2 - Tempered Representations of Semisimple Lie Groups
Published online by Cambridge University Press: 06 November 2024
Summary
We start by describing the fundamental theory of representation theory of a semisimple Lie group. This is followed by a classification of almost all irreducible tempered representations of a connected, semisimple, linear Lie group. We apply this classification to describe the structure of the reduced group C*-algebras of such groups.
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- K-Theory and Representation Theory , pp. 37 - 98Publisher: Cambridge University PressPrint publication year: 2024
References
Adams, J. and Taïbi, O.. Galois and Cartan cohomology of real groups. Duke Math. J. 167 (6) (2018), 1057–1097.Google Scholar
Adams, J., van Leeuwen, M., Trapa, P., and Vogan, D. A., Jr. Unitary representations of real reductive groups. arXiv:1212.2192, 2012.Google Scholar
Adams, J., van Leeuwen, M., Trapa, P., Vogan, D. A., Jr., et al. Atlas software package. www.liegroups.org.Google Scholar
Adams, J. and Vogan, D. A., Jr. L-groups, projective representations, and the Langlands classification. Amer. J. Math. 114 (1) (1992), 45–138.CrossRefGoogle Scholar
van den Ban, E. P.. Induced representations and the Langlands classification. In Representation theory and automorphic forms (Edinburgh, 1996), volume 61 of Proceedings of Symposia in Pure Mathematics, pages 123–155. American Mathematical Society, Providence, RI, 1997.Google Scholar
Baum, P., Connes, A., and Higson, N.. Classifying space for proper actions and K-theory of group C∗-algebras. In C∗-algebras: 1943–1993 (San Antonio, TX, 1993), volume 167 of Contemporary Mathematics, pages 240–291. American Mathematical Society, Providence, RI, 1994.Google Scholar
Cartan, E.. Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. France, 41 (1913), 53–96.Google Scholar
Clare, P., Crisp, T., and Higson, N.. Parabolic induction and restriction via C∗algebras and Hilbert C∗-modules. Compos. Math. 152 (6) (2016), 1286–1318.CrossRefGoogle Scholar
Clare, P., Higson, N., Song, Y., and Tang, X.. On the Connes-Kasparov isomorphism, I: The reduced C∗-algebra of a real reductive group and the K-theory of the tempered dual. Jpn. J. Math. 19 (1) (2024), 67–109.CrossRefGoogle Scholar
Cowling, M., Haagerup, U., and Howe, R.. Almost L2 matrix coefficients. J. Reine Angew. Math. 387 (1988), 97–110.Google Scholar
Dixmier, J.. C∗-algebras. In: North-Holland mathematical library, volume 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett.Google Scholar
Harish-Chandra. Representations of a semisimple Lie group on a Banach space. I. Trans. Amer. Math. Soc. 75 (1953), 185–243.Google Scholar
Harish-Chandra. Discrete series for semisimple Lie groups. II. Explicit determination of the characters. Acta Math. 116 (1966), 1–111.Google Scholar
Harish-Chandra. Harmonic analysis on real reductive groups. I. The theory of the constant term. J. Funct. Anal. 19 (1975), 104–204.Google Scholar
Harish-Chandra. Harmonic analysis on real reductive groups. II. Wavepackets in the Schwartz space. Invent. Math. 36 (1976), 1–55.Google Scholar
Harish-Chandra. Harmonic analysis on real reductive groups. III. The MaassSelberg relations and the Plancherel formula. Ann. Math. 104 (1) (1976), 117–201.Google Scholar
Kaniuth, E. and Taylor, K. F.. Induced representations of locally compact groups, volume 197, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2013.Google Scholar
Knapp, A. W. and Stein, E. M.. Intertwining operators for semisimple groups. II. Invent. Math. 60 (1) (1980), 9–84.CrossRefGoogle Scholar
Knapp, A. W.. Commutativity of intertwining operators for semisimple groups. Compositio Math. 46 (1) (1982), 33–84.Google Scholar
Knapp, A. W.. Representation theory of semisimple groups. An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, 1986.Google Scholar
Knapp, A. W.. Lie groups beyond an introduction, 2nd edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002.Google Scholar
Knapp, A. W. and Zuckerman, G. J.. Classification of irreducible tempered representations of semisimple groups. Ann. Math. 116 (2) (1982), 389–455.CrossRefGoogle Scholar
Knapp, A. W. and Zuckerman, G. J.. Classification of irreducible tempered representations of semisimple groups. II. Ann. Math. 116 (3) (1982), 457–501.CrossRefGoogle Scholar
Koosis, P.. An irreducible unitary representation of a compact group is finite dimensional. Proc. Amer. Math. Soc. 8 (1957), 712–715.Google Scholar
Langlands, R. P.. On the classification of irreducible representations of real algebraic groups. In Representation theory and harmonic analysis on semisimple Lie groups, volume 31 of Mathematical Surveys and Monographs, pages 101–170. American Mathematical Society, Providence, RI, 1989.Google Scholar
Matsuki, T.. The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan. 31 (2) (1979), 331–357.CrossRefGoogle Scholar
Mautner, F. I.. Unitary representations of locally compact groups. I. Ann. Math. 51 (2) (1950), 1–25.CrossRefGoogle Scholar
Mautner, F. I.. Unitary representations of locally compact groups. II. Ann. Math. 52 (2) (1950), 528–556.CrossRefGoogle Scholar
Milicˇic’, D.. The dual spaces of almost connected reductive groups. Glasnik Mat. Ser. III. 9 (29) (1974), 273–288.Google Scholar
Nachbin, L.. On the finite dimensionality of every irreducible unitary representation of a compact group. Proc. Amer. Math. Soc. 12 (1961), 11–12.Google Scholar
Penington, M. G. and Plymen, R. J.. The Dirac operator and the principal series for complex semisimple Lie groups. J. Funct. Anal. 53 (3) (1983), 269–286.CrossRefGoogle Scholar
Rieffel, M. A.. Actions of finite groups on C∗-algebras. Math. Scand., 47(1) (1980), 157–176.CrossRefGoogle Scholar
Segal, I. E.. An extension of Plancherel's formula to separable unimodular groups. Ann. Math. 52 (2) (1950), 272–292.CrossRefGoogle Scholar
Forrest Stinespring, W.. A semi-simple matrix group is of type I. Proc. Amer. Math. Soc. 9 (1958), 965–967.Google Scholar
A. Wassermann. Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs. C. R. Acad. Sci. Paris Sér. I Math. 304 (18) (1987), 559–562.Google Scholar
Weyl, H.. Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I. Math. Z. 23 (1) (1925), 271–309.CrossRefGoogle Scholar
Williams, D.. Crossed products of C∗-algebras, volume 134 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007.Google Scholar