The Unitary Dual Problem and Hall-Llttlewood Polynomials

数学中最重要的公开问题之一是酉对偶问题：对一个群的不可约酉表示进行分类。这个问题是如此重要，如果不是被错误地认为这个问题已经得到解决，它将是一个粘土千禧年奖的问题。酉对偶问题是盖尔芬德抽象调和分析方案中的一个必要组成部分，该方案建议按如下方式解决不同科学领域的问题。对于任何数学领域的难题，都可以附加一个等价的代数问题。把代数问题分解成更简单的代数问题。解决那些更简单的代数问题，并将答案重新组合成全局解。将这个解决方案转化为原始问题的解决方案。这项建议是关于酉对偶问题的，该问题自1930年代以来一直存在。对酉表示进行分类的一般方法如下。确定允许不变厄米特形式的表示，计算形式的签名，然后确定哪些形式是正定的，从而酉正的。厄米特表示的签名字符的公式已经存在。这个想法是变形表示并跟踪当可缩减性点相交时的变化。不幸的是，由于递归，这些公式涉及符号、2的幂和平移的乘积，非常笨拙。幸运的是，最近的工作表明，在不可约Verma模的情况下，所有的复杂性都可以由仿射Hecke代数编码：签名特征仅仅是以Q=-1乘以Weyl分母的形式求值的Hall-Littlewood多项式的和的“负”。寻找么正表示的当前技术状态是用于确定给定表示是否么正的计算机算法。Hall-Littlewood的结果表明，寻找整个酉对偶的封闭形式的答案是可以得到的，原因有两个。首先，Hall-Littlewood多项式是q=0的有限维不可约最高权模和q=1的单项对称函数的特征，寻找酉对偶等价于确定签名特征标和特征标何时重合。第二，Hall-Littlewood结果包含奇异无穷小特征标的情形。Salamanca-Riba的工作已经知道，任何强正则无穷小特征标的酉表示都同构于AQ(？)模和任意AQ(？)是酉性的，留下了酉对偶问题的奇异无穷小特征情形。我建议将带符号Kazhdan-Lusztig多项式的工作推广到奇异情形，利用签名特征标公式对酉不可约最高权模进行分类，简化和更好地理解范畴O的签名特征标，确定上同调诱导表示的签名特征标，并确定哪些上同调诱导模是酉模。

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.