Unitary representations of affine Hecke algebras and reductive p-adic groups

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The project focuses on the study of unitary representations of reductive p-adic groups via affine Hecke algebras. The general approach is motivated by the local Langlands program which predicts a correspondence between the larger class of admissible representations of p-adic groups and certain geometric categories defined in terms of the dual complex group. The first direction of research is the development of an algorithm for computing signatures of invariant hermitian forms for the affine Hecke algebra, motivated by the recent results obtained in the setting of real reductive groups. A second direction concerns basic abstract harmonic analysis problems for (graded) affine Hecke algebras, and certain applications of Dirac operator theory in this setting.The project falls in the area of representation theory of Lie groups. Lie groups, named after the Norwegian mathematician Sophus Lie, are mathematical objects underlying the symmetries inherent in a system, and their representations, i.e., the ways in which the Lie groups can manifest themselves, have had an important impact in theoretical physics and number theory. This research will generate topics that can constitute bases for Ph.D. or master theses; some problems of combinatorial nature related to the project are suitable for undergraduate research through REU programs.