Representations of groups over close local fields. (English) Zbl 0634.22010

Kazhdan, D.

J. Anal. Math. 47, 175-179 (1986).

Let F and F’ be nonarchimedean local fields, with rings of integers O and O’, respectively, and \(P\subset O\) and P’\(\subset O'\) the maximal ideals. Say that F and F’ are m-close if the quotient rings \(O/P^ m\) and \(O'/P^{'m}\) are isomorphic. Let \({\mathbb{G}}\) be a reductive Z-group, assumed split for convenience. For any ring A, set \(G_ A={\mathbb{G}}(A)\). For any positive integer l, let \(A_ l\) be the quotient \(O/P^ l\), and let \(K_ l(F)\) be the congruence subgroup of \(G_ O mod P^ l\). Identify \(G_{A_ l}\) with \(G_ O/K_ l(F)\). Let \(H_ l(G,K)\) be the Hecke algebra of \(K_ l(F)\)-biivariant measures, under convolution, on G.
The main theorem is that given a local nonarchimedean field F and a positive integer l, there exists an \(m\geq l\) such that, if F’ is m-close to F, then the natural morphism \(H_ l(G,F)\to H_ l(G,F')\) is an algebra isomorphism. As a corollary, if F is a local field of positive characteristic, and l is a positive integer, then there exists a local field F’ of characteristic zero such that the Hecke algebras \(H_ l(G,F)\) and \(H_ l(G,F')\) are isomorphic. Since irreducible \(H_ l\)- modules are in one-to-one correspondence with irreducible representations of G which have non-zero \(K_ l\)-fixed vectors, this result can be used, for example, to extend results proved for groups over fields of characteristic zero, to groups over fields of positive characteristic.
The present paper contains the following application. In the article reviewed above, the author shows for groups over fields of characteristic zero, that the characters of tempered admissible representations of G are dense in the space of all invariant distributions on G. The corresponding result for groups over fields of positive characteristic is shown using a result of Bernstein: The following are equivalent. (1) Suppose h is an element of the Hecke algebra H(G) for which \(<\pi,h>=0\) for all irreducible representations \(\pi\) of G. Then h is annihilated by all invariant distributions on G. (2) For each positive l, the algebra \(H_ l\) has the property, that if \(h\in H_ l\) has trace zero in every irreducible \(H_ l\)-module, then h is in the span \([H_ l,H_ l]\) of all commutators in \(H_ l\).

Reviewer: C.D.Keys

MSC:

22E35	Analysis on \(p\)-adic Lie groups
22E50	Representations of Lie and linear algebraic groups over local fields
43A80	Analysis on other specific Lie groups
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
22E20	General properties and structure of other Lie groups
43A05	Measures on groups and semigroups, etc.

Keywords:

nonarchimedean local fields; reductive Z-group; congruence subgroup; biivariant measures; Hecke algebras; irreducible representations; characters of tempered admissible representations; invariant distributions

Citations:

Zbl 0634.22009

References:

[1]	Allan, Nelo D., Hecke rings of congruence subgroups, Bull. Am. Math. Soc., 78, 541-545 (1972) · Zbl 0264.22006
[2]	Bernstein, I.; Deligne, P., Le “centre” de Bernstein, inReprésentations des groupes reductifs sur un corps local (1985), Paris: Hermann, Paris
[3]	Kazhdan, D., Cuspidal geometry of p-adic groups, J. Analyse Math., 47, 1-36 (1986) · Zbl 0634.22009 · doi:10.1007/BF02792530