A Plancherel formula for parabolic subgroups

Nakata, Mie

doi:10.1090/S0002-9947-1990-1019522-6

A Plancherel formula for parabolic subgroups
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Trans. Amer. Math. Soc. 319 (1990), 243-256 Request permission

Abstract:

We obtain explicit Plancherel formulas for the parabolic subgroups $P$ of $p$-adic unitary groups which fix one dimensional isotropic subspaces. By means of certain limits of difference operators (called strong derivatives), we construct a Dixmier-Pukanszky operator which compensates for the nonunimodularity of the group $P$. Then, we compute the Plancherel formula of $N \cdot A$, where $N$ is the nilradical of $P$ and $A = {Q’_p}$, the multiplicative group of nonzero $p$-adic numbers, by formulating a $p$-adic change of variable formula and using the strong derivative.

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