Journal of Lie Theory, Vol. 10, No. 1, pp. 157-170 (2000)

Ladder representation norms for hermitian symmetric groups

Mark G. Davidson and Ronald J. Stanke

Mathematics Department
Baton Rouge, LA 70809

Abstract: Let $G$ be a connected noncompact simple Hermitian symmetric group with finite center. Let $\cal{H}(\lambda )$ denote the geometric realization of an irreducible unitary highest weight representation with highest weight $\lambda $. Then $\cal{H}(\lambda )$ consists of vector-valued holomorphic functions on $G/K$ and the action of $G$ on $ \cal{H}(\lambda )$ is given in terms of a factor of automorphy. For highest weights $\lambda $ corresponding to ladder representations, we obtain the $G$-invariant inner product on $\cal{H}(\lambda ).$ This inner product arises as the pullback of an isometry $\Phi _{\cal{\lambda }}:\cal{H} (\lambda )\cal{\rightarrow H}(\tilde{\lambda})\otimes Y_{\lambda},$ where $Y_{\lambda }$ is finite dimensional and the weight $\tilde{\lambda}$ corresponds to a scalar valued representation. In all but finitely many cases the $G$-invariant inner product on $\cal{H }(\tilde{\lambda})$ is known and is used to express the $G$-invariant inner product on $\cal{H}(\lambda ).$ Explicit examples are given for families of ladder representations of $SU(p,q)$ and $SO^{\ast }(2n)$. Finally, inversion formulas for unitary intertwining operators between $\cal{H}(\lambda )$ and any equivalent realization are exhibited.

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