Real Shintani Functions on U(n, 1)

Abstract

n 2. We realize H as a closed subgroup of G, so that (G,H) forms a semisimple symmetric pair of rank one. For irreducible rep-resentations π and η of G and H respectively, we consider the space Iη,π = HomgC,K(π, IndGH(η)) with K a maximal compact subgroup in G and gC the complexified Lie algebra of G. The functions that be-long to Im(Φ) for some Φ ∈ Iη,π will be called the Shintani functions. We prove that dimCIη,π 1 for any π and any η, giving an explicit formula of the Shintani functions that generate a ‘corner ’ K-type of π in terms of Gaussian hypergeometric series. We also give an explicit formula of corner K-type matrix coefficients of π in the usual sense.