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The Dirac operator on certain homogenous spaces and representations of some lie groups.
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Slebarski, Stephen (1983) The Dirac operator on certain homogenous spaces and representations of some lie groups. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3072569~S15
Abstract
Let G be a real non-compact reductive Lie group and L a compact subgroup. Take a maximal compact subgroup K of G containing L, and suppose that G/L is Riemannian via a bi-invariant metric and that there is a spin structure. Then there is the Dirac operator D over G/L, on spinors with values in a unitary vector bundle. D is a first order, G-invariant, elliptic, essentially self-adjoint differential operator.
It has been shown by R. Parthasarathy that with G semi-simple, rank K = rank G, 'discrete-series' representations of G can be realized geometrically on, the kernel of D (i.e. the L2-solutions of Df = 0). Following this, we are interested in how the kernel of D decomposes into irreducible representations of G, when L is any compact subgroup. In future work we expect to reduce this problem to the compact case i.e. to considering the Dirac operator on K/L.
Therefore, in this Thesis, we consider the Dirac operator on a compact, Riemannian, spin homogeneous space K/L. And determine the decomposition of the kernel into irreducible representations of K. We consider the tensor product of an induced representation and a finite-dimensional representation, and apply 'inducing in stages' to the Dirac operator.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Differential operators, Lie groups, Tensor products | ||||
| Official Date: | May 1983 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Rawnsley, John H. (John Howard), 1947- | ||||
| Sponsors: | Science and Engineering Research Council (Great Britain) | ||||
| Format of File: | |||||
| Extent: | 168 leaves : illustrations | ||||
| Language: | eng |
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