maximal torus contained in B; recall that this is a finite Coxeter group, with a generating set S consisting of elements of order 2. We will assume in this section that F acts trivially on W. (The general case is technically slightly more complicated.) The standard example of this set-up is the case where G = GL_{n}(\overline{𝔽}_{q}), B consists of the upper triangular matrices, and W ≅ 𝔖_{n}; moreover, F is given by (a_{ij}) ↦ (a_{ij}^{q}).

5.A. The permutation module on the cosets of B^{F}. The group G^{F} acts on the set of cosets G^{F} /B^{F} and we form the corresponding permutation module ℂ[G^{F} /B^{F}]. Then the Hecke algebra of G^{F} with respect to B^{F} is defined by

H_{q} := End_{ℂG^{F}}(ℂ[G^{F} /B^{F}]).

Since ℂG^{F} is semisimple, there exist mutually non-isomorphic simple ℂG^{F} -modules V_{1}, …, V_{r} such that

ℂ[G^{F} /B^{F}] ≅ ⨁_{i=1}^{r} n_{i}V_{i} where n_{i} > 0.

It follows that

H_{q} ≅ ⨁_{i=1}^{r} End_{ℂG^{F}}(n_{i}V_{i}) ≅ ⨁_{i=1}^{r} M_{n_{i}}(End_{ℂG^{F}}(V_{i})) ≅ ⨁_{i=1}^{r} M_{n_{i}}(ℂ)

is a semisimple algebra. This also shows that we have a bijection

Irr(H_{q}) ↔ {χ ∈ Irr(G^{F}) | χ occurs in ℂ[G^{F} /B^{F}]}
ϕ ↔ [ϕ].

In fact, we have that the character of ℂ[G^{F} /B^{F}] regarded as a (H_{q}, ℂG^{F})-bimodule is given by ∑_{ϕ∈Irr(H_{q})} ϕ ⊗ [ϕ].

Now the Bruhat decomposition of (3.D) gives rise to the decomposition G^{F} = ∐_{w∈W} B^{F} ẇB^{F} (note that we assume that F acts trivially on W). Since ℂ[G^{F} /B^{F}] can also be regarded as the result of inducing the trivial module from B^{F} to G^{F}, an easy application of the Mackey formula yields that dim H_{q} = |W |. A much more precise result is given by:

5.B. Iwahori’s Theorem. It states that H_{q} has a basis {T_{w} | w ∈ W } such that the following holds.

(1) T_{s}^{2} = qT_{1} + (q − 1)T_{s} for any s ∈ S.

(2) Let w ∈ W and write w = s_{1} ⋯ s_{m} with s_{i} ∈ S. Assume that m is minimal possible with this property. (We shall set l(w) = m and call this the length of w.) Then T_{w} = T_{s_{1}} ⋯ T_{s_{m}}.

(3) Regard ℂ[G^{F} /B^{F}] as a module for H_{q} and let ρ be its character. Then

ρ(T_{1}) = [G^{F} : B^{F}] and ρ(T_{w}) = 0 for w ≠ 1.

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(Proofs can be found in [8, §67] or [5, Chap. 10].) The statement in (3) can be used to give H_{q} the structure of a symmetric algebra. Indeed, if we define a linear map τ: H_{q} → ℂ by

τ(T_{1}) = 1 and τ(T_{w}) = 0 for w ≠ 1,

then we see that τ is a symmetrizing trace of H_{q}. Moreover, the basis of H_{q} which is dual to {T_{w}} is given by T_{w}^{∨} = q^{−l(w)}T_{w^{−1}} for w ∈ W. The fact that H_{q} is a symmetric algebra implies that the irreducible characters of H_{q} satisfy orthogonality relations (see A. Marcus’ lecture [this issue 5.10, 5.11]). Indeed, if ϕ, ϕ′ are irreducible characters of H_{q}, we have

∑_{w∈W} ϕ(T_{w})ϕ′(T_{w}^{∨}) = \{ \begin{matrix}0 & if ϕ ≠ ϕ′, \\ d_{ϕ} ≠ 0 & if ϕ = ϕ′. \end{matrix}

In the case of group algebras, the constant d_{ϕ} appearing in the above formula is just given by the order of the underlying group. We now show that, in the present case, d_{ϕ} is related to the degree of [ϕ]. Let ρ be as in (3). Using the bimodule decomposition mentioned in (5.A), we can write ρ = ∑_{ϕ} deg([ϕ])ϕ. Now fix ϕ ∈ Irr(H_{q}). Then we have

deg(ϕ)[G^{F} : B^{F}] = ϕ(T_{1})ρ(T_{1}) = ∑_{w∈W} ϕ(T_{w})ρ(T_{w}^{∨})

= ∑_{ϕ′∈Irr(H_{q})} deg([ϕ′])( ∑_{w∈W} ϕ(T_{w})ϕ′(T_{w}^{∨})) = deg([ϕ])d_{ϕ},

and so

deg([ϕ]) = \frac{deg(ϕ)[G^{F} : B^{F}]}{d_{ϕ}} = \frac{deg(ϕ)( ∑_{w∈W} q^{l(w)})}{d_{ϕ}}

where we use the identity [G^{F} : B^{F}] = ∑_{w∈W} q^{l(w)} (see [4, §8.6]). Thus, in order to find deg([ϕ]), we have to determine deg(ϕ) and the constants d_{ϕ} appearing in the above orthogonality relations.

5.C. The generic algebra and its specializations. Let A = ℂ[v, v^{−1}] be the ring of Laurent polynomials in an indeterminate v over ℂ. Then it is possible to define an associative A-algebra H_{v} with a basis {T_{w} | w ∈ W } such that the multiplication is determined by similar rules as in (5.B)(1) and (2), i.e., we have T_{w} = T_{s_{1}} ⋯ T_{s_{m}} if w = s_{1} ⋯ s_{m} with m = l(w), and T_{s}^{2} = v^{2}T_{1} + (v^{2} − 1)T_{s} for s ∈ S. (The existence of such an algebra, without any reference to a realization as endomorphism algebra as in (5.B), is a non-trivial fact which is due to N. Bourbaki; see [14, §7.2].) The algebra H_{v} is called the generic Iwahori–Hecke algebra associated with W.

IV. FINITE REDUCTIVE GROUPS 117

The reason why we want to have v^{2} in the above quadratic relations is the following. Let K = ℂ(v) be the field of fractions of A, and consider the K-algebra H_{K} = K⊗_{A}H_{v}. Then Lusztig [17] has shown that there is a canonical isomorphism H_{K} ≅ ℂW, and so H_{K} is a split semisimple algebra. The point of introducing H_{v} is that we can obtain all the algebras H_{q} (any q) from the single object H_{v} by the process of specialization, as follows.

Let 0 ≠ ζ ∈ ℂ and consider the ring homomorphism f_{ζ}: A → ℂ, v ↦ ζ. Then we can regard ℂ as an A-module via f_{ζ} and form the tensor product

H_{ζ} := ℂ ⊗_{A} H_{v}.

We sometimes call f_{ζ} a specialization and H_{ζ} the corresponding specialized algebra. Thus, H_{ζ} is an algebra over ℂ, it has a basis {T_{w}} such that similar relations hold as before, but now the quadratic relations read

T_{s}^{2} = ζ^{2}T_{1} + (ζ^{2} − 1)T_{s} for s ∈ S.

Note that H_{ζ} is not necessarily semisimple. Example 5.1 (a) Let ζ = 1. Then the quadratic relations in H_{ζ} read T_{s}^{2} = T_{1}, and so H_{ζ} is canonically isomorphic to ℂW.

(b) Choose a square root \sqrt{q} and set ζ = \sqrt{q}. Then H_{ζ} is canonically isomorphic to the endomorphism algebra H_{q} = End_{ℂG^{F}}(ℂ[G^{F} /B^{F}]).

Exercise 5.2 Assume that W = ⟨s⟩ has order 2 and let ζ = −1. Check that T_{1} + T_{s} spans a 1-dimensional nilpotent ideal in H_{ζ} and, hence, H_{ζ} is not semisimple.

5.D. Tits’ Deformation Theorem. We now turn to the relations between the irreducible characters of H_{K} and those of its specializations. Since A is a principal ideal domain, every representation of H_{K} can be realized over A. Hence, the triple (K, A, ℂ) or, rather, the triple (K, A_{I}, ℂ) where A_{I} denotes the localization of A in I = ker(f_{ζ}), may be regarded as a modular system, analogous to p-modular systems in the case of group algebras (see A. Marcus’ lecture [this issue 4.10, 4.11, 4.12]). It follows that, given χ ∈ Irr(H_{K}), we have χ(T_{w}) ∈ A for all w ∈ W, and the map

χ_{ζ}: H_{ζ} → ℂ, T_{w} ↦ f_{ζ}(χ(T_{w})),

is the character of a representation of H_{ζ}. Now Tits’ Deformation Theorem states that if H_{ζ} is semisimple, then the correspondence χ ↦ χ_{ζ} defines a bijection between Irr(H_{K}) and Irr(H_{ζ}). The proof, in the present context, is very easy: the Cartan matrix C of H_{ζ} is the identity matrix since H_{ζ} is semisimple. Denoting by D the decomposition matrix, we have C = D^{tr}D and so D must be a permutation matrix,

118 MEINOLF GECK

which is what we wanted to prove. (The Deformation Theorem actually holds under more general assumptions and then the proof is more complicated; see [8, §68A].)

Summarizing, we now see that the specializations v ↦ 1 and v ↦ \sqrt{q} define bijections

Irr(W) ↔ Irr(H_{K}) ↔ Irr(H_{q})
ϕ_{1} ↔ ϕ ↔ ϕ\sqrt{q}

Hence, in order to study the irreducible characters of H_{q}, we are reduced to work with the generic algebra H_{K} and, in some sense, even to W. Note that H_{K} also is a symmetric algebra, with a symmetrizing trace defined similarly as that for H_{q}. So we also have orthogonality relations for the irreducible characters of H_{K}, and these “specialize” to the corresponding relations for H_{q}. For any ϕ ∈ Irr(H_{K}), we define

D_{ϕ} := \frac{deg(ϕ)( ∑_{w∈W} v^{2l(w)})}{∑_{w∈W} ϕ(T_{w})ϕ(T_{w}^{∨})} ∈ K,

and call it the generic degree of ϕ. A comparison with the formula for deg([ϕ\sqrt{q}]) in (5.B) shows that D_{ϕ}(q) = deg([ϕ\sqrt{q}]) for all ϕ ∈ Irr(H_{K}). Benson and Curtis (1972) (see the exposition of their work in [8, §68C]) have used this relation to show that D_{ϕ} is actually a polynomial in u = v^{2} (and not just a fraction of two polynomials in v). The generic degrees have been computed in all cases, and can be found in the tables in [5, §13.5]. Example 5.3 Let G = GL_{n}(\overline{𝔽}_{q}) and W = 𝔖_{n}. The irreducible characters of W and, hence, also of H_{q} and H_{K}, are parametrized by partitions of n. For example, the trivial character corresponds to the partition (n) and the sign character corresponds to the partition (1^{n}). Now let λ be such a partition, with parts λ_{1} ≥ λ_{1} ≥ ⋯ ≥ λ_{r} > 0 such that ∑_{i} λ_{i} = n. For 1 ≤ i ≤ r and 1 ≤ j ≤ λ_{i}, we set h_{ij} := r + λ_{i} − i − j + 1. If we visualize λ in the usual way by a tableau consisting of left-justified rows of boxes, with λ_{i} boxes in the i-th row, then h_{ij} is the length of the hook made up of all boxes below and to the right of the box at position (i, j). Let D_{λ} be the generic degree of the character labelled by λ; then we have

D_{λ} := u^{0·λ_{1}+1·λ_{2}+···+(r−1)·λ_{r}} \frac{[1]_{u}[2]_{u} ⋯ [n]_{u}}{∏_{\begin{matrix}1≤i≤r \\ 1≤j≤λ_{i} \end{matrix}}[h_{ij}]_{u}}

where u = v^{2} and [i]_{n} = 1 + u + u^{2} + ⋯ + u^{i−1} for any i ≥ 1. For example, if n = 3, we have three partitions (3), (21), (1^{3}) and the generic degrees are

D_{(3)} = 1, D_{(21)} = u(u + 1), D_{(1^{3})} = u^{3}.

So, if we specialize v ↦ \sqrt{q}, we now see that the module ℂ[G^{F} /B^{F}] has three irreducible constituents, with degrees 1, q(q + 1), q^{3}, respectively.

6. Deligne-Lusztig characters

We keep our usual set-up, that is, G is a connected reductive group defined over 𝔽_{q}, with corresponding Frobenius map F. In the last section, we have studied the permutation module ℂ[G^{F} /B^{F}] but note that we did not really make use of the fact that the finite group G^{F} comes from an underlying algebraic group; the whole theory could have been developed abstractly, in the framework of finite groups with a “BNpair” (see [5, Chap. 10] or [8, §67]). We will now go on making substancial use of that underlying algebraic group structure. The idea is to look for a generalization of the construction of permutation modules, where we also take into account the action of G^{F} on spaces which are “invisible” in the finite group G^{F}, i.e., which are varieties attached to G but they may not be stable under F. Deligne and Lusztig (1976) discoverd that such a generalization does exist, using the theory of ℓ-adic cohomology (see [5, Chap. 7]).

6.A. ℓ-adic cohomology. Let k be an algebraic closure of the field with p elements (where p is a prime) and X be any algebraic variety over k. (Here, we will allow more general types of varieties than considered before; for example, X may be any quasiprojective variety, see [23]). Let ℓ be a prime not equal to p. Then, for each i ≥ 0, there exists a finite-dimensional vector space H_{c}^{i}(X) over \overline{ℚ}_{ℓ} (an algebraic closure of the field of ℓ-numbers), called the ℓ-adic cohomology with compact support, which was introduced by Artin and Grothendieck. In [5, Chap. 7], we find 12 basic properties, listed in the form of axioms on which the subsequent development is based. Here are some of them:

(1) We have H_{c}^{i}(X) = {0} unless i ∈ {1, …, 2 dim X} where dim X denotes the dimension of X as an algebraic variety. If X = k^{n}, then H_{c}^{i}(X) = \overline{ℚ}_{l} for i = 2n and H_{c}^{i}(X) = {0} otherwise.

(2) Let g: X → X be a morphism satisfying some extra conditions which hold, for example, if g is an automorphism or if g is a Frobenius map. Then g induces a linear map g^{∗}: H_{c}^{i}(X) → H_{c}^{i}(X) and g ↦ g^{∗} is functorial in the sense that if h is another such morphism, then we have (g ∘ h)^{∗} = h^{∗} ∘ g^{∗}.

(3) Let F: X → X be the Frobenius map corresponding to some 𝔽_{q}-rational structure of X (where q is a power of p). Then the linear maps F^{∗}: H_{c}^{i}(X) → H_{c}^{i}(X)

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are non-singular and we have the following identity, called Grothendieck’s trace formula,

|X^{F} | = ∑_{i≥0}(−1)^{i} trace(F^{∗} | H_{c}^{i}(X)).

For more details on the definition of H_{c}^{i}(X) and on the proofs of the above statements, see Chap. V of Srinivasan’s book [27].

6.B. Lefschetz numbers. Let g: X → X be an automorphism of finite order. Then

L(g, X) := ∑_{i≥0} trace(g^{∗} | H_{c}^{i}(X))

is called the Lefschetz number of g on X. This Lefschetz number can also be defined in a completely elementary way, as follows. The morphism g is defined over some finite subfield of k. So there exists some power q of p such that X is defined over 𝔽_{q} and the corresponding Frobenius map F satisfies F ∘ g = g ∘ F. Then, for any n ≥ 1, the maps g and F^{n} commute with each other and, hence, by Exercise 4.1, g ∘ F^{n} also is a Frobenius map. Let t be an indeterminate and consider the formal power series

R(t) := − ∑_{n=1}^{∞} |X^{g∘F^{n}}|t^{n}.

We claim that R(t) is a rational function for which the limit t → ∞ exists (i.e., we have R(t) = P/Q, where P, Q are polynomials in t such that deg(P) ≤ deg(Q)) and that the limit in fact equals L(g, X).

This is proved as follows. Since g and F^{n} commute with each other, the same holds for the induced maps g^{∗} and (F^{∗})^{n} on H_{c}^{i}(X). Hence the matrices of these maps with respect to a suitable basis of H_{c}^{i}(X) can be put simultaneously into a triangular shape. (This is just elementary linear algebra.) Now let k_{i} = dim H_{c}^{i}(X). Moreover, let (x_{j,i})_{1≤j≤k_{i}} be the eigenvalues of g^{∗} on H_{c}^{i}(X) and (λ_{j,i})_{1≤j≤k_{i}} be the eigenvalues of F^{∗}. Then (x_{j,i}λ_{j,i}^{n})_{1≤j≤k_{i}} are the eigenvalues of (g ∘ F^{n})^{∗} = (F^{∗})^{n} ∘ g^{∗} on H_{c}^{i}(X) and so we have

trace((g ∘ F^{n})^{∗} | H_{c}^{i}(X)) = ∑_{j=1}^{k_{i}} x_{j,i}λ_{j,i}^{n}.

Using Grothendieck’s trace formula, we compute

R(t) = − ∑_{n=1}^{∞} |X^{g∘F^{n}}|t^{n} = − ∑_{n=1}^{∞} ∑_{i≥0}(−1)^{i} trace((g ∘ F^{n})^{∗} | H_{c}^{i}(X))t^{n}

= − ∑_{n=1}^{∞} ∑_{i≥0}(−1)^{i} ∑_{j=1}^{k_{i}} x_{j,i}λ_{j,i}^{n}t^{n} = − ∑_{i≥0} ∑_{j=1}^{k_{i}}(−1)^{i}x_{j,i}(∑_{n=1}^{∞}(λ_{j,i}t)^{n})

= − ∑_{i≥0} ∑_{j=1}^{k_{i}}(−1)^{i}x_{j,i} \frac{λ_{j,i}t}{1 − λ_{j,i}t}@ > t → ∞ >> ∑_{i≥0} ∑_{j=1}^{k_{i}}(−1)^{i}x_{j,i} = L(g, X),

as desired; note that λ_{j,i} ≠ 0 for all i, j by (6.A)(3). This characterization of Lefschetz numbers also shows, for example, that L(g, X) ∈ ℤ and that L(g, X) = L(g^{−1}, X).

Now, why not just work with this elementary definition of L(g, X)? The point is, for example, that the interpretation of L(g, X) in terms of ℓ-adic cohomology immediately implies, using (6.A)(2), that if H is any finite subgroup of automorphisms of X, then the map g ↦ L(g^{−1}, X) (g ∈ H) is a virtual character of H. No other proof of this fact is known! Example 6.1 In the above set-up, assume that X is a finite set on which F acts trivially. Then X^{g∘F^{n}} = X^{g} for any n ≥ 1 and so

R(t) = − ∑_{n=1}^{∞} |X^{g∘F^{n}}|t^{n} = −|X^{g}| ∑_{n=1}^{∞} t^{n} = −|X^{g}|\frac{t}{1 − t}@ > t → ∞ >> |X^{g}|.

Hence, we conclude that L(g, X) = |X^{g}|.

6.C. The construction of Deligne and Lusztig. Let us now return to our connected reductive group G, with corresponding Frobenius map F. Let B ⊆ G be an F -stable Borel subgroup, T_{0} ⊆ G be an F -stable maximal torus contained in B, and W = N_{G}(T_{0})/T_{0} be the Weyl group with respect to T_{0}.

Consider once more the permutation module ℂ[G^{F} /B^{F}]. Its character is simply given by counting the number of fixed points of g ∈ G^{F} on G^{F} /B^{F}. Hence Example 6.1 shows that this character equals the map g ↦ L(g, G^{F} /B^{F}). Following Deligne-Lusztig we now consider, for any w ∈ W, the variety

X_{w} := {xB ∈ G/B | x^{−1}F (x) ∈ BẇB},

where ẇ denotes a representative of w in N_{G}(T_{0}) (but note that BẇB is independent of the choice of ẇ). Now left multiplication with any g ∈ G^{F} defines an automorphism of finite order of X_{w}. Hence we obtain a virtual character R_{w} of G^{F} by setting, for any g ∈ G^{F},

R_{w}(g) := L(g, X_{w}) = ∑_{i≥0}(−1)^{i} trace((g^{∗})^{−1} | X_{c}^{i}(X_{w})).

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If we take w = 1 then R_{1} is just the character of ℂ[G^{F} /B^{F}], since

X_{1} = {xB ∈ G/B | x^{−1}F (x) ∈ B} = (G/B)^{F} ≅ G^{F} /B^{F} .

A similar elementary definition for R_{w} (w ≠ 1) is not known!

We can even go one step further. The cohomology groups H_{c}^{i}(X) are obtained by taking cohomology with coefficients in the constant \overline{ℚ}_{ℓ}-sheaf, but we can also allow more general sheafs. For this purpose, fix w ∈ W and consider pairs (n, λ) consisting of an integer n ≥ 1 such that p does not divide n and a homomorphism λ: T → k^{×} such that the following holds. There exists a homomorphism λ_{1} : T → k^{×} such that

λ(F (t)) = λ(ẇ^{−1}tẇ)λ_{1}^{n}(t) for all t ∈ T .

As shown in [18, (2.1) and (2.3)], a pair (n, λ) as above and a choice of a representative ẇ give rise to a \overline{ℚ}_{ℓ}-sheaf F_{ẇ,n}^{λ} of rank 1 over X_{w}. Hence we obtain a virtual character of G^{F} by setting, for any g ∈ G^{F},

R_{ẇ,n}^{λ}(g) := ∑_{i≥0}(−1)^{i} trace((g^{∗})^{−1} | X_{c}^{i}(X_{w}, F_{ẇ,n}^{λ})).

Let T be an F -stable maximal torus of G which is parametrized by the F -conjugacy class of w ∈ W (see Example 4.5). Then F_{ẇ,n}^{λ} can be interpreted as an irreducible character θ ∈ Irr(T^{F}) (see the proof of [18, Lemma 6.2]). Hence we may also denote R_{ẇ,n}^{λ} by R_{T,θ}. In this form, R_{T,θ} only depends on the G^{F} -conjugacy class of the pair (T, θ). The virtual characters R_{T,θ}, for various pairs (T, θ) as above, are the famous Deligne–Lusztig characters, which are of fundamental importance to the representation theory of G^{F}.

For example, there is a scalar product formula (see [5, Theorem 7.3.4]) which shows that if (T, θ) is a pair such that no non-identity element of N_{G}(T^{F})/T^{F} fixes θ, then R_{T,θ} is up to sign an irreducible character of G^{F}. It can be shown that “most” irreducible characters of G^{F} arise in this way, in the following sense: the number of irreducible characters of the form ±R_{T,θ} (for pairs (T, θ) satisfying the above condition) divided by the total number of all irreducible characters is ≤ Cq^{−1}, where C is a constant depending only on the type of G but not on q.

7. A guide to further reading

As a general reference about groups of Lie type, we recommend Carter’s book [5] which includes, among others, chapters on groups with a BN-pair, conjugacy classes in algebraic groups, the definition and basic properties of R_{T,θ}’s, Gelfand-Graev characters, the Alvis–Curtis–Kawanaka duality operation on characters, the theory of Harish–Chandra series and Hecke algebras and, last but not least, a final chapter with

IV. FINITE REDUCTIVE GROUPS 123

tables containing a variety of detailed information on conjugacy classes, characters of finite Weyl groups and character degrees of finite groups of Lie type. It also contains a quite exhaustive bibliography which covers much of the literature about finite groups of Lie type up to around 1985. Several survey articles can also be found in the proceedings volumes [6, 11].

The CHEVIE system [12] makes a considerable part of the information in the tables in Carter’s book electronically accessible; it also includes all the known character tables of finite groups of Lie type, finite Coxeter groups and Iwahori–Hecke algebras, as well as programs to work with them.

Let us now turn to more specific aspects of the representation theory of finite groups of Lie type.

7.A. Hecke algebras. We have seen that Hecke algebras arise naturally as endomorphism algebras of certain induced (ordinary) representations of finite groups of Lie type. But a generic Hecke algebra as in (5.C) also has amazing connections with the geometry of the flag variety of the underlying algebraic group. This can be used to obtain a number of deep results about Hecke algebras for which no elementary proof is known; for further reading, see for example Chapters 5 and 12 of [18], the articles [17, 20] and Lusztig’s paper ”Leading coefficients of the character values of Hecke algebras” in [11].

In another direction, we can also consider induced representations for p-adic Chevalley groups, in which case the endomorphism algebras turn out to be Hecke algebras associated with affine Weyl groups. The most far-reaching results in this direction have been recently obtained by Lusztig [22].

7.B. The theory of R_{T,θ}’s. An exposition of the basic results about the Deligne-Lusztig characters R_{T,θ} can be found in Lusztig’s monograph [16] and in the books by Srinivasan [27], Carter [5] and Digne-Michel [9].

A special feature of Srinivasan’s book is that it contains a quite detailed chapter about ℓ-cohomology. Digne and Michel also discuss the general Lusztig functor R_{L}^{G} where L is an F -stable Levi subgroup of G.

The ultimate reference about R_{T,θ} is Lusztig’s book [18] which contains, among many other results, a classification of the irreducible characters of G^{F} in the case where G has a connected center. (For example, this includes GL_{n}(k) but not SL_{n}(k).) This is achieved in Main Theorem 4.23 in terms of a multiplicity formula which determines explicitly the multiplicities of the irreducible characters of G^{F} as constituents of the

124 MEINOLF GECK

virtual characters R_{T,θ}. In [19], Lusztig has extended these results to the case where the center of G is not connected.

We did not even mention a number of related subjects like the Shintani descent and the theory of Green functions. The interested reader can find out more about this in articles by Asai, Digne, Kawanaka and Shoji in [11].

7.C. The computation of character values. After the completion of the classification of the irreducible characters of a finite group of Lie type in [18, 19], Lusztig has developed a theory of “character sheaves” with the aim of finding a strategy for computing the values of the irreducible characters. Such a strategy has eventually been found in [21]; see also [25]. The theory provides an orthonormal basis of class functions whose values are explicitly computable but where the transition matrix to the basis consisting of irreducible characters contains some roots of unity as unknown quantities. These roots of unity have been determined in a number of cases (see, for example, Shoji’s article on “Unipotent characters of finite classical groups” in [3]) but a general scheme is still missing.

7.D. Modular representations in defining characteristic. These are representations over a field of the same characteristic as that of the field over which the group is defined. A basic part of the theory, including a classification of the simple modules in terms of certain “weights”, can be found in Curtis–Reiner [8, §72B]. The main open problem is the determination of the (Brauer) characters (or, at least, the degrees) of the simple modules. A conjecture due to Lusztig predicts a solution for the case where the characteristic is bigger than a certain constant which is explicitly determined in terms of the Weyl group W. This conjecture has been proved in [1], but without specifying a bound for that constant. This means that Lusztig’s conjecture is known to be true whenever the characteristic p is “large enough”, but if we are given a particular prime p and a group over a field of characteristic p we do not know if p is actually “large enough” or not. For an introduction to Lusztig’s conjecture, see Donkin’s article in [6].

7.E. Modular representations in non-defining characteristic. These are representations over a field of characteristic ℓ > 0 where ℓ is a prime which is not equal to the defining characteristic of the group. Since the fundamental papers by Fong and Srinivasan on the blocks of finite classical groups and by Dipper and James on the decomposition numbers of GL_{n}(q), a growing confluence has emerged between general “abstract” concepts of block theory and “geometric” properties of finite groups of

IV. FINITE REDUCTIVE GROUPS 125

Lie type. For example, it has turned out that the block distribution of characters is compatible with the distribution of the characters into so-called geometric conjugacy classes, which are defined in the framework of Deligne-Lusztig theory. For further reading, see the volume by Broué, Malle and Michel [2] on generic blocks, the survey article [10] and the proceedings volume [3] which contains a number of articles on various aspects of the modular representation theory of finite groups of Lie type, ranging from the block distribution of characters to modular Harish–Chandra series and Hecke algebras.

References

[1] H. H. Andersen, J. C. Jantzen and W. Soergel, Representations of quantum groups at p-th roots of unity and of semisimple groups in characteristic p: independence of p, Astérisque 220 (1994).

[2] M. Broué, G. Malle and J. Michel, Représentations unipotentes génériques et blocs des groupes réductifs finis, avec un appendice de G. Lusztig, Astérisque 212 (1993).

[3] M. Cabanes (ed.), Finite reductive groups, related structures and representations, Progress in Math., vol. 141, Birkhäuser, Basel, 1997.

[4] R. W. Carter, Simple groups of Lie type, Wiley, New York, 1972; reprinted in the Wiley Classics Edition, 1989.

[5] R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York, 1985.

[6] R. W. Carter and M. Geck (ed.), Representations of reductive groups, Proceedings of an instructional conference at the Isaac Newton Institute, Cambridge, U.K., January 1997, to appear at Cambridge University Press.

[7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Oxford University Press, London/New York, 1985.

[8] C. W. Curtis and I. Reiner, Methods of Representation Theory Vol. I and II, Wiley, New York, 1981 and 1987.

[9] F. Digne and J. Michel, Representations of finite groups of Lie type, London Math. Soc. Students Texts vol. 21, Cambridge University Press, 1991.

[10] R. Dipper, M. Geck, G. Hiss and G. Malle, Representations of Hecke algebras and finite groups of Lie type, to appear in a proceedings volume on the DFG project ”Algorithmic algebra and number theory”.

[11] P. Fong (ed.), The Arcata conference on representations of finite groups, Proc. Symp. Pure Math., vol. 47, Amer. Math. Soc., 1987.

[12] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE–A system for computing and processing generic character tables, AAECC 7 (1996), 175–210; electronically

available under http://www.iwr.uni-heidelberg.de/iwr/compalg/chevie

[13] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, Math. Surveys and Monographs vol. 40, no. 1, Amer. Math. Soc., 1994.

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[14] J. Humphreys, Finite Reflection Groups, Cambridge studies in advanced mathematics,vol. 29, Cambridge University Press (1990).

[15] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York (1976); corrected reprint by Dover Publ., New York, 1994.

[16] G. Lusztig, Representations of finite Chevalley groups, C.B.M.S. Regional Conference Series in Mathematics, vol. 39, Amer. Math. Soc., 1978.

[17] G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71, 490–498 (1981).

[18] G. Lusztig, Characters of reductive groups over a finite field, Annals Math. Studies, vol. 107, Princeton University Press, 1984.

[19] G. Lusztig, On the representations of reductive groups with disconnected centre, in: Orbites Unipotentes et Représentationes, I. Groupes finis et Algèbres de Hecke, Astérisque 168 (1988).

[20] G. Lusztig, Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematics, Kyoto, Japan, 1990, Springer, 1991, pp. 155–174.

[21] G. Lusztig, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5, 971–986 (1992).

[22] G. Lusztig, Classification of unipotent representations of simple p-adic groups, Internat. Math. Research Notices 11 (1995), 517–589.

[23] D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer, 1988.

[24] M. Schönert et al., GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathematik, RWTH Aachen, Germany, fourth ed., (1994); see also

    http://www.math.rwth-aachen.de/LDFM

[25] T. Shoji, Character sheaves and almost characters of reductive groups, Advances in Math. 111 (1995), 244–313; part II, ibid., 314–354.

[26] T. A. Springer, Linear algebraic groups, Progress in Math., vol. 9, Birkhäuser, Boston, 1981.

[27] B. Srinivasan, Representations of finite Chevalley groups, Lecture Notes in Mathematics, vol. 764, Springer, 1979.

[28] R. Steinberg, Lectures on Chevalley groups, Yale University (1967).

[29] R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the Amer. Math. Soc. 80 (1968).

[30] R. Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, vol. 366, Springer, 1974.

UFR de Mathématiques, Université Paris VII, 2 Place Jussieu, F–75251 Paris 05, France

   E-mail address: geck@math.jussieu.fr