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Le ”centre” de Bernstein. (The Bernstein ”centre”). (French) Zbl 0599.22016
Représentations des groupes réductifs sur un corps local, 1-32 (1984).
[For the entire collection see Zbl 0544.00007.]
This paper is concerned with the category Alg G of smooth (also termed ”algebraic”) representations of a reductive group G over a local non- Archimedean field. It extends earlier work of the author and A. V. Zelevinsky [Usp. Mat. Nauk 31, No.3(189), 5-70 (1976; Zbl 0342.43017); Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 441-472 (1977; Zbl 0412.22015)] and differs from related works on the subject in the fact that no admissibility condition is assumed on the representations.
The notion of ”support” of an irreducible representation developed in these papers yields a structure of complex variety on a (finite-to-one) quotient \(\Omega(G)\) of the set \(\hat G\) of classes of irreducible representations of G. The category Alg G is decomposed into subcategories corresponding to the connected components of \(\Omega(G)\). The center of the category Alg G is described in several equivalent ways: (a) as the algebra of multipliers of the convolution algebra H of locally constant compact supported functions on G; (b) as a convolution algebra of invariant distributions on G; and (c) as the algebra of regular functions on \(\Omega(G)\). Some applications of these results are given, e.g.: (i) the description of finitely generated representations of G; (ii) for a family of arbitrarily small compact subgroups K of G, the category of representations generated by their K-fixed elements is equivalent to the category of \(H(G,K)\)-modules, where \(H(G,K)\) is the subalgebra of H consisting of K-bi-invariant elements.
Further application of this theory is given in a recent paper by the author, P. Deligne and D. Kazhdan [”Trace Paley-Wiener theorem for reductive p-adic groups”, Preprint, Harvard Univ., Cambridge, Mass., 1984; per. revr.], where they prove Paley-Wiener type theorems for representations of G.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.