The Cauchy Harish-Chandra integral and the invariant eigendistributions (Q2929671)

The present paper finishes the study of the Cauchy Harish-Chandra integral which was initiated in [\textit{F. Bernon} and \textit{T. Przebinda}, J. Lie Theory 21, No. 3, 615--702 (2011; Zbl 1238.22007); ibid. 499--613 (2011; Zbl 1241.22016)].NEWLINENEWLINELet \( W \) be a finite dimensional vector space over the reals, with a non-degenerate symplectic form. Let \( J \) be a positive compatible complex structure on \( W \), \( \mathrm{Sp}(W) \) (resp. \( \mathrm{sp}(W) \)) the corresponding symplectic group (resp. the symplectic Lie algebra). The conjugation by \( J \) is a Cartan involution \(\theta\) on \(\mathrm{ sp}(W) \).NEWLINENEWLINELet \( \widetilde {\mathrm{Sp}}(W) \) be the metaplectic group with the canonical surjection \( \widetilde {\mathrm{Sp}}(W)\to\mathrm{Sp}(W)\). For a subset \( E \) of \(\mathrm{Sp}(W) \), denote by \( \tilde E \) the preimage of \( E \) in \( \widetilde {\mathrm{Sp}(W)} \).NEWLINENEWLINELet \( (G, G' ) \) be a reductive dual pair of \( \mathrm{Sp}(W) \) [\textit{R. Howe}, J. Am. Math. Soc. 2, No. 3, 535--552 (1989; Zbl 0716.22006)] with the rank of \( G' \) less or equal to the rank of \( G \). Denote by \( \mathfrak g,\mathfrak g' \) the Lie algebras of \( G \), \( G' \) respectively. The Cartan involution \(\theta\) can be lifted to the group and one can assume that \( G \) and \( G' \) are preserved by \(\theta\). Let \( H' \) be a Cartan subgroup of \( G' \) preserved by \(\theta\).NEWLINENEWLINEIn [\textit{T. Przebinda}, Invent. Math. 141, No. 2, 299--363 (2000; Zbl 0953.22014)] the Cauchy Harish-Chandra integrals \( \widetilde {\mathrm{chc}} \) and \( \widetilde {\mathrm{Chc}} \) have been defined, which are functions on \( \mathfrak {h'}^{\text{reg}}\times \mathfrak g \) resp. on \( \tilde {H'}^{\text{reg}}\times G \). The authors use also the normalized versions chc and Chc of both mappings and they define for a test function \( \varphi:\mathfrak g\to \mathbb C \) the function \(\mathrm{chc}(\varphi)(x' ) =\int_{\mathfrak g} \mathrm{chc}(x' + x)\varphi(x) dx \) on \( {\mathfrak h'}^{\text{reg}} \). Similarly one defines \(\mathrm{Chc}(\varphi ) \) on \( \widetilde {H'}^{\text{reg}} \) for a test function \(\varphi \) on \( \widetilde G \). The authors prove that for any \( \varphi \in \mathcal D(\mathfrak g) \) the function \( \mathrm{chc}(\varphi) \) is contained in the space \( \tilde {\mathcal I(\mathfrak g')} \) of unnormalized orbital integrals on \( {\mathfrak g'}^{\text{reg}} \) and similarly for \( \varphi \in\mathcal D(\widetilde { G'}) \) the function \( \mathrm{Chc}(\varphi) \) is an unormalized orbital integral on \( \widetilde G' \). One can then also consider the transpose \(\mathrm{Chc}^t: {\mathcal D'}^{\tilde G'}(\tilde G')\to \mathcal {\mathcal D'}^{\tilde G}(\tilde G) \) of the mapping \(\mathrm{Chc} \). Denote by \( U(\mathfrak g_\mathbb C) \) (resp. \( U(\mathfrak g'_\mathbb C) \)) the enveloping algebra of \( \mathfrak g_\mathbb C \) (resp. \( \mathfrak g'_\mathbb C \)). Consider the Capelli Harish-Chandra homomorphism \( \mathcal C_{\mathfrak g,\mathfrak g'}:U(\mathfrak g_\mathbb C)\to U(\mathfrak g'_\mathbb C) \) and let \( L \) be the left regular representation of \( U(\mathfrak g_\mathbb C) \) on \(\mathcal D(\tilde G) \). It is proved that for \(z\in U(\mathfrak g_\mathbb C)\) and \( \varphi \in \mathcal D(G) \) one has \( \mathrm{Chc}(L( \check z )\varphi ) = L(\mathcal C_{\mathfrak g,\mathfrak g'})\mathrm{Chc}(\varphi), \) where \( z \to \check z \) is the canonical involution on the universal enveloping algebra. The main result of this series of papers on the Cauchy Harish-Chandra integral states then that for \( z\in U(\mathfrak g_\mathbb C) \) and \(u' \in \mathcal D^{\tilde G'} {(\tilde G' )'} \) one has that \(\mathrm{Chc}^t(L(\mathcal C_{\mathfrak g,\mathfrak g'} (z))u' ) = L(\check z)\mathrm{Chc}^t(u' ). \) As in the other two papers the proofs are based on long computations.