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Distribution Algebras on padic Groups and Lie Algebras


Published:20110425
Printed: Oct 2011
Allen Moy,
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
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Abstract
When $F$ is a $p$adic field, and $G={\mathbb
G}(F)$ is the group of $F$rational points of a connected algebraic
$F$group, the complex vector space ${\mathcal H}(G)$ 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, $\mathcal{H}(G)$ has drawbacks such as the lack of
an identity element, and the process $G \mapsto \mathcal{H}(G)$
is not a functor. Bernstein introduced an enlargement
$\mathcal{H}\,\hat{\,}(G)$
of $\mathcal{H}(G)$. The algebra
$\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left
essentially compact. We show that the process $G \mapsto
\mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow
H$ is a morphism of $p$adic groups, let $F(\tau) \colon
\mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be
the morphism of $\mathbb{C}$algebras. We identify the kernel of
$F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$adic
Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra,
$\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the
natural projection, we show that $F(\tau)$ maps $G$invariant distributions
on $\mathcal{G}$ to $N_G (\mathfrak{m})$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 nondegenerate symmetric bilinear form on ${\mathfrak g}$
and in the case of $SL(2)$ show how certain members of the family can
be moved to the group.