Representations over p-adic fields of the finite general linear and unitary groups

Abstract

Let p be a prime and let K denote the field of p-adic numbers. Let q =pm be a power of p and let GF(q) denote the finite field with q elements. K has a unique unramified extension K, of degree m, whose associated residue class field is isomorphic to GF(q). K, is obtained by adjoining a primitive root of unity of order q 1 to K. (For a description of these facts, see, for example, Theorem 2, p. 54 and Proposition 16, p. 77 of [S].) Now consider the general linear group G = GL(n, q) of degree n over GF(q). Part of the purpose of this paper is to show that if we consider representations of G over extension fields of K, the field K, is a minimal splitting field for G. Similarly, if U = U(n, q*) is the corresponding unitary group, the characters of U take values in K,,. Moreover, if p and q are sufficiently large, the representations themselves are realizable in K,, and are unitary with respect to the involution of K,, over K,. We also discuss the number of representations of G and U that are realizable in given subfields of Km and K,,. Our proofs rely mainly on a well-known permutation lemma of Brauer and give no indication of how these K,,,-representations might be constructed in practice. We also note here that analogous results on the p-adic representations of other classical groups do not appear to hold in complete generality. This can already be seen in the case of the group SL(2,p), certain of whose irreducible characters take values in a quadratic ramified extension of K.