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Journal of Lie Theory 21 (2011), No. 3, 615--702
Copyright Heldermann Verlag 2011

Normalization of the Cauchy Harish-Chandra Integral

Florent Bernon
Dép. de Mathématiques, Université de Versailles, 78035 Versailles, France

Tomasz Przebinda
Dept. of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A.

This is the second of our three articles showing that the Cauchy Harish-Chandra integral maps invariant eigendistributions to invariant eigendistributions with the correct infinitesimal character. In this paper, we define a normalization of this integral for all real reductive dual pairs. Then we prove that the normalized Cauchy Harish-Chandra integral maps orbital integrals to orbital integrals for the pairs (Up,q, U1,1), (Op,q, Sp2(R)) and (Sp2n(R), O1,2).

Keywords: Dual pairs, orbital integrals.

MSC: 22E45; 22E46

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