When F is a p-adic field, and is the group of F-rational points of a connected algebraic F-group, the complex vector space 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, has drawbacks such as the lack of an identity element, and the process is not a functor. Bernstein introduced an enlargement . The algebra consists of the distributions that are left essentially compact. We show that the process is a functor. If is a morphism of p-adic groups, let be the morphism of ℂ-algebras. We identify the kernel of in terms of Ker. In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and the natural projection, we show that maps G-invariant distributions on to NG (m)-invariant distributions on m. 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.
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