Archimedean zeta integrals on U(2,1)

TL;DR

This paper explicitly computes archimedean zeta integrals for the dual pair U(2,1), involving detailed calculations of matrix coefficients of Weil representations and discrete series, with implications for automorphic L-functions.

Contribution

It provides explicit formulas for archimedean zeta integrals on U(2,1), advancing the understanding of their role in automorphic L-functions and arithmetic applications.

Findings

Explicit formulas for archimedean zeta integrals on U(2,1)

Calculation of matrix coefficients of Weil representations

Evaluation of discrete series matrix coefficients

Abstract

It is known that for a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions, which have applications to arithmetic. In this paper we explicitly calculate archimedean zeta integrals of this type for U(2,1). In particular we compute the matrix coefficients of Weil representations using joint harmonics in the Fock model, and those of discrete series using Schmid operators.