A finite Linear Dependence of Discrete Series Multiplicities

TL;DR

This paper proves a finite linear dependence property of discrete series multiplicities for certain Lie groups, refining multiplicity one results and linking cusp form dimensions across weights.

Contribution

It establishes that a finite subset of discrete series multiplicities determines all multiplicities, refining the strong multiplicity one theorem for discrete series.

Findings

Finite subset of multiplicities determines all multiplicities.

Refinement of the strong multiplicity one theorem.

Equality of cusp form dimensions across weights inferred.

Abstract

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $\Gamma$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations of $G$. We prove that for any finite subset of $\widehat{G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(\Gamma \backslash G)$ determines all discrete series multiplicities in $L^2(\Gamma \backslash G)$. This allows us to obtain a refinement of the strong multiplicity one result for discrete series representations. As an application, we deduce that for two given levels, the equality of the dimensions of the spaces of cusp forms over a suitable finite set of weights implies the equality of the dimensions of the spaces of cusp forms for all weights.