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

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##### Abstract

If G is a connected Lie group, the Kasparov representation ring KKG(C, C) contains a singularly important element—the 'y-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 = SO0(n, 1) ([Kas84]) and G = SU(n, 1) ([JK95]), a key component is an explicit construction of the 'y-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 -y in the case of 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 OX and AY 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 'y-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 A and ideals KX and KY which are related to the canonical fibrations. We explain why these are likely natural homes for the operators which would appear in a construction of the 'y-element.