UNIPOTENT CHARACfERS OF THE EVEN ORTHOGONAL GROUPS OVER A FINITE FIELD

The characters of unipotent representations of a simple algebraic group over Fq of type oF D. on any regular semisimple element are explicitly known for large q. This paper deals with the remaining case: type D •. The purpose of this paper is to give explicit formulas for the character of the unipotent representations of the special orthogonal group of a quadratic form of even dimension over a finite field Fq on any regular semisimple element, provided that q is sufficiently large. The methods used in this paper are those of [4], where the case of symplectic and odd orthogonal groups was considered. To avoid repetitions, I have only given proofs for the results which differ essentially from those in [4]. 1. Characters of Weyl groups and Heeke algebras of type Dn' 1.1. Let Dn (n ;;;. 2) be the Coxeter group with diagram b (Here SI' S2" .. ,sn-2' b, b' are the simple reflections.) For n = 2, this is the group generated by the commuting involutions b, b'. We make the convention that DI is the group with 1 element. We may regard Dn (n ;;;. I) as a subgroup of index 2 of the group Dn generated by Dn and an element cp of order 2 which commutes with all Si (I ,..;; i ,..;; n 2) and satisfies cpbcp = b' (if n ;;;. 2). 1.2. The group Dn is isomorphic in two ways with the Coxeter group of type Bn (denoted w" in [4]): One choice of simple reflections is {SI' S2" ",sn-2' b, cp}; the other is {SI' S2.· •• ,sn-2' b', cpl. Each of these two choices gives rise to an isomorphism Dn R< w" of Coxeter systems. We may thus parametrize the irreducible Q[Dn]-modules by ordered pairs 0'1,0'2 of irreducible representations of the symmetric groups 6 k , 6, (k + 1= n), as in [4, 2.1] and this parametrization will be independent of the choice of simple reflections for Dn (the two choices are conjugate by cp). It will be convenient to param~trize the irreducible Q[Dn]-modules by ordered pairs (f) where S, T are finite subsets of {O, 1,2, ... } defined as follows: if a I' 0'2 Received by the editors May 19, 1981. 1980 Mathematics Subject Classification. Primary 20040. 1 Supported in part by the National Science Foundation. 733 ©1982 American Mathematical Society 0002-9947/82/0000-0430/$06.50 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use