Semisimple orbital integrals on the symplectic space for a real reductive dual pair

TL;DR

This paper establishes a Weyl Harish-Chandra integration formula for reductive dual pairs acting on symplectic spaces, introducing new concepts like Cartan subspaces and almost semisimple elements, and providing orbital integral estimates.

Contribution

It introduces the notion of Cartan subspaces and almost semisimple elements in symplectic spaces, and proves a Weyl Harish-Chandra integration formula for reductive dual pairs.

Findings

Almost semisimple elements are dense in the symplectic space.

Orbital integrals can be estimated for different Cartan subspaces.

The integration formula generalizes classical results to the setting of dual pairs.

Abstract

We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space $W$. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space $W$. We prove that the almost semisimple elements are dense in $W$. Finally, we provide estimates for the orbital integrals associated with the different Cartan subspaces in $W$.