Deligne–Lusztig theory

Suppose that G is a reductive group defined over a finite field, with Frobenius map F.

Ian G. Macdonald conjectured that there should be a map from general position characters of F-stable maximal tori to irreducible representations of ${\displaystyle G^{F}}$ (the fixed points of F). For general linear groups this was already known by the work of J. A. Green (1955). This was the main result proved by Pierre Deligne and George Lusztig; they found a virtual representation for all characters of an F-stable maximal torus, which is irreducible (up to sign) when the character is in general position.

When the maximal torus is split, these representations were well known and are given by parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction can be constructed using functions on a space, which can be thought of as elements of a suitable zeroth cohomology group. Deligne and Lusztig's construction is a generalization of parabolic induction to non-split tori using higher cohomology groups. (Parabolic induction can also be done with tori of G replaced by Levi subgroups of G, and there is a generalization of Deligne–Lusztig theory to this case too.)

Vladimir Drinfeld proved that the discrete series representations of SL₂(F_q) can be found in the ℓ-adic cohomology groups

${\displaystyle H_{c}^{1}(X,\mathbb {Q} _{\ell })}$

of the affine curve X defined by

${\displaystyle xy^{q}-yx^{q}=1}$.

The polynomial ${\displaystyle xy^{q}-yx^{q}}$ is a determinant used in the construction of the Dickson invariant of the general linear group, and is an invariant of the special linear group.

The construction of Deligne and Lusztig is a generalization of this fundamental example to other groups. The affine curve X is generalized to a ${\displaystyle T^{F}}$ bundle over a "Deligne–Lusztig variety" where T is a maximal torus of G, and instead of using just the first cohomology group they use an alternating sum of ℓ-adic cohomology groups with compact support to construct virtual representations.

The Deligne-Lusztig construction is formally similar to Hermann Weyl's construction of the representations of a compact group from the characters of a maximal torus. The case of compact groups is easier partly because there is only one conjugacy class of maximal tori. The Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar.

For real semisimple groups there is an analogue of the construction of Deligne and Lusztig, using Zuckerman functors to construct representations.

The construction of Deligne-Lusztig characters uses a family of auxiliary algebraic varieties X_T called Deligne–Lusztig varieties, constructed from a reductive linear algebraic group G defined over a finite field F_q.

If B is a Borel subgroup of G and T a maximal torus of B then we write

W_T,B

for the Weyl group (normalizer mod centralizer)

N_G(T)/T

of G with respect to T, together with the simple roots corresponding to B. If B₁ is another Borel subgroup with maximal torus T₁ then there is a canonical isomorphism from T to T₁ that identifies the two Weyl groups. So we can identify all these Weyl groups, and call it 'the' Weyl group W of G. Similarly there is a canonical isomorphism between any two maximal tori with given choice of positive roots, so we can identify all these and call it 'the' maximal torus T of G.

By the Bruhat decomposition

G = BWB,

the subgroup B₁ can be written as the conjugate of B by bw for some b∈B and w∈W (identified with W_T,B) where w is uniquely determined. In this case we say that B and B₁ are in relative position w.

Suppose that w is in the Weyl group of G, and write X for the smooth projective variety of all Borel subgroups of G. The Deligne-Lusztig variety X(w) consists of all Borel subgroups B of G such that B and F(B) are in relative position w [recall that F is the Frobenius map]. In other words, it is the inverse image of the G-homogeneous space of pairs of Borel subgroups in relative position w, under the Lang isogeny with formula

g.F(g)⁻¹.

For example, if w=1 then X(w) is 0-dimensional and its points are the rational Borel subgroups of G.

We let T(w) be the torus T, with the rational structure for which the Frobenius is wF. The G^F conjugacy classes of F-stable maximal tori of G can be identified with the F-conjugacy classes of W, where we say w∈W is F-conjugate to elements of the form vwF(v)⁻¹ for v∈W. If the group G is split, so that F acts trivially on W, this is the same as ordinary conjugacy, but in general for non-split groups G, F may act on W via a non-trivial diagram automorphism. The F-stable conjugacy classes can be identified with elements of the non-abelian Galois cohomology group of torsors

${\displaystyle H^{1}(F,W)}$.

Fix a maximal torus T of G and a Borel subgroup B containing it, both invariant under the Frobenius map F, and write U for the unipotent radical of B. If we choose a representative w′ of the normalizer N(T) representing w, then we define X′(w′) to be the elements of G/U with F(u)=uw′. This is acted on freely by T(F), and the quotient is isomorphic to X(T). So for each character θ of T(w)^F we get a corresponding local system F_θ on X(w). The Deligne-Lusztig virtual representation

R^θ(w)

of G^F is defined by the alternating sum

${\displaystyle R^{\theta }(w)=\sum _{i}(-1)^{i}H_{c}^{i}(X(w),F_{\theta })}$

of l-adic compactly supported cohomology groups of X(w) with coefficients in the l-adic local system F_θ.

If T is a maximal F-invariant torus of G contained in a Borel subgroup B such that B and FB are in relative position w then R^θ(w) is also denoted by R^θ_T⊂B, or by R^θ_T as up to isomorphism it does not depend on the choice of B.

• The character of R^θ_T does not depend on the choice of prime l≠p, and if θ=1 its values are rational integers.
• Every irreducible character of G^F occurs in at least one character R^θ(w).
• The inner product of R^θ_T and R^θ′_T′ is equal to the number of elements of W(T,T′)^F taking θ to θ′. The set W(T,T′) is the set of elements of G taking T to T′ under conjugation, modulo the group T^F which acts on it in the obvious way (so if T=T′ it is the Weyl group). In particular the inner product is 0 if w and w′ are not F-conjugate. If θ is in general position then R^θ_T has norm 1 and is therefore an irreducible character up to sign. So this verifies Macdonald's conjecture.
• The representation R^θ_T contains the trivial representation if and only if θ=1 (in which case the trivial representation occurs exactly once).
• The representation R^θ_T has dimension

${\displaystyle \dim(R_{T}^{\theta })={\pm |G^{F}| \over |T^{F}||U^{F}|}}$

where U^F is a Sylow p-subgroup of G^F, of order the largest power of p dividing |G^F|.

• The restriction of the character R^θ_T to unipotent elements u does not depend on θ and is called a Green function, denoted by Q_T,G(u) (the Green function is defined to be 0 on elements that are not unipotent). The character formula gives the character of R^θ_T in terms of Green functions of subgroups as follows:

${\displaystyle R_{T}^{\theta }(x)={1 \over |G_{s}^{F}|}\sum _{g\in G^{F},g^{-1}sg\in T}Q_{gTg^{-1},G_{s}}(u)\theta (g^{-1}sg)}$

where x=su is the Jordan–Chevalley decomposition of x as the product of commuting semisimple and unipotent elements s and u, and G_s is the identity component of the centralizer of s in G. In particular the character value vanishes unless the semisimple part of x is conjugate under G^F to something in the torus T.

• The Deligne-Lusztig variety is usually affine, in particular whenever the characteristic p is larger than the Coxeter number h of the Weyl group. If it is affine and the character θ is in general position (so that the Deligne-Lusztig character is irreducible up to sign) then only one of the cohomology groups Hⁱ(X(w),F_θ) is non-zero (the one with i equal to the length of w) so this cohomology group gives a model for the irreducible representation. In general it is possible for more than one cohomology group to be non-zero, for example when θ is 1.

Lusztig classified all the irreducible characters of G^F by decomposing such a character into a semisimple character and a unipotent character (of another group) and separately classifying the semisimple and unipotent characters.

Suppose that q is an odd prime power, and G is the algebraic group SL₂. We describe the Deligne–Lusztig representations of the group SL₂(F_q). (The representation theory of these groups was well known long before Deligne–Lusztig theory.)

The irreducible representations are:

• The trivial representation of dimension 1.
• The Steinberg representation of dimension q
• The (q − 3)/2 irreducible principal series representations of dimension q + 1, together with 2 representations of dimension (q + 1)/2 coming from a reducible principal series representation.
• The (q − 1)/2 irreducible discrete series representations of dimension q − 1, together with 2 representations of dimension (q − 1)/2 coming from a reducible discrete series representation.

There are two classes of tori associated to the two elements (or conjugacy classes) of the Weyl group, denoted by T(1) (cyclic of order q−1) and T(w) (cyclic of order q + 1). The non-trivial element of the Weyl group acts on the characters of these tori by changing each character to its inverse. So the Weyl group fixes a character if and only if it has order 1 or 2. By the orthogonality formula, R^θ(w) is (up to sign) irreducible if θ does not have order 1 or 2, and a sum of two irreducible representations if it has order 1 or 2.

The Deligne-Lusztig variety X(1) for the split torus is 0-dimensional with q+1 points, and can be identified with the points of 1-dimensional projective space defined over F_q. The representations R^θ(1) are given as follows:

• 1+Steinberg if θ=1
• The sum of the 2 representations of dimension (q+1)/2 if θ has order 2.
• An irreducible principal series representation if θ has order greater than 2.

The Deligne-Lusztig variety X(w) for the non-split torus is 1-dimensional, and can be identified with the complement of X(1) in 1-dimensional projective space. So it is the set of points (x:y) of projective space not fixed by the Frobenius map (x:y)→ (x^q:y^q), in other words the points with

${\displaystyle xy^{q}-yx^{q}\neq 0}$

Drinfeld's variety of points (x,y) of affine space with

${\displaystyle xy^{q}-yx^{q}=1}$

maps to X(w) in the obvious way, and is acted on freely by the group of q+1th roots λ of 1 (which can be identified with the elements of the non-split torus that are defined over F_q), with λ taking (x,y) to (λx,λy). The Deligne Lusztig variety is the quotient of Drinfeld's variety by this group action. The representations −R^θ(w) are given as follows:

• Steinberg−1 if θ=1
• The sum of the 2 representations of dimension (q−1)/2 if θ has order 2.
• An irreducible discrete series representation if θ has order greater than 2.

The unipotent representations are the trivial representation and the Steinberg representation, and the semisimple representations are all the representations other than the Steinberg representation. (In this case the semisimple representations do not correspond exactly to geometric conjugacy classes of the dual group, as the center of G is not connected.)

Lusztig (1985) replaced the ℓ-adic cohomology used to define the Deligne-Lusztig representations with intersection ℓ-adic cohomology, and introduced certain perverse sheaves called character sheaves. The representations defined using intersection cohomology are related to those defined using ordinary cohomology by Kazhdan–Lusztig polynomials. The F-invariant irreducible character sheaves are closely related to the irreducible characters of the group G^F.

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