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Journal of Lie Theory 30 (2020), No. 2, 489--512
Copyright Heldermann Verlag 2020

Derivatives of Elliptic Orbital Integrals on a Symplectic Space

Mark McKee
338 Hammond Lane, Providence, UT 84332, U.S.A.

Angela Pasquale
Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, 57045 Metz, France

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

For a real reductive dual pair with one member compact we study the orbital integrals on the corresponding symplectic space that occur in the Weyl-Harish-Chandra integration formula on that space. We obtain estimates of the derivatives of such integrals. These estimates are needed for expressing the intertwining distribution attached to a pair of representations in Howe's correspondence in terms of the orbital integrals. This is in analogy to Harish-Chandra's theory, where the distribution character of an irreducible admissible representation of a real reductive group factors through the semisimple orbital integrals on the group.

Keywords: Reductive dual pairs, Howe duality, Weyl calculus, Lie superalgebras.

MSC: 22E45, 22E46, 22E30.

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