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The Dirac operator and the principal series for complex semisimple Lie groups. (English) Zbl 0542.22013
A. Connes and G. Kasparov [Proc. Japan USA Conf. geom. methods in operator alg.; Sov. Math., Dokl. 27, 105-109 (1983; Zbl 0526.22007); translation from Dokl. Akad. Nauk SSSR 268, 533-537 (1983)] have conjectured that the elements constructed from twisted Diract operators on a homogeneous space G/H, where G is a connected Lie group and H is a maximal compact subgroup, form a basis for Kasparov’s K-theory of the reduced $$C^*$$-algebra $$C^*_ r(G)$$. The authors prove the conjecture for all connected complex semisimple Lie groups. The technique used can be extended to all real semisimple Lie groups with only one conjugacy class of Cartan subgroups, e.g. $$SO(2n+1,1)$$ and SL(n,Q), the special linear group of $$n\times n$$ matrices with entries in the quaternions Q.
Reviewer: P.Rudva

##### MSC:
 22E46 Semisimple Lie groups and their representations 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations
Zbl 0526.22007
Full Text:
##### References:
 [1] Atiyah, M.F; Schmid, W, A geometric construction of the discrete series for semisimple Lie groups, Invent. math., 42, 1-62, (1977) · Zbl 0373.22001 [2] Atiyah, M.F; Singer, I.M, The index of elliptic operators: III, Ann. of math., 87, 546-604, (1968) · Zbl 0164.24301 [3] Baaj, S; Julg, P, Théorie bivariante de kasparov et opérateurs non bornés dans LES C∗-modules hilbertiens, C. R. acad. sci. (Paris), 296, 875-878, (1983) · Zbl 0551.46041 [4] {\scP. Baum and A. Connes}, Geometric K-theory for Lie groups and foliations, Proceedings of Japan—U.S.A. Conference on Geometric Methods in Operator Algebras, to appear. · Zbl 0985.46042 [5] Bourbaki, N, Groupes et algèbres de Lie, (1981), Masson Paris, Chapter V · Zbl 0483.22001 [6] Bruhat, F, Sur LES représentations induites des groupes de Lie, Bull. soc. math. France, 84, 97-205, (1956) · Zbl 0074.10303 [7] Connes, A, An analogue of the thom isomorphism for crossed products of a C∗-algebra by an action of $$R$$, Adv. in math., 39, 31-55, (1981) · Zbl 0461.46043 [8] Connes, A, A survey of foliations and operator algebras, (), 521-628 [9] Connes, A, The Chern character in K-homology, and de Rham homology and noncommutative algebra, (1983), I.H.E.S, preprints [10] Dixmier, J, LES C∗-algèbres et leurs représentations, (1964), Gauthier-Villars Paris · Zbl 0152.32902 [11] Fell, J.M.G, The dual spaces of C∗-algebras, Trans. amer. math. soc., 94, 365-403, (1960) · Zbl 0090.32803 [12] Harish-Chandra, The Plancherel theorem for complex semi-simple Lie groups, Trans. amer. math. soc., 76, 485-528, (1954) · Zbl 0055.34003 [13] Kasparov, G.G, The operator K-functor and extensions of C∗-algebras, Math. USSR-izv., 16, 513-572, (1981) · Zbl 0464.46054 [14] Kasparov, G.G, K-theory, group C∗-algebras and higher signatures (conspectus), parts 1 and 2, (1981), preprint, Chernogolovka · Zbl 0957.58020 [15] Kasparov, G.G, The index of invariant elliptic operators, K-theory and Lie group representations, Dokl. akad. nauk. USSR, 268, 533-537, (1983) · Zbl 0526.22007 [16] Kobayashi, S; Nomizu, K, () [17] Lipsman, R.L, Group representations, () · Zbl 0724.22007 [18] Lipsman, R.L, The dual topology for the principal and discrete series on semi-simple groups, Trans. amer. math. soc., 152, 399-417, (1970) · Zbl 0208.37902 [19] Parthasarathy, R, Dirac operator and the discrete series, Ann. of math., 96, 1-30, (1972) · Zbl 0249.22003 [20] Rosenberg, J, Group C∗-algebras and topological invariants, (), in press · Zbl 0524.22009 [21] Wallach, N, Cyclic vectors and irreducibility for principal series representations, Trans. amer math. soc., 158, 107-113, (1971) · Zbl 0227.22016 [22] Warner, G, () [23] Schmid, W, On the characters of the discrete series. the Hermitian symmetric case, Invent. math., 30, 47-144, (1975) · Zbl 0324.22007 [24] Kasparov, G.G, Operator K-theory and its applications: elliptic operators, group representations, higher signatures, C∗-extensions, (1983), Preprint, Chernogolovka · Zbl 0571.46047
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