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Zbl 0999.22013
DeBacker, Stephen
Homogeneity results for invariant distributions of a reductive $p$-adic group. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 35, No. 3, 391-422 (2002). ISSN 0012-9593

Let $G$ denote the group of $k$-points of a connected reductive group defined over a nonarchimedean local field $k$, and let $\frak g$ denote the Lie algebra of $G$. Let $\pi$ be an admissible representation of $G$, and $\varphi$ a suitable map from a subset of $\frak g$ into $G$ (such as the exponential map, when it exists). If the characteristic of $k$ is zero or $G=GL_n(k)$ then the character $\Theta_\pi$ of $\pi$ has a Harish-Chandra-Howe local character expansion, which takes the form $$ \Theta_\pi(\varphi(X)) = \sum_{O} c_{O}(\pi) \widehat\mu_{O}(X), $$ where $O$ ranges over the set of nilpotent orbits in $\frak g$, the $c_{O}(\pi)$ are complex coefficients, and the $\widehat\mu_{O}$ are certain functions on $\frak g$. The expansion is valid for all regular $X\in\frak g$ sufficiently close to zero. Thus, the set $\{\widehat\mu_O\}$ is a natural basis for the space of germs of admissible characters. \par A conjecture of Hales, Moy and Prasad [see {\it A. Moy} and {\it G. Prasad}, Comment. Math. Helvetici 71, 98-121 (1996; Zbl 0860.22006)] offers a quantitative version of this qualitative result, namely, an explicit neighborhood of values $X$ for which the expansion is valid. Vastly generalizing his previous works [The Hales-Moy-Prasad conjecture for $\text{Sp}_4$, in: Analyse harmonique sur le group $\text{Sp}_4$ (CIRM Luminy, 1998), University of Chicago Lecture Notes in Representation Theory (1999), On supercuspidal characters of $\text{GL}_\ell$, $\ell$ a prime, Ph.D. thesis, The University of Chicago (1997), Compos. Math. 124, No. 1, 11-16 (2000; Zbl 0964.22015)], the author proves this conjecture (as well as two analogues concerning invariant distributions on $G$ and $\frak g$), and offers a geometric algorithm for computing the coefficients $c_O(\pi)$. \par The proofs depend on several hypotheses, which can fail when the characteristic of $k$ is positive and small. On the other hand, the local expansion was not previously known for general groups over fields of any positive characteristic, large or small. \par The main techniques are inspired by those of {\it J.-L. Waldspurger} [Publ. Math., Inst. Hautes Étud. Sci. 81, 25-72 (1995; Zbl 0841.22009), Quelques résultats de finitude concernant les distributions invariantes sur les algèbres de Lie $p$-adiques, preprint (1993)] (who proved important special cases of the conjecture), {\it D. Barbasch} and {\it A. Moy} [J. Am. Math. Soc. 13(3), 639-650 (2000; Zbl 0976.22008), Ann. Sci. Éc. Norm. Supér. (4) 30(5), 553-567 (1997; Zbl 0885.22021)], and others. A key ingredient is the author's parameterization of nilpotent orbits in terms of Bruhat-Tits theory, given elsewhere [Ann. Math. (2) 156, 295-332 (2002)].
[Jeffrey Adler (Akron)]

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

Keywords: $p$-adic groups; homogeneity; local character expansion; Hales-Moy-Prasad conjecture

Citations: Zbl 0860.22006; Zbl 0964.22015; Zbl 0841.22009; Zbl 0976.22008; Zbl 0885.22021

Cited in: Zbl 1128.22008 Zbl 1015.20033

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