Sur les caractères des groupes de Lie réductifs non connexes. (On the characters of non-connected reductive Lie groups). (French) Zbl 0622.22009

The paper deals with the following class of Lie groups G: the Lie algebra \({\mathfrak g}\) of G is reductive; G contains an abelian normal subgroup \(\Gamma\) so that \(\Gamma G_ 0\) is of finite index in G. (An essential point of the paper is that G is not assumed to being connected, nor even of Harish-Chandra class.) For such a group Duflo has described certain representations \(\Pi (f_ 0,\tau_ 0)\), parametrized by pairs \((f_ 0,\tau_ 0)\), where \(f_ 0\) is an element of \({\mathfrak g}^*\) whose centralizer \({\mathfrak g}(f_ 0)\) is a Cartan subalgebra and \(\tau_ 0\) is a unitary irreducible representation of a canonical double cover of \(G(f_ 0)\) so that \(\tau_ 0(\exp X)=e^{if_ 0(X)}\) and \(\tau_ 0(\epsilon)=-1\), \(\epsilon\) the non-trivial element of the kernel of the covering. (If G is of Harish-Chandra class, these are the tempered representations with regular infinitesimal character.)
The paper gives a formula for the character \(\Theta (f_ 0,\tau_ 0)\) of such a representation in a neighbourhood of 1 in the centralizer Z of each semisimple element s of G. The formula may be viewed as a formula of the Kirillov type adapted to Harish-Chandra’s method of descent: it represents the character in exponential coordinates as an integral over \(G\cdot f_ 0\cap {\mathfrak z}^*\), which is a union of a finite number of Z-orbits. For \(s=1\) it reduces to the usual Kirillov-type formula; for s regular to Harish-Chandra’s formula. (The proof in general uses these cases.) When G is connected the formula (for relative discrete series) is due to M. Duflo, G. Heckman, and M. Vergne [Mém. Soc. Math. Fr., Nouv. Sér. 15, 65-128 (1984; Zbl 0575.22014)].
Reviewer: W.Rossmann


22E46 Semisimple Lie groups and their representations


Zbl 0575.22014
Full Text: DOI


[1] Bouaziz, A., Sur les représentations des groupes de Lie réductifs non connexes, Math. Ann., 268, 539-555 (1984) · Zbl 0528.22013
[2] Bouaziz, A., Sur les caractères des groupes de Lie réductifs non connexes, C.R. Acad. Sci. Paris Sér. I Math., 301, n∘ 4, 93-96 (1985) · Zbl 0589.22013
[3] Clozel, L., Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4), 15, 45-115 (1982) · Zbl 0516.22010
[4] Dixmier, J., Algèbres enveloppantes, (Cahiers Scientifiques (1974), Gauthiers-Villars: Gauthiers-Villars Paris), Fasicule XXXVII · Zbl 0422.17003
[5] Duflo, M., Fundamental-series representations of a semisimple Lie group, Functional Anal. Appl., 4, 122-126 (1970) · Zbl 0254.22007
[6] Duflo, M., Constructions de représentations unitaires d’un groupe de Lie, (Cours d’été au C.I.M.E. Cartona (1980) (1982), Liguori: Liguori Naples) · Zbl 0522.22011
[7] Duflo, M.; Heckman, G.; Vergne, M., Projection d’orbites, formule de Kirillov et formule de Blattner, Ser. Mat. Fis. (2), Mémoire n∘ 15, 65-128 (1984) · Zbl 0575.22014
[8] Duistermaat, J. J., Fourier Integral Operators (1973), Courant Institute, New York University: Courant Institute, New York University New York · Zbl 0272.47028
[9] Gantmacher, F., Canonical representation of automorphisms of a complex semisimple Lie group, Mat. Sb., 5, 47, 101-144 (1939) · JFM 65.1131.02
[10] Guillemin, V.; Sternberg, S., Geometric Asymptotics, (Math. Surveys n∘ 14 (1977), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0503.58018
[11] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc., 119, 457-508 (1965) · Zbl 0199.46402
[12] Harish-Chandra, Representations of semisimple Lie groups, III, Trans. Amer. Math. Soc., 76, 234-253 (1954) · Zbl 0055.34002
[13] Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc., 83, 98-163 (1956) · Zbl 0072.01801
[14] Harish-Chandra, Two theorems on semisimple Lie groups, Ann. of Math., 83, 74-128 (1966) · Zbl 0199.46403
[16] Hiraï, T., The characters of some induced representations of semisimple Lie groups, J. Math. Kyoto Univ., 8, 313-363 (1968) · Zbl 0185.21503
[17] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math., 74, 329-387 (1976) · Zbl 0134.03501
[18] Kirillov, A. A., Éléments de la théorie des représentations (1974), Éditions MIR: Éditions MIR Moscou
[19] Rossmann, W., Kirillov’s character formula for reductive Lie groups, Invent. Math., 48, 207-220 (1978) · Zbl 0372.22011
[20] Trombi, P. C., The tempered spectrum of a real semisimple Lie group, Amer. J. Math., 99, 57-75 (1977) · Zbl 0373.22007
[21] Trombi, P. C.; Varadarajan, V. S., Asymptotic behaviour of eigenfunctions on a semisimple Lie group; the discrete spectrum, Acta Math., 129, 237-280 (1972) · Zbl 0244.43006
[22] Varadarajan, V. S., Harmonic Analysis on Real Reductive Lie Groups, (Lecture Notes in Mathematics, Vol. 576 (1977), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0354.43001
[23] Vogan, D., Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures, Duke Math. J., 46, n∘ 4, 805-859 (1979) · Zbl 0421.22008
[24] Wolf, J. A., Unitary Representations on Partially Holomorphic Cohomology Spaces, (Mem. Amer. Math. Soc. n∘ 138 (1974), Amer. Math. Soc: Amer. Math. Soc Providence, R.I)
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