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Distribution Algebras on p-Adic Groups and Lie Algebras

Authors Moy, Allen View this author's profile
Issue Date 2011
Source Canadian Journal of Mathematics , v. 63, (5), October 2011, p. 1137-1160
Summary When F is a p-adic field, and G = G(F) is the group of F-rational points of a connected algebraic F-group, the complex vector space H(G) of compactly supported locally constant distributions on G has a natural convolution product that makes it into a 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, H(G) has drawbacks such as the lack of an identity element, and the process G bar right arrow H(G) is not a functor. Bernstein introduced an enlargement H(G) of H(G). The algebra H(G) consists of the distributions that are left essentially compact. We show that the process G bar right arrow H(G) is a functor. If tau: G -> H is a morphism of p-adic groups, let F(tau):H(G) -> (H)(H) be the morphism of C-algebras. We identify the kernel of F(tau) in terms of Ker(tau). In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and tau: g -> m the natural projection, we show that F(tau) maps G-invariant distributions on G to N(G)(III)-invariant distributions on III. Finally, we exhibit a natural family of G-invariant essentially compact distributions on g associated with a G-invariant non-degenerate symmetric bilinear form on g and in the case of SL(2) show how certain members of the family can be moved to the group.
ISSN 0008-414X
Language English
Format Article
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