A Cauchy Harish-Chandra Integral, for a Real Reductive Dual Pair

For a real irreducible dual pair G, G' in a metaplectic group, with the rank of G less or equal to the rank of G', we construct an integral kernel operator from the space of invariant eigen-distributions on G to the space of invariant distributions on G', and conjecture that this operator is compatible with Howe's correspondence on the level of characters.

The construction indicates a direct link between the Cauchy determinantal identity and Howe's correspondence. The estimates involved are based on Harish-Chandra's theory of orbital integrals.

Appeared in Invent. Math. 141 (2000), no. 2, 299-363