Real Reductive Groups I (Pure and Applied Mathematics by Nolan R. Wallach

By Nolan R. Wallach

Genuine Reductive teams I is an creation to the illustration conception of genuine reductive teams. it's according to classes that the writer has given at Rutgers for the prior 15 years. It additionally had its genesis in an test of the writer to accomplish a manuscript of the lectures that he gave on the CBMS local convention on the college of North Carolina at Chapel Hill in June of 1981.
This ebook includes 10 chapters and starts off with a few heritage fabric as an advent. the subsequent chapters then talk about straight forward illustration idea; actual reductive teams; the fundamental thought of (g, K)-modules; the asymptotic habit of matrix coefficients; The Langlands category; a development of the basic sequence; cusp types on G; personality conception; and unitary representations and (g, K)-cohomology.
This ebook may be of curiosity to mathematicians and statisticians.

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The above considerations imply that n extends to a bounded linear map of L'( U ) into End(H) and that (1) is satisfied. Assume that 1 E U. If V is an open subset of U containing 1 and having the properties that V V is contained in U and that if u E V then up' E V then the map V x Li(V) to L ' ( U ) given b y x, f H L ( x ) f is continuous. f)vis continuous for f E L ' ( V ) and u E H. Let vj be a decreasing sequence of open relatively compact subsets of G such that V, = (1). Let { u j } be a sequence of non-negative, continuous, functions on G such that supp uj is contained in and 0 j uj(g)dg = 1.

Since this is ridiculous, we conclude that if u E D then u and Tu are linearly dependent. This easily implies that T is a scalar multiple of I on D. 3. Square integrable representations Let G be a locally compact, separable group. Fix, dg, a right invariant measure on G. Let L z ( G )denote the space of all square integrable functions with respect to dg. 1. R ( x ) f ( g )= f ( 9 4 for 9 E G. Since dg is right invariant R(x)is a unitary operator for all x E G . Furthermore, ( W u , 0) = J u(gx)o(g)dg, G which is easily seen to be a continuous function of x .

Let T be as in the statement of the result we are proving. Assume that u E H and that u and To are linearly independent. (2) implies that there exists a sequence { U j } in A such that lim q u =u lim ~ T = vu. and Now, if w E D' then ( u , w) = lim ( ~ T uw), = lim ( T q u , w) = lim (uju, Sw) = ( u , S w ) = (Tu, w). Since D' is dense in H this implies that Tu = u. Since this is ridiculous, we conclude that if u E D then u and Tu are linearly dependent. This easily implies that T is a scalar multiple of I on D.

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