Matrix coefficients of discrete series representations of SU(3,1) as reproducing kernel
主 题: Matrix coefficients of discrete series representations of SU(3,1) as reproducing kernel
报告人: Professor Takayuki Oda (University of Tokyo, Japan)
时 间: 2016-03-11 15:00 - 16:00
地 点: 理科一号楼 1114(数学所活动）
Matrix coefficients play important roles in the representation theory of Lie groups as well as in the theory of automorphic forms. When the associated Riemannian symmetric space G/K is of Hermitian type, a semisimple Lie group G has holomorphic discrete series representations, and matrix coefficients of these representations at the minimal K-type are known explicitly as the Bergman kernel. This expression played the crucial role in the calculation of the dimensions of the spaces of holomorphic automorphic forms on G/K via the Godement-Selberg expression of the dimensions of them. On the contrary, we know little about automorphic forms belonging to non-holomorphic discrete series representations. We want to have some case study of the counterpart of the Bergman kernel in a case of a non-holomorphic discrete series representation. In this talk, for large discrete series representations of SU(3; 1), we give expressions of the radial parts of their matrix coefficients in terms of the generalized hypergeometric series, and describe their asymptotic behavior, explicitly. Geometrically speaking, this is to obtain an explicit formula for some Hilbert space of non-holomorphic harmonic L^2-sections in an SU(3; 1)-equivariant vector bundle over the complex hyperball of dimension 3.