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

Research paper by **Yoav Ben Shalom**

Indexed on: **28 Oct '12**Published on: **28 Oct '12**Published in: **Mathematics - Representation Theory**

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#### 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.