Admissible representations, multiplicity-free representations and visible actions on non-tube type Hermitian symmetric spaces
The following two remarkable facts along with some first corollaries concerning the strong action visibility concept of \textit{T. Kobayashi} [Acta Appl. Math. 81, No. 1--3, 129--146 (2004; Zbl 1050.22018)] are proved: {\parindent=8mm \begin{itemize} \item[(1)] Given a noncompact irreducible Hermitian symmetric space \(G/K\), the statements (i)-(vi) below are equivalent:\item [(i)] \(G/K\) is of non-tube type;\item [(ii)] \(G_\mathbb C/K^s_\mathbb C\) is spherical; \item [(iii)] the action of any compact real form \(G_u\) of \(G_\mathbb C\) on \(G_\mathbb C/K^s_\mathbb C\) is strongly visible; \item [(iv)] the \(K^s\) action on the Hermitian symmetric space \(G/K\) is strongly visible; \item [(v)] the restriction \(\pi|_{K^s}\) is \(K^s\)-admissible for any holomorphic discrete series representation \(\pi\) of \(G\); \item [(vi)] for any holomorphic discrete series representation \(\pi\) of \(G\) of scalar type, \(\pi|_{K^s}\) is multiplicity free. \item [(2)] Given a non-tube type Hermitian symmetric space \(G/K\), there exists an Abelian subgroup \(B\) of \(G_\mathbb C\) with \(\text{dim}(B) = 1+\text{rank}(G/K)\) giving rise to a decomposition of the form \(G_\mathbb C = G_u B K^s_\mathbb C\). \end{itemize}}
scientific article; zbMATH DE number 6486385