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We give a description of the Bernstein center of a reductive p-adic group G in terms of invariant locally integrable functions and compute a basis of these functions for the group SL(2). 1.1. The Lie algebra g and associated enveloping algebra U(g) of a connected Lie group G are fundamental tools in the study of the group's representations. The center Z(U(g)) of the enveloping algebra is particularly useful for a number of pur- poses. There are three useful descriptions of the center: (i) an algebraic description of the center in terms of the Lie algebra, (ii) the Harish-Chandra homomorphism identication of the center with the space of Weyl invariant regular functions on the symmetric algebra of a Cartan subalgebra, and (iii) as invariant distributions on the group supported at the identity. 1.2. By following J. Bernstein, one can also consider the center Z(U(g)) from a ring and categorical point of view. If A is an algebra with identity, then the center of A is naturally isomorphic to the algebra of endomorphisms of the identity functor of the category of all A-modules. In particular, Z(U(g)) is isomorphic to the algebra of endomorphisms of the identity functor of the category of modules of the Lie algebra. This formulation of the center of the enveloping algebra is the starting point for Bernstein's construction of an analogue of the center of the enveloping algebra for reductive p-adic groups. If G is a reductive p-adic group G ,l etS(G) denote the category of smooth representations of G. The Bernstein center Z(G )o f G is dened to be the algebra of endomorphisms of the identity functor ofS(G) There are two realizations of Bernstein center. Both are much less concrete than in the real case.

The Bernstein center in terms of invariant locally integrable functions