Analytic structures for the index theory of SL(3,C)

Abstract : If G is a connected Lie group, the Kasparov representation ring KK^G(C,C) contains a singularly important element---the gamma-element---which is an idempotent relating the Kasparov representation ring of G with the representation ring of its maximal compact subgroup K. In the proofs of the Baum-Connes conjecture with coefficients for the groups G=SO(n,1) [Kasparov] and G=SU(n,1) [Julg-Kasparov], a key component is an explicit construction of the gamma-element as an element of G-equivariant K-homology for the space G/B, where B is the Borel subgroup of G. In this thesis, we describe some analytical constructions which may be useful for such a construction of $\gamma$ for the rank-two Lie group G=SL(3,C). The inspiration is the Bernstein-Gel'fand-Gel'fand complex---a natural differential complex of homogeneous bundles over G/B. The reasons for considering this complex are explained in detail. For G=SL(3,C), the space G/B admits two canonical fibrations, which play a recurring role in the analysis to follow. The local geometry of G/B can be modeled on the geometry of the three-dimensional complex Heisenberg group H in a very strong way. Consequently, we study the algebra of differential operators on H. We define a two-parameter family H^(m,n)(H) of Sobolev-like spaces, using the two fibrations of G/B. We introduce fibrewise Laplacian operators $\Delta_X$ and $\Delta_Y$ on $H$. We show that these operators satisfy a kind of directional ellipticity in terms of the spaces H^(m,n)(H) for certain values of (m,n), but also provide a counterexample to this property for another choice of (m,n). This counterexample is a significant obstacle to a pseudodifferential approach to the gamma-element for SL(3,C). Instead we turn to the harmonic analysis of the compact subgroup K=SU(3). Here, using the simultaneous spectral theory of the K-invariant fibrewise Laplacians on G/B, we construct a C*-category $\mathcal{A}$ and ideals $\mathcal{K}_X$ and $\mathcal{K}_Y$ related to the canonical fibrations. We explain why these are likely natural homes for the operators which would appear in a construction of the gamma-element.
Document type :
Mathematics. Penn State University, 2006. English
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Contributor : Robert Yuncken <>
Submitted on : Monday, March 7, 2011 - 12:08:03 PM
Last modification on : Monday, March 7, 2011 - 3:06:58 PM


  • HAL Id : tel-00574163, version 1


Robert Yuncken. Analytic structures for the index theory of SL(3,C). Mathematics. Penn State University, 2006. English. <tel-00574163>




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