On a multiplicity formula of weights of representations of ${\rm SO}\sp *(2n)$ and reciprocity theorems for symplectic groups

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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