On the Restriction of Deligne-lusztig Characters

results of Hagedorn gave me the courage to attempt such calculations for general groups and to obtain closed multiplicity formulas for orthogonal groups. It is a pleasure to thank Dick Gross for initiating the work in [12] which led to this paper, for helpful remarks on an earlier version, and for aquainting me with Hagedorn’s thesis. The referee read the original version of this paper with care and insight, made valuable comments and simplified some of the arguments. In particular, the proof of Lemma 3.1 given below is due to the referee and is much shorter than the original one. Some general notation: The cardinality of a finite set X is denoted by |X|. Equivalence classes are generally denoted by [ · ], sometimes with ornamentation. If g is an element of a group G, we write Ad(g) for the conjugation map Ad(g) : x → gxg−1, and also write T := gTg−1 for a subgroup T ⊂ G. The center of G is denoted Z(G) and the centralizer of g ∈ G is denoted CG(g). We write 〈 , 〉H for the pairing on the space of class functions on a finite group H, for which the irreducible characters of H are an orthonormal basis. If G,G′ ⊃ H are finite overgroups of H and ψ, ψ′ are class functions on G,G′ respectively, then 〈ψ, ψ〉H is understood to mean 〈ψ|H , ψ|H〉H , where |H denotes restriction to H. 2. Remarks on maximal tori Let G be a connected reductive algebraic F-group. We assume G is defined over f and has Frobenius F . If T is a maximal torus in G we denote its normalizer in G by NG(T ) and write WG(T ) = NG(T )/T for the Weyl group of T in G. If T is F -stable, we have W (T ) = NG(T ) /T , by the Lang-Steinberg theorem. The reduction formula for Deligne-Lusztig characters (recalled in section 4 below) involves a sum over the following kind of subset of G . Fix an F -stable maximal torus T ⊂ G, and let s be a semisimple element in G . We must sum over the set NG(s, T ) := {γ ∈ G : s ∈ T}. Note that NG(s, T ) , if nonempty, is a union of Gs ×NG(T ) double cosets, where Gs := CG(s) is the identity component of the centralizer CG(s) of s in G. To say that s ∈ T is to say that T ⊂ Gs, so determining the Gs × NG(T ) double cosets in NG(s, T ) amounts to determining the Gs -conjugacy classes of F -stable maximal tori in Gs which are contained in a given G -conjugacy class. Such classes of tori are parameterized by twisted conjugacy classes in Weyl groups of Gs and G. The aim of this section is to parameterize the Gs × NG(T ) double cosets in NG(s, T ) in terms of the fiber of a natural map between twisted conjugacy classes in the Weyl groups of Gs and G. This parameterization will be fundamental to our later calculations with Deligne-Lusztig characters. We begin by recalling the classification of F -stable maximal tori in G. See [5, chap. 3] for more details in what follows. Fix an F -stable maximal torus T0 in G contained in an F -stable Borel subgroup of G, and abbreviate NG = NG(T0), WG = WG(T0). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON THE RESTRICTION OF DELIGNE-LUSZTIG CHARACTERS 577 Let T (G) denote the set of all F -stable maximal tori in G. Then T (G) is a finite union of G -orbits. For any T ∈ T (G), let [T ]G := {T : γ ∈ G } denote the G -orbit of T . There is g ∈ G such that T = T0. Since T is F -stable, we have g−1F (g) ∈ NG. This gives an element w := g−1F (g)T0 ∈ WG. The map Ad(g)t = gtg−1 is an f-isomorphism Ad(g) : (T0, wF ) −→ (T, F ), where the second component denotes the action of Frobenius under an f-structure. For any finite group A with F -action, we let H(F,A) denote the set of F conjugacy classes in A. These are the orbits of the action of A on itself via (a, b) → abF (a)−1. Let [b] ∈ H(F,A) denote the F -conjugacy class of an element b ∈ A. For g, T, w as above, the F -conjugacy class of w is independent of the choice of g. Hence we have a well-defined class cl(T,G) := [w] ∈ H(F,WG). For each ω ∈ H(F,WG), the set Tω(G) := {T ∈ T (G) : cl(T,G) = ω} is a single G -orbit in T (G), and all G -orbits are of this form. Thus, the partition of the set of F -stable maximal tori into G -orbits is given by T (G) = ∐