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Zbl 0885.22021
Barbasch, Dan; Moy, Allen
Local character expansions. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 30, No. 5, 553-567 (1997). ISSN 0012-9593

Let $G$ be a reductive group defined over a nonarchimedean local field $k$ of characteristic zero, and let $\Theta_\pi$ be the character of an irreducible admissible representation $\pi$ of the group $G(k)$ of $k$-rational points of $G$. Then $\Theta_\pi$ can be represented as a locally constant integrable function on the regular set. In turn there is the local character expansion, due to Harish-Chandra and Howe, which expresses $\Theta_\pi$ as a linear combination of Fourier transforms of nilpotent orbits in a sufficiently small neighborhood of zero in the Lie algebra ${\germ g} =\text {Lie} (G)(k)$. Much qualitative and quantitative information on the representation $\pi$ can be obtained from this expansion. The main focus of this paper is to develop a method to compute the local character expansion of a depth zero representation $\pi$, i.e. to actually calculate the coefficients $c_{\cal O} (\pi)$, one for each nilpotent orbit ${\cal O}$ in ${\germ g}$, in the expansion. The main idea is to use the generalized Gelfand-Graev characters for finite groups as test functions to plug into the character formula. Results of Waldspurger on the validity of the local character expansion in a large enough neighborhood of the identity permit to do so. This method leads to a classification of the unipotent orbits in terms of parahoric subalgebras.
[J.Schwermer (Eichstätt)]

MSC 2000:: *22E50 Repres. of Lie and linear algebraic groups over local fields

Keywords: representation; Fourier transforms; orbits; Lie algebra; local character expansion; Gelfand-Graev characters

Cited in: Zbl 0999.22013

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