The Littelmann path model via the affine Grassmannian

Kazhdan-Lusztig理论中最重要的公开问题之一是给出Kazhdan-Lusztig多项式的封闭公式。这些多项式连接数学的各个领域。到目前为止，封闭公式只存在于一些特殊情况下，如著名的Kostka-Foulkes多项式，它是仿射Kazhdan-Lusztig多项式。它们在对称函数理论中起着重要的作用，并用于仿射格拉斯曼的几何研究。有一个著名的封闭公式，这些多项式的电荷统计与杨tableaux。这些tableaux是经典的组合对象，在一般线性群的表示理论中发挥着重要作用，线性群是代数群的最基本的例子。该项目的主要目标之一是给出电荷统计的几何解释，从而重新统一组合学和几何学。对于任何复约化群，都存在Kostka-Foulkes多项式的自然推广。然而，封闭公式仅存在于特殊线性群的情况下。我们相信电荷的几何解释将提供这样一个公式。Littelmann路径模型是Young tableaux对所有复杂约化群的推广。如果说Littelmann路径在过去的20年里吸引了大量的关注，那将是一个巨大的轻描淡写。我们的目标是分配一个电荷统计的任何Littelmann path. Recently，Littelmann路径模型也被解释在几何方面的仿射格拉斯曼，使用对象称为建筑物。然而，这种解释是不完整的。我们的目的是完成这一解释，这将加深利特尔曼路径之间的现有联系，几何的仿射格拉斯曼，和理论的建筑物。我们相信这将导致电荷的几何定义。与此相关的是本项目的第二个目标。在最近的工作与舒曼，我们已经证明了一个猜想Naito-Sagaki给出了一个分支规则的分解限制的不可约表示的特殊线性李代数的辛李代数。这个猜想已经公开了十多年，新规则提供了一个新的方法来分支规则的非Levi子代数的Littelmann道路。我们考虑的子代数是那些作为有限阶自同构的不动点集而得到的子代数。如果自同构是半单的无限阶的，则其不动点集是Levi子代数.我们的目标是描述“Levi分支”的建立理论几何，并在更神秘、更困难的非Levi情形下提供一个类似.我们相信，这样的解释将不仅提供分支规则，但也限制函子的几何佐竹等价，这只存在于Levi子代数的情况下，设置。

One of the most important open problems in Kazhdan-Lusztig theory is to provide a closed formula for Kazhdan-Lusztig polynomials. These polynomials connect various areas of mathematics. So far, closed formulas exist only in some special cases, such as the well-known Kostka-Foulkes polynomials, which are affine Kazhdan-Lusztig polynomials. They play important roles in the theory of symmetric functions and are used in the geometric study of affine Grassmannians. There is a celebrated closed formula for these polynomials in terms of the charge statistic associated to Young tableaux. These tableaux are classical combinatorial objects which play an important role in the representation theory of the general linear group, the most fundamental example of an algebraic group. One of the main targets of this project is to give a geometric interpretation of the charge statistic, thus reuniting the combinatorics and the geometry. There exists a natural generalisation of Kostka-Foulkes polynomials for any complex reductive group. However, a closed formula exists only in the case of the special linear group. We believe that a geometric interpretation of charge will provide such a formula. The Littelmann path model is a generalisation of Young tableaux for all complex reductive groups. To say that Littelmann paths have attracted a lot of attention in the last twenty years would be a vast understatement. We aim to assign a charge statistic to any Littelmann path. Recently, the Littelmann path model has also been interpreted in terms of the geometry of the affine Grassmannian, using objects called buildings. This interpretation is, however, not complete. The aim is to complete this interpretation, which would deepen the existing connection between Littelmann paths, the geometry of the affine Grassmannian, and the theory of buildings. We believe that this will lead to a geometric definition of charge.Related to this is the second aim of this project. In recent work with Schumann we have proven a conjecture ofNaito-Sagaki giving a branching rule for the decomposition of the restriction of an irreduciblerepresentation of the special linear Lie algebra to the symplectic Lie algebra. This conjecture had been open for over ten years,and the new rule provides a new approach to branching rules for non-Levi subalgebras in termsof Littelmann paths. The subalgebras that we consider are those obtained as fixed-point sets of an automorphism of finite order. If the automorphism is semisimple and of infinite order, the set of fixed points is a Levi subalgebra.Our aim in this project is to describe the building theoretical geometry of "Levi branching", and provide an analogue in the more enigmatic and difficult non-Levi case. We believe that such an interpretation would not only provide branching rules, but also restriction functors in the set-up of the geometric Satake equivalence, which only exist in the case of Levi subalgebras.