A REMARK ON GELFAND PAIRS OF FINITE-GROUPS OF LIE TYPE

We show how to deduce multiplicity one theorems for cuspidal represent ations of finite groups of Lie type from analogous results for p-adic groups. We then look at examples where the latter is known. One such e xample is the restriction of irreducible representations of SO(n) to S O(n -1) [S. Rallis, preprint]. We show that the multiplicity of a cusp idal representation of the finite group SO(n -1) in the restriction of a cuspidal representation of SO(n) to SO(n -1) is at most one. (C) 19 98 Academic Press.