Research in Representation Theory

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Representation theory and invariant theory have important applications to, in particular, number theory, physics, combinatorics and geometry. This proposal emphasizes both the analytic and the algebraic theory. The main thrust is an extension of the analytic theory of generalized Whittaker models. These include the Bessel models that have played an important role in the study of automorphic forms and L-functions stemming from attempts at extending the theory of the local Langlands correspondence to groups beyond inner twists of GL(n). The importance of these Whittaker models is that they give a representation theoretic interpretation of the Fourier coefficients of multivariate automorphic forms. The mathematical literature, to date, on Bessel models studies only models for subrepresentations of degenerate principal series. The most general work of this type is the PI?s on previous grants. In this grant he proposes to do a complete analysis of these models for generic principal series. The work will be in part carried out in collaboration with his student Raul Gomez. He expects to have solved the problem for the case of compact stabilizer before the end of the first summer. The general case will be much harder and will be emphasized in the next two years of the grant (after Gomez graduates). The other work will involve more algebraic geometric problems in invariant theory including the structure of the singular set in a semisimple Lie algebra and the Hilbert series of rings of invariants of actions of reductive groups on affine varieties. This work, if successful, will impact applications to geometry and physics.General Abstract. Representations and symmetry have played key roles in the recent solutions of some of the most profound problems in mathematics. These include the proof of Fermat?s Last Theorem and proof of The Poincaré Conjecture in three dimensions. The first example uses representation theory to calculate local factors needed in the proof of a special case of the Langlands Program known as the Taniyama-Shimura conjecture. The second involves the study of the various invariants of the Riemann Curvature tensor and how they evolve under the Ricci Flow. This proposal aims to extend the representation theory involved in these major breakthroughs leading to applications to further developments in number theory (more cases of the Langlands Program) , extensions of applicability of the Ricci flow and related geometry, applications to some amazing recent developments in Conformal Field theory related to exceptional Lie groups.