4-Homogeneous groups by Kantor W.M.

By Kantor W.M.

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Let W Let G be a t. d. group, H a closed subgroup, and U a vector be a finite-dimensional PH-stable subspace of C 00(G:U), where PH denotes the restriction to H of the right regular representation p of G on C 00(G:U). Proof. Then W is a smooth H-module under Let dimW = r. For x such that cp , , ••• , ~,ul (if; <{J map to a basis for W'. E W). E , xr ,ur G and u' E PH. U' define cp x,u Obviously, we may choose is a basis for W'. •. , (xr, u~) Let (x1 , u 11 ) , ••• , (xr' ur') Let rf;1, •..

D. V is called admissible if: 7r l Omitted. O,n-oo. is smooth, and group. Groups. A representation 7r of G in a vector space 36 (2) for every open (compact) subgroup K of G, the space of K-fixed vectors, V K' is finite-dimensional. As remarked earlier, 7r is smooth over all open compact subgroups of G. if V = U V K' where K ranges K If, in addition, dim V K < oo for all K, then 7r is admissible. Let 7r be irreducible and admissible. representation has the same properties. Then, clearly, every equivalent We write e

A(a)x(a)da. l = ker x. l (i = 1, ••• , n). Let /J1t = n/1'1/. and i=l l since defines a faithful representation of 1J. , dim °B= n. l The lemma is now an immediate consequence of a well-known theorem of Dedekind. Let c; be a smooth representation of A. in a vector space W. ) write W x for the set of all w e W for all t e A. :_ O. ). For any w e W we have xi xi d(w) < dim W • Whether W is finite-dimensional or not, one sees that w e W xi d x - that W = @ W if and only if there exists d = d(w) > 0 Corollary 1.

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