Distribution Algebras on p-adic Groups and Lie Algebras

Moy, Allen

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 1137-1160

Résumé

When $F$ is a $p$ -adic field, and $G\,=\,\mathbb{G}\left( F \right)$ is the group of $F$ -rational points of a connected algebraic $F$ -group, the complex vector space $\mathcal{H}\left( G \right)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a $\mathbb{C}$ -algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$ -adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}\left( G \right)$ has drawbacks such as the lack of an identity element, and the process $G\,\mapsto \,\mathcal{H}\left( G \right)$ is not a functor. Bernstein introduced an enlargement ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ of $\mathcal{H}\left( G \right)$ . The algebra ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ consists of the distributions that are left essentially compact. We show that the process $G\,\mapsto \,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ is a functor. If $\tau \,:\,G\,\to \,H$ is a morphism of $p$ -adic groups, let $F\left( \tau\right):\,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)\,\to \,{{\mathcal{H}}^{\hat{\ }}}\left( H \right)$ be the morphism of $\mathbb{C}$ -algebras. We identify the kernel of $F\left( \tau\right)$ in terms of $\text{Ker}\left( \tau\right)$ . In the setting of $p$ -adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \,:\,\mathfrak{g}\,\to \,\mathfrak{m}$ the natural projection, we show that $F\left( \tau\right)$ maps $G$ -invariant distributions on $\mathcal{G}$ to ${{N}_{G}}\left( \mathfrak{m} \right)$ -invariant distributions on $\mathfrak{m}$ . Finally, we exhibit a natural family of $G$ -invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$ -invariant non-degenerate symmetric bilinear form on $\mathfrak{g}$ and in the case of $SL\left( 2 \right)$ show how certain members of the family can be moved to the group.

Détail

DOI : 10.4153/CJM-2011-025-3

Mots-clés : 22E50, 22E35, distribution algebra, p-adic groups