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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
Citations:
Zbl 0526.22007
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