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

Abstract (Summary)

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If G is a connected Lie group, the Kasparov representation ring KK G(C, C) contains
a singularly important element—the ?-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 ?-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 ? 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 ?X and ?Y on H. We show that these
operators satisfy a kind of directional ellipticity in terms of the spaces H (m,n)(H) for
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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 ?-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 ?-element.
Bibliographical Information:

Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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