Canonical representations of \(Sp(1, n)\) associated with representations of \(Sp(1)\)
Canonical representations for \(SU(1,1)\) have been introduced by \textit{I. M. Gel'fand, M. I. Graev} and \textit{A. M. Vershik} [Russ. Math. Surveys 28, 87-132 (1973; Zbl 0297.22003)] and have now received more interest. They are defined by a certain positive definite kernel on the unit disk in the complex plane, as the symmetric space of \(SU(1,1)\). In the present paper the authors introduce certain matrix-valued positive definite kernels which act on an irreducible representation \(\tau\) of \(Sp(1)\). They prove that the corresponding canonical representations can be realized as the restriction of certain maximal degenerate representations of \(SL(n+1, \mathbb H)\). (There are also some other realizations, for example as the restrictions of holomorphic representations of a larger group of Hermitian type.) The authors find their irreducible decomposition into \(\tau\)-spherical representations; this is done by identifying the \(\tau\)-spherical functions as hypergeometric functions and by calculating the spherical transform of the kernel. For certain \(\tau\) there appear some discrete parts in the decomposition, this in turn gives an interesting embedding of those discrete spherical representations into the canonical representations.
scientific article; zbMATH DE number 1335323