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Zbl 0856.22025
Murnaghan, Fiona
Characters of supercuspidal representations of classical groups. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 29, No. 1, 49-105 (1996). ISSN 0012-9593

Let $G$ be a classical (not necessarily connected) $p$-adic group (with compact center). An irreducible supercuspidal representation $\pi$ of $G$ is called (in the paper) Kirillov if there exists a neighborhood $V_\pi$ of zero in Lie $G$, and a regular elliptic element $X_\pi$ in Lie $G$, such that $\Theta_\pi(\exp X)= d(\pi) \widehat\mu_{{\cal O}(X_\pi)}(X)$ at any $X$ in $V_\pi\cap \text{Lie } G_{\text{reg}}$. Here $\Theta_\pi$ denotes the character of $\pi$; $d(\pi)$ is the formal degree of $\pi$; $\widehat\mu_{{\cal O}(X_\pi)}$ is the Fourier transform of the invariant measure $\mu_{{\cal O}(X_\pi)}$ on the $G$-orbit of $X_\pi$. Not all $\pi$ are Kirillov. Harish-Chandra's local character expansion is: $\Theta_\pi(\exp X)= \sum c_{{\cal O}} \widehat\mu_{{\cal O}}(X)$, $\cal O$ ranges over the set of nilpotent $G^0$-orbits in Lie $G$. Since $\widehat\mu_{{\cal O}(Y)}(X)= \sum_{\cal O} \Gamma_{{\cal O}}(Y) \widehat\mu_{{\cal O}}(X)$, Kirillov $\pi$ have $c_{{\cal O}}(\pi)= d(\pi)\Gamma_{{\cal O}}(X_\pi)$. The paper shows that $\pi$ is Kirillov if it is compactly induced of the form $\text{Ind}^G_{P_\Psi} \rho_\Psi$, where $\Psi$ is a ``uniform'' (p. 77) ``cuspidal datum'' (p. 68), $P_\Psi$ is a compact open subgroup and $\rho_\Psi$ is an irreducible finite-dimensional representation thereof (p. 70); in this case $X_\pi$ can be taken to be $c_\Psi$ (p. 69). Much of the paper concerns these definitions (due to {\it L. Morris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, 233-274 (1992; Zbl 0782.22012)]) and their properties.
[Y.Flicker (Columbus/Ohio)]

MSC 2000:: *22E50 Repres. of Lie and linear algebraic groups over local fields
22E35 Analysis on p-adic Lie groups

Keywords: germ expansion; formal degree; supercuspidal representation; Fourier transform; measure; character expansion

Citations: Zbl 0782.22012

Cited in: Zbl 1031.22008

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