A Finite Linear Dependence of Discrete Series Multiplicities

Abstract

Let G be the real points of a connected semisimple simply connected algebraic group over

such that G has a compact Cartan subgroup and let

Γ $\Gamma $ normal upper Gamma

be a uniform lattice in G . Let

G ^ d $\widehat {G}_d$ ModifyingAbove upper G With caret Subscript d

denote the set of equivalence classes of unitary discrete series representations of G . We prove that for any finite subset of

G ^ d $\widehat {G}_d$ ModifyingAbove upper G With caret Subscript d

satisfying a certain condition, the associated finite set of discrete series multiplicities in

L 2 ( Γ ∖ G ) $L^2(\Gamma \backslash G)$ upper L squared left parenthesis normal upper Gamma minus upper G right parenthesis

determines all discrete series multiplicities in

L 2 ( Γ ∖ G ) $L^2(\Gamma \backslash G)$ upper L squared left parenthesis normal upper Gamma minus upper G right parenthesis

. 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.