On a multiplicity formula of weights of representations of 𝑆𝑂*(2𝑛) and reciprocity theorems for symplectic groups

Leung, Eric Y.; Ton-That, Tuong

doi:10.1090/S0002-9939-1995-1227520-2

On a multiplicity formula of weights of representations of $\textrm {SO}^ *(2n)$ and reciprocity theorems for symplectic groups
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by Eric Y. Leung and Tuong Ton-That PDF

Proc. Amer. Math. Soc. 123 (1995), 1281-1288 Request permission

Abstract:

A formula relating the multiplicity of a weight of a holomorphic discrete series of the group $S{O^ \ast }(2n)$ to the frequency of occurrence of an irreducible holomorphically induced representation of the group $Sp(2n,\mathbb {C})$ in an n-fold tensor product of irreducible symmetric representations of $Sp(2n,\mathbb {C})$ is given. Reciprocity theorems relating holomorphic discrete series of $S{O^ \ast }(2n)$ (resp. $Sp(2n,\mathbb {R})$) to holomorphically induced representations of $Sp(2k,\mathbb {C})$ (resp. $SO(N,\mathbb {C})$) are also derived.

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