A Remark on Gelfand Pairs of Finite Groups of Lie Type

We show how to deduce multiplicity one theorems for cuspidal representations of finite groups of Lie type from analogous results forp-adic groups. We then look at examples where the latter is known. One such example is the restriction of irreducible representations ofSO(n) toSO(n−1)9. We show that the multiplicity of a cuspidal representation of the finite groupSO(n−1) in the restriction of a cuspidal representation ofSO(n) toSO(n−1) is at most one.