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Parabolic induction and restriction via $C^{\ast }$ -algebras and Hilbert $C^{\ast }$ -modules

  • Pierre Clare (a1), Tyrone Crisp (a2) and Nigel Higson (a3)


This paper is about the reduced group $C^{\ast }$ -algebras of real reductive groups, and about Hilbert $C^{\ast }$ -modules over these $C^{\ast }$ -algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$ -algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$ -algebra to determine the structure of the Hilbert $C^{\ast }$ -bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.



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[Art75]Arthur, J., A theorem on the Schwartz space of a reductive Lie group, Proc. Natl. Acad. Sci. USA 72 (1975), 47184719; MR 0460539 (57 #532).
[Art83]Arthur, J., A Paley–Wiener theorem for real reductive groups, Acta Math. 150 (1983), 189; MR 697608 (84k:22021).
[BCH94]Baum, P., Connes, A. and Higson, N., Classifying space for proper actions and K-theory of group C -algebras, C -algebras: 1943–1993 (San Antonio, TX, 1993), Contemporary Mathematics, vol. 167 (American Mathematical Society, Providence, RI, 1994), 240291; MR 1292018 (96c:46070).
[BdlHV08]Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s property (T), New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008).
[Ber92]Bernstein, J., Representations of p-adic groups, Lecture Notes (Harvard University, 1992); written by K. E. Rumelhart.
[BK15]Bezrukavnikov, R. and Kazhdan, D., Geometry of second adjointness for p-adic groups, Represent. Theory 19 (2015), 299332.
[BM76]Boyer, R. and Martin, R., The regular group C -algebra for real-rank one groups, Proc. Amer. Math. Soc. 59 (1976), 371376; MR 0476913 (57 #16464).
[Bru56]Bruhat, F., Sur les reprsentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97205.
[Cla13]Clare, P., Hilbert modules associated to parabolically induced representations, J. Operator Theory 69 (2013), 483509; MR 3053351.
[Cla14]Clare, P., C -algebraic intertwiners for degenerate principal series of special linear groups, Chin. Ann. Math. Ser. B 35 (2014), 691702; MR 3246931.
[Cla15]Clare, P., C -algebraic intertwiners for principal series: case of SL (2), J. Noncommut. Geom. 9 (2015), 119; MR 3337952.
[CCH14]Clare, P., Crisp, T. and Higson, N., Adjoint functors between categories of Hilbert $C^{\ast }$-modules, Preprint (2014), arXiv:1409.8656.
[CHH88]Cowling, M., Haagerup, U. and Howe, R., Almost L 2 matrix coefficients, J. Reine Angew. Math. 387 (1988), 97110; MR 946351 (89i:22008).
[CH16]Crisp, T. and Higson, N., Parabolic induction, categories of representations and operator spaces, in Operator algebras and their applications: a tribute to Richard V. Kadison, Contemporary Mathematics, vol. 671 (American Mathematical Society, Providence, RI, 2016), to appear.
[Dix57]Dixmier, J., Sur les représentations unitaires des groupes de Lie algébriques, Ann. Inst. Fourier (Grenoble) 7 (1957), 315328; MR 0099380 (20 #5820).
[Dix77]Dixmier, J., C -algebras, North-Holland Mathematical Library, vol. 15 (North-Holland, Amsterdam, 1977); translated from the French by Francis Jellett; MR 0458185 (56 #16388).
[Fel60]Fell, J. M. G., The dual spaces of C -algebras, Trans. Amer. Math. Soc. 94 (1960), 365403; MR 0146681 (26 #4201).
[God52]Godement, R., A theory of spherical functions I, Trans. Amer. Math. Soc. 73 (1952), 496556.
[HC53]Harish-Chandra, Representations of semisimple Lie groups on a Banach space I, Trans. Amer. Math. Soc. 75 (1953), 185243.
[HC66]Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1111; MR 0219666 (36 #2745).
[HC72]Harish-Chandra, On the theory of the Eisenstein integral, Lecture Notes in Mathematics, vol. 266(Springer, New York, 1972), 123149.
[HC76]Harish-Chandra, Harmonic analysis on real reductive groups III. The Maaß–Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), 117201.
[Hum75]Humphreys, J. E., Linear algebraic groups, Graduate Texts in Mathematics, vol. 21 (Springer, New York, 1975); MR 0396773 (53 #633).
[Kna86]Knapp, A. W., Representation theory of semisimple groups, Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 1986).
[Kna02]Knapp, A. W., Lie groups beyond an introduction, Progress in Mathematics, vol. 140, second edition (Birkhäuser, Boston, 2002).
[KS72]Knapp, A. W. and Stein, E. M., Irreducibility theorems for the principal series, in Conference on Harmonic Analysis (College Park, Maryland, 1971), Lecture Notes in Mathematics, vol. 266 (Springer, Berlin, 1972), 197214; MR 0422512 (54 #10499).
[KS80]Knapp, A. W. and Stein, E. M., Intertwining operators for semisimple groups II, Invent. Math. 60 (1980), 984.
[Lan95]Lance, E. C., Hilbert C*-modules, LMS Lecture Note Series (Cambridge University Press, Cambridge, 1995).
[Lan89]Langlands, R. P., On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical Surveys Monographs, vol. 31 (American Mathematical Society, Providence, RI, 1989), 101170; MR 1011897 (91e:22017).
[Lip70]Lipsman, R. L., The dual topology for the principal and discrete series on semisimple groups, Trans. Amer. Math. Soc. 152 (1970), 399417; MR 0269778 (42 #4673).
[Mil73]Miličić, D., On C -algebras with bounded trace, Glas. Mat. Ser. III 8(28) (1973), 722; MR 0324429 (48 #2781).
[MP82]Miličić, D. and Primc, M., On the irreducibility of unitary principal series representations, Math. Ann. 260(4) (1982), 413421; MR 670190 (84c:22019).
[Ped79]Pedersen, G. K., C -algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14 (Academic Press, London, 1979); MR 548006 (81e:46037).
[PP83]Penington, M. G. and Plymen, R. J., The Dirac operator and the principal series for complex semisimple Lie groups, J. Funct. Anal. 53 (1983), 269286; MR 724030 (85d:22016).
[Rie74]Rieffel, M. A., Induced representations of C*-algebras, Adv. Math. 13 (1974), 176257.
[Sti58]Stinespring, W. F., A semi-simple matrix group is of type I, Proc. Amer. Math. Soc. 9 (1958), 965967; MR 0104756 (21 #3509).
[Tro77]Trombi, P. C., The tempered spectrum of a real semisimple Lie group, Amer. J. Math. 99 (1977), 5775; MR 0453929 (56 #12182).
[Val85]Valette, A., Dirac induction for semi-simple Lie groups having one conjugacy class of Cartan subgroups, in Operator algebras and their connections with topology and ergodic theory, Lecture Notes in Mathematics, vol. 1132, eds Araki, H., Moore, C. C., Stratila, S.-V. and Voiculescu, D.-V. (Springer, Berlin, 1985), 526555.
[Vog81]Vogan, D. A. Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15 (Birkhäuser, Boston, 1981); MR 632407 (83c:22022).
[Wal88]Wallach, N. R., Real reductive groups. I, Pure and Applied Mathematics, vol. 132 (Academic Press, Boston, 1988); MR 929683 (89i:22029).
[Wal92]Wallach, N. R., Real reductive groups. II, Pure and Applied Mathematics, vol. 132-II (Academic Press, Boston, 1992); MR 1170566 (93m:22018).
[Was87]Wassermann, A., Une démonstration de la conjecture de Connes–Kasparov pour les groupes de Lie linaires connexes rductifs, C. R. Acad. Sci. Paris 18 (1987), 559562.
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Parabolic induction and restriction via $C^{\ast }$ -algebras and Hilbert $C^{\ast }$ -modules

  • Pierre Clare (a1), Tyrone Crisp (a2) and Nigel Higson (a3)


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