Real Shintani Functions on $U(n,1)$

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 4, Page 609--688.

Tsuzuki, Masao
Real Shintani Functions on $U(n,1)$
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Let $G=\sU(n,1)$ and $H=\sU(n-1,1) × \sU(1)$ with $n\geqslant 2$. We realize $H$ as a closed subgroup of $G$, so that $(G,H)$ forms a semisimple symmetric pair of rank one. For irreducible representations $π$ and $η$ of $G$ and $H$ respectively, we consider the space ${\cal I}_{η,π}={\rm Hom}_{\g_\C,K} (π,{\rm Ind}_H^G(η))$ with $K$ a maximal compact subgroup in $G$ and $\g_\C$ the complexified Lie algebra of $G$. The functions that belong to ${\rm Im}(Φ)$ for some $Φ\in {\cal I}_{η,π}$ will be called the {\it Shintani functions}. We prove that ${\rm dim}_\C{\cal I}_{η,π}\leqslant 1$ for any $π $ and any $η$, giving an explicit formula of the Shintani functions that generate a \lq corner\rq\ $K$-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.

Mathematics Subject Classification (1991): Primary 11F70; Secondary 22E46, 33C05, 11F67
Mathematical Reviews Number: MR1868294

Received: 2000-11-06