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。这个问题是如此重要，它本来是一个粘土千年奖的问题，除非它被错误地认为这个问题已经解决了。酉对偶问题是I.M. Gelfand的程序在抽象的谐波分析，提出解决问题的不同领域的科学如下。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。解决那些简单的代数问题，并将答案重新组合成一个全局解。将该解决方案转化为原始问题的解决方案。 这个提议是关于自20世纪30年代以来一直公开的酉对偶问题。 对酉表示进行分类的一般方法如下。识别承认不变的埃尔米特形式的表示，计算形式的签名，然后确定哪些形式是正定的，因此是酉的。厄米特表示的签名字符的公式存在。这个想法是变形表示和跟踪变化，因为减少点交叉。不幸的是，由于递归，公式是非常笨拙的，涉及产品的迹象，2的权力，和翻译。幸运的是，在最近的工作中，证明了在不可约Verma模的情况下，所有的复杂性都可以用仿射Hecke代数编码：签名字符只是霍尔-利特尔伍德多项式的被加数的"负数"，其值为q =-1乘以Weyl分母的版本。用于寻找酉表示的现有技术是用于确定给定表示是否是酉的计算机算法。霍尔-利特尔伍德的结果表明，一个封闭形式的答案找到整个酉对偶是可实现的原因有两个。首先，Hall-Littlewood多项式是有限维q = 0的不可约最高权模和q = 1的单项式对称函数的特征标，求其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结果包括奇异无穷小特征标的情况。由Salamanca-Riba的工作可知，任何强正则无穷小特征标的酉表示同构于一个Aq（？）模和任意Aq（？）是酉的，留下奇异无穷小字符的情况下，酉对偶问题开放。我建议扩展工作签署Kazhdan-Lusztig多项式的奇异情况下，分类酉不可约的最高权重模块使用签名字符公式，简化和更好地理解签名字符的类别O，确定签名字符的上同调诱导表示，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.