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Osborne M.S., Warner G. the speculation of Eisenstein platforms (AP, 1981)(ISBN 0125292503)

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L,,,(a) = j;(X)~,/~(x) MX) is an integral operator on L 2 ( C / r )with (differentiable) kernel Consequently, L,,r(a) is a Hilbert-Schmidt operator, hence is completely continuous. Now recall the following elementary representation-theoretic generality. 34 Let G be a Lie group; let U be a unitary representation o f G on the separable Hilbert space E. Suppose that INTRODUCTION 19 is completely continuous,for all ci E C,"(G)-then U is discretely decomposable, u given irreducible irnifury representation qf G occurring at most a ,finitenumber of'times in the direct sum decomposition of U .

Internut. C o n y . , Stockholm (1962) 177-189. Selberg did not provide complete proofs for many of his results. Details in the uniform case (G/rcompact) have been supplied by T. TAMAGAWA: On Selberg’s trace formula, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 8(1960), 363-386. Compare too I. M. GELFAND, M. I. GRAEV, A N D 1 . 1 . ” W. B. , Philadelphia, 1969. 26 CHAPTER 1 For a survey of more recent results, consult N. WALLACH: On the Selberg trace formula in the case of compact quotient, Bull. Amer.

B. VENKOV:The Selberg trace formula for SL(3, Z), J . Soviet Math. 12( 1979), 384-424. As a blanket convention, any reference to Harish-Chandra or Langlands in the text proper is understood to be to that author’s respective Springer Lecture Notes cited above. S. OSBORNE: “Spectral Theory and Uniform Lattices,” Lecture Notes in Representation Theory. Univ. of Chicago Press, Chicago, Illinois, 1977. To this end, it is necessary to make some assumptions on both G and r. The assumptions on G are standard.

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