Unitary Representations and Generalized Harish-Chandra Modules

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Representation theory is a subject with numerous applications to other areas of mathematics, physics, and more generally the sciences. The two problems covered by this proposal are unitary representations and generalized Harish-Chandra modules. The Unitary Dual Problem (UDP) is one of the most important open problems in mathematics. For example, solvability of certain differential equations by Fourier analysis techniques corresponds to groups for which the UDP has been solved. Philosophically, the idea behind Fourier analysis is to decompose function spaces into subspaces where the problem is tractable. More generally, I.M. Gelfand's abstract harmonic analysis programme (for which a solution to the UDP is necessary) transfers difficult problems in diverse areas of mathematics to more tractable (though possibly infinitely many) problems in algebra. In addition to numerous areas of mathematics, the UDP has applications in physics and engineering. I propose to extend work that I have done on unitarity of representations by: 1) Simplifying formulas for signature characters. 2) Studying the singular infinitesimal character case. 3) Applying cohomological induction to obtain results for Harish-Chandra modules. Since representations of abelian Lie algebras and compact groups are well-understood, one of the basic philosophies for studying more complicated groups is to restrict one's attention to such a subgroup or subalgebra. This leads to Category O and Harish-Chandra modules. By restricting instead to mixed subgroups, one generalizes these categories and permits the philosophies of these categories to be combined. I propose to continue work on generalized Harish-Chandra modules for mixed subgroups (joint with Paul-Sahi) by studying Kazhdan-Lusztig-Vogan polynomials for mixed subgroups, relating them to polynomials for the Harish-Chandra and parabolic Cateogory O cases, and by implementing a generalized induction functor, thereby establishing a module-theoretic bijection of what is known as S-equivalence.