A Cauchy Harish-Chandra integral, for a real reductive dual pair. (English) Zbl 0953.22014

Przebinda, Tomasz

Invent. Math. 141, No. 2, 299-363 (2000).

For a real irreducible dual pair \(G\), \(G'\) in a metaplectic group, with \(\text{rank }G\leq \text{rank }G'\), the author constructs an integral kernel operator from the space of invariant eigendistributions on \(G\) to the space of invariant distributions on \(G'\), and conjectures that this operator is compatible with Howe’s correspondence on the level of characters. The construction indicates a direct link between the Cauchy determinant identity and the Howe correspondence.

Reviewer: Vladimir L.Popov (Wien)

MSC:

22E45

Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

Cauchy identity; action; character; trace; classical invariant theory; real irreducible dual pair; metaplectic group; Cauchy determinant identity; Cauchy Harish Chandra integral