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Fourier Inversion on Borel Subgroups of Chevalley Groups: The Symplectic Group Case

Ronald L. Lipsman
Transactions of the American Mathematical Society
Vol. 260, No. 2 (Aug., 1980), pp. 607-622
DOI: 10.2307/1998026
Stable URL: http://www.jstor.org/stable/1998026
Page Count: 16
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Fourier Inversion on Borel Subgroups of Chevalley Groups: The Symplectic Group Case
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Abstract

In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups and nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Bore subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the "pseudodifferential" Dixmier-Pukanszky operator in the nonarchimedean situation.

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