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.
mark.mckee.zoso@gmail.com
Angela Pasquale
Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, 57045 Metz, France
angela.pasquale@univ-lorraine.fr
Tomasz Przebinda
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A.
tprzebinda@gmail.com
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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