The Unitary Dual Problem and Hall-Llttlewood Polynomials

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One of the most important open problems in mathematics is the Unitary Dual Problem: classify a group's irreducible unitary representations. The problem is so important that it would have been a Clay Millenium Prize Problem except it was incorrectly thought that the problem had already been solved. The Unitary Dual Problem is a necessary component in I.M. Gelfand's programme in abstract harmonic analysis which proposes to solve problems in disparate areas of the sciences as follows. To a difficult problem in any area of mathematics, attach an equivalent algebraic problem. Decompose the algebraic problem into simpler algebraic problems. Solve those simpler algebraic problems and reassemble the answers into a global solution. Translate that solution to a solution to the original problem. This proposal is about the Unitary Dual Problem which has been open since the 1930s. The general approach to classifying unitary representations has been as follows. Identify representations admitting an invariant Hermitian form, compute the signatures of the forms, and then determine which forms are positive definite and hence unitary.******Formulas for signature characters of Hermitian representations exist. The idea is to deform representations and track changes as reducibility points are crossed. Unfortunately, due to recursion, the formulas are highly unwieldy involving products of signs, powers of 2, and translations. Fortunately, in recent work, it was shown that in the case of irreducible Verma modules, all of the complexity can be encoded by the affine Hecke algebra: the signature character is simply the "negative" of a summand of a Hall-Littlewood polynomial evaluated at q=-1 times a version of the Weyl denominator. The current state of the art for finding unitary representations is a computer algorithm for determining if a given representation is unitary. The Hall-Littlewood result suggests that a closed form answer to finding the entire unitary dual is attainable for two reasons. First, Hall-Littlewood polynomials are characters of finite dimensional irreducible highest weight modules at q=0 and monomial symmetric functions at q=1 and finding the unitary dual is equivalent to determining when signature characters and characters coincide. Second, the Hall-Littlewood result includes the case of singular infinitesimal character. It is known by work of Salamanca-Riba that any unitary representations of strongly regular infinitesimal character is isomorphic to an Aq() module and any Aq() is unitary, leaving the singular infinitesimal character case of the Unitary Dual Problem open.******I propose to extend work on signed Kazhdan-Lusztig polynomials to the singular case, classify unitary irreducible highest weight modules using signature character formulas, simplify and better understand signature characters for Category O, determine signature characters for cohomologically induced representations, and identify which cohomologically induced modules are unitary.**