Abstract
For G = SL ( 3 , C ) , we construct an element of G -equivariant analytic K-homology from the Bernstein–Gelfand–Gelfand complex for G . This furnishes an explicit splitting of the restriction map from the Kasparov representation ring R ( G ) to the representation ring R ( K ) of its maximal compact subgroup SU ( 3 ) , and the splitting factors through the equivariant K-homology of the flag variety X of G . In particular, we obtain a new model for the γ-element of G . The construction is made using SU ( 3 ) -harmonic analysis associated to the canonical fibrations of X . On this matter, we prove results which demonstrate the compatibility of both the G -action and the order zero longitudinal pseudodifferential operators with the SU ( 3 ) -harmonic analysis.
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