Distribution Algebras on p-adic Groups and Lie Algebras

Abstract

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.