The Unitary Dual Problem and Hall-Llttlewood Polynomials

数学中最重要的开放问题之一是酉对偶问题：对群的不可约酉表示进行分类。这个问题非常重要，如果不是错误地认为这个问题已经被解决了，它就会成为克莱千禧年奖问题。统一对偶问题是Gelfand抽象谐波分析计划的必要组成部分，该计划提出解决以下不同科学领域的问题。对任何数学领域的难题，附上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。解决那些简单的代数问题，并将答案重新组合成一个全局解决方案。将该解决方案转化为原始问题的解决方案。这个建议是关于自20世纪30年代以来一直开放的统一对偶问题。对酉表示进行分类的一般方法如下。识别承认不变厄米形式的表示，计算形式的签名，然后确定哪些形式是正定的，因此是酉的。存在用于厄米表示的签名字符的公式。其思想是在可简化点交叉时变形表示并跟踪变化。不幸的是，由于递归的原因，这些公式涉及到符号的乘积、2的幂和平移，非常笨拙。幸运的是，在最近的工作中，证明了在不可约的Verma模块的情况下，所有的复杂性都可以用仿射Hecke代数来编码：签名字符仅仅是Hall-Littlewood多项式在q=-1处的和的“负”乘以Weyl分母的一个版本。寻找酉表示的当前技术状态是用于确定给定表示是否为酉的计算机算法。Hall-Littlewood的结果表明，找到整个酉对偶的封闭形式答案是可以实现的，有两个原因。首先，Hall-Littlewood多项式是有限维不可约的最高权模在q=0处和单项式对称函数在q=1处的特征，寻找酉对偶等价于确定签名字符和字符重合的时间。其次，Hall-Littlewood结果包含了奇异无穷小特征的情况。由Salamanca-Riba的工作可知，任何强正则无穷小字符的幺正表示都与Aq（?）模同构，并且任何Aq（?）都是幺正的，从而使得幺正对偶问题的无穷小字符奇异情况打开。我建议将有符号Kazhdan-Lusztig多项式的工作扩展到奇异情况，使用签名字符公式对酉不可约的最高权模块进行分类，简化和更好地理解O类的签名字符，确定上同调诱导表示的签名字符，并确定哪些上同调诱导模块是酉的。

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.