Analytic structures for the index theory of SL(3, C ) [microform]
- Bib ID:
- 4225097
- Format:
- Book and Microform
- Author:
- Yuncken, Robert
- Online Version:
- http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3231922
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- Description:
- 253 p.
- ISBN:
- 9780542843990
- Summary:
-
If G is a connected Lie group, the Kasparov representation ring KKG( 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 = SO0(n, 1) ([Kas84]) and G = SU(n, 1) ([JK95]), 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 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 DeltaX and DeltaY 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 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 gamma-element.
- Notes:
-
- (UMI)AAI3231922
- Adviser: Nigel Higson.
- Source: Dissertation Abstracts International, Volume: 67-08, Section: B, page: 4468.
- Thesis (Ph.D.)--The Pennsylvania State University, 2006.
- Reproduction:
- Microfiche. Ann Arbor, Mich.: University Microfilms International
- Subject:
- Mathematics
- Other authors/contributors:
- The Pennsylvania State University
- Copyright:
-
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Copyright status was determined using the following information:
- Material type:
- Literary Dramatic Musical
- Published status:
- Unpublished
- Creation date:
- 2006
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