Semisimple orbital integrals on the symplectic space for a real reductive dual pair
The authors introduce the notion of Cartan subspace for a symplectic real vector space. The main result of this paper is an analog of the Weyl Harish-Chandra formula for the symplectic real vector space \(W\). It gives the integral on \(W\) of a continuous compactly supported function in terms of semisimple orbital integrals parametrized by the mutually non-conjugate Cartan subspaces of \(W\). The authors also show that semisimple elements in \(W\) are dense in \(W\) (with the exception of two cases of dual pairs, namely \(O_{p,q}\times\mathrm{Sp}_{2n}(\mathbb{R})\) with \(p+q\) odd and \(p+q<2n\) and \(O_p(\mathbb{C})\times\mathrm{Sp}_{2n}(\mathbb{C})\) with \(p\) odd and \(p<2n\)). For these two remaining cases, the authors introduce the notion of almost semi-simple elements and prove that these are dense. Finally, the paper provides some estimates for the orbital integrals associated with the different Cartan subspaces in \(W\).
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