Weak Gelfand Pair Property And Application To GL(n+1),GL(n) Over Finite Fields

Abstract

Let F_q be the finite field with q elements. Consider the standard embedding GL(n,F_q) -> GL(n+1,F_q). In this paper we prove that for every irreducible representation pi of GL(n+1,F_q) over algebraically closed fields of characteristic different from 2 we have dim\pi^GL(n,F_q)<=2. To do that we define a property of weak Gelfand pair and prove a generalization of Gelfand trick for weak Gelfand pairs, using the anti-involution transpose to get the result for GL(n+1,F_q),GL(n,F_q). In a similar manner we show that for q not a power of 2 O(n+1,F_q),O(n,F_q) is a Gelfand pair over algebraically closed fields of characteristic different from 2.