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Highly cuspidal pseudocoefficients and \(K\)-theory. (Pseudo-coefficients très cuspidaux et \(K\)-théorie.) (French) Zbl 0789.22028

The author generalizes and sharpens results given by L. Clozel and P. Delorme for linear groups to real reductive Lie groups \(G\) of Harish- Chandra class. Let \(K\) be a maximal compact subgroup of \(G\), \(\gamma\) a spinor representation of \(K\) and \(A_ \gamma\) the Dirac operator associated to \(\gamma\). By regularization with respect to a multiplier of J. Arthur the author exhibits the existence of a smooth function \(f\) of compact support, being an index-function for the Dirac operator \(A^ +_ \gamma\), such that for each irreducible unitary representation \(\pi\), \(\text{tr }\pi (f_ \gamma)=\text{ind } \pi(A^ +_ \gamma)\). It is proved that this function is highly cuspidal, i.e. its constant terms vanish along any proper parabolic subgroup.

MSC:

22E46 Semisimple Lie groups and their representations
19K99 \(K\)-theory and operator algebras
22E41 Continuous cohomology of Lie groups
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References:

[1] [A 1] Arthur, J.: A Paley-Wiener theorem for real reductive groups. Acta Math.150, 1-89 (1983) · Zbl 0514.22006 · doi:10.1007/BF02392967
[2] [A 2] Arthur, J.: The invariant trace formula. II. Global theory. J. Am. Math. Soc.1, 501-554 (1988) · Zbl 0667.10019 · doi:10.1090/S0894-0347-1988-0939691-8
[3] [AS] Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semi-simple Lie groups. Invent. Math.42, 1-62 (1977) · Zbl 0373.22001 · doi:10.1007/BF01389783
[4] [BM] Barbasch, D., Moscovici, H.:L 2-index and the Selberg trace formula. J. Funct. Anal.53, 151-201 (1983) · Zbl 0537.58039 · doi:10.1016/0022-1236(83)90050-2
[5] [CD 1] Clozel, L., Delorme, P.: Pseudo-coefficients et cohomologie des groupes r?ductifs. C.R. Acad. Sci. Paris300, 385-387 (1985) · Zbl 0593.22010
[6] [CD 2] Clozel, L., Delorme, P.: Le th?or?me de Paley-Wiener invariant pour les groupes de Lie r?ductifs. II. Ann. Sci. ?c. Norm. Sup. 4e s?rie23, 193-228 (1990) · Zbl 0724.22012
[7] [CM] Connes, A., Moscovici, H.: TheL 2-index for homogeneous spaces of Lie groups. Ann. Math.115, 291-330 (1982) · Zbl 0515.58031 · doi:10.2307/1971393
[8] [D] Delorme, P.: Multipliers for the convolution algebra of left and rightK-finite compactly supported smooth functions on a semi-simple Lie group. Invent. Math.75, 9-23 (1984) · Zbl 0536.43007 · doi:10.1007/BF01403087
[9] [HC] Harish-Chandra: Harmonic analysis on real reductive groups. I. J. Funct. Anal.19, 104-204 (1975) · Zbl 0315.43002 · doi:10.1016/0022-1236(75)90034-8
[10] [K] Knapp, A.: Representation theory of semisimple groups. Princeton Math. Series36, (1986) · Zbl 0604.22001
[11] [KZ] Knapp, A., Zuckerman, G.: Classification of irreducible tempered representations of semisimple groups. Ann. Math.116, 389-501 (1982) · Zbl 0516.22011 · doi:10.2307/2007066
[12] [Lau] Laumon, G.: Lettre ? J. Arthur 3 Mai 1989
[13] [P] Parthasarathy, R.: Dirac operators and the discrete series. Ann. Math.96, 1-30 (1972) · Zbl 0249.22003 · doi:10.2307/1970892
[14] [W] Wallach, N.: Real reductive groups. I. London New York: Academic Press 1988 · Zbl 0666.22002
[15] [V] Vogan, D.: Unitarizability of certain series of representations. Ann. Math.120, 141-187 (1984) · Zbl 0561.22010 · doi:10.2307/2007074
[16] [Was] Wassermann, A.: Une d?monstration de la conjecture de Connes-Kasparov pour les groupes de Lie lin?aires connexes r?ductifs. C.R. Acad. Sci. Paris304, 559-562 (1987) · Zbl 0615.22011
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