The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

Higson, Nigel; Román, Angel

doi:10.1090/ert/554

The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

Authors: Nigel Higson and Angel Román
Journal: Represent. Theory 24 (2020), 580-602
MSC (2020): Primary 22E45, 46L99
DOI: https://doi.org/10.1090/ert/554
Published electronically: November 9, 2020
MathSciNet review: 4171564
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Abstract: The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group $G$, between the tempered dual of $G$ and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the $C^*$-algebra of the motion group into the reduced $C^*$-algebra of $G$, and use it to characterize the continuous field of reduced group $C^*$-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.

References

A. Afgoustidis and A.-M. Aubert, Continuity of the Mackey-Higson bijection, Preprint, arXiv:1901.00144 (2019).

Alexandre Afgoustidis, On the analogy between real reductive groups and Cartan motion groups: contraction of irreducible tempered representations, Duke Math. J. 169 (2020), no. 5, 897–960. MR 4079418, DOI https://doi.org/10.1215/00127094-2019-0071

Alexandre Afgoustidis, On the analogy between real reductive groups and Cartan motion groups: a proof of the Connes-Kasparov isomorphism, J. Funct. Anal. 277 (2019), no. 7, 2237–2258. MR 3989145, DOI https://doi.org/10.1016/j.jfa.2019.02.023

Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $K$-theory of group $C^\ast $-algebras, $C^\ast $-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, DOI https://doi.org/10.1090/conm/167/1292018

Alexander Braverman and David Kazhdan, Remarks on the asymptotic Hecke algebra, Lie groups, geometry, and representation theory, Progr. Math., vol. 326, Birkhäuser/Springer, Cham, 2018, pp. 91–108. MR 3890206

Pierre Clare, Tyrone Crisp, and Nigel Higson, Parabolic induction and restriction via $C^*$-algebras and Hilbert $C^*$-modules, Compos. Math. 152 (2016), no. 6, 1286–1318. MR 3518312, DOI https://doi.org/10.1112/S0010437X15007824

Alain Connes and Nigel Higson, Déformations, morphismes asymptotiques et $K$-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106 (French, with English summary). MR 1065438

Jacques Dixmier, $C^ *$-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR 0458185

William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620

I. M. Gel′fand and M. A. Naĭmark, Unitarnye predstavleniya klassičeskih grupp, Trudy Mat. Inst. Steklov. no. 36, Izdat. Nauk SSSR, Moscow-Leningrad, 1950 (Russian). MR 0046370

Harish-Chandra, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485–528. MR 63376, DOI https://doi.org/10.1090/S0002-9947-1954-0063376-X

Nigel Higson, The Mackey analogy and $K$-theory, Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, Contemp. Math., vol. 449, Amer. Math. Soc., Providence, RI, 2008, pp. 149–172. MR 2391803, DOI https://doi.org/10.1090/conm/449/08711

Nigel Higson, The tangent groupoid and the index theorem, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 241–256. MR 2732053

Ahmad Reza Haj Saeedi Sadegh and Nigel Higson, Euler-like vector fields, deformation spaces and manifolds with filtered structure, Doc. Math. 23 (2018), 293–325. MR 3846057

Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389

George W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. MR 31489, DOI https://doi.org/10.1073/pnas.35.9.537

George W. Mackey, On the analogy between semisimple Lie groups and certain related semi-direct product groups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 339–363. MR 0409726

George W. Mackey, The theory of unitary group representations, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Mathematics. MR 0396826

Gert K. Pedersen, $C^{\ast } $-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006

M. G. Penington and R. J. Plymen, The Dirac operator and the principal series for complex semisimple Lie groups, J. Funct. Anal. 53 (1983), no. 3, 269–286. MR 724030, DOI https://doi.org/10.1016/0022-1236%2883%2990035-6

Angel Roman-Martinez, Mackey Bijection for Some Reductive Groups and Continuous Fields of Reduced Group C*-Algebras, ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–The Pennsylvania State University. MR 4132485

Maarten Solleveld, On the classification of irreducible representations of affine Hecke algebras with unequal parameters, Represent. Theory 16 (2012), 1–87. MR 2869018, DOI https://doi.org/10.1090/S1088-4165-2012-00406-X

Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107–113. MR 281844, DOI https://doi.org/10.1090/S0002-9947-1971-0281844-2