On the Discrete Series Representations of the Classical Groups over Finite Fields

1. One of the main unsolved problems in the representation theory of the finite Chevalley groups is the determination of the discrete series representations; these are the most basic, but least accessible, representations of such groups. Let me recall that an (ordinary) representation D of a finite Chevalley group G is said to be in the discrete series, or cuspidal, if, for any proper parabolic subgroup P c G, the restriction of D to the "unipotent radical" of P does not contain the unit representation. The theme of this section is that we encounter discrete series representations in the process of decomposing the Brauer lifting of some very natural modular representations. The idea of the Brauer lifting was introduced by J. A. Green in 1955 in his well-known work on the characters of GLn(Fq)9 and revived by Quillen in connection with his solution of the Adams conjecture. Part of the results presented here, namely the ones on GLW, are proved in my publication The discrete series of GLW over a finite field, Annals of Mathematics Studies, No. 81, Princeton University Press, 1974. Let F be a finite field with q elements, q = p\ let WF be the ring of Witt vectors associated to F and QF its quotient field. The canonical homomorphism F* -• Qf will be denoted by À -> 1. Let V be a vector space of dimension n ^ 1 over F. Define S to be the set of all pairs (d9 P) where d = (V\ <= V% c — cz Vn-\) is a complete flag in K(dim V{ = 0 and Pe V—Vn-\. The group Gh(V) acts transitively on S. Let V be a vector space of dimension 2n ^ 2 over F endowed with a nondegenerate symplectic form. Define S" to be the set of all pairs (d9 P) where d = (V\ <= V2 a ... c Vn) is a complete isotropic flag in V\ (dim Vt= /) and P e V^-xVn. Let CSp(K') be the