The Littelmann path model via the affine Grassmannian

Kazhdan-Lusztig理论中最重要的开放问题之一是提供Kazhdan-Lusztig多项式的封闭公式。这些多项式把数学的各个领域联系起来。到目前为止，封闭公式只存在于一些特殊情况下，例如众所周知的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子的几何研究。对于这些多项式，有一个著名的封闭公式，是根据与杨氏表相关的电荷统计量来计算的。这些表是经典的组合对象，在一般线性群的表示理论中起着重要的作用，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量的几何解释，从而重新统一组合学和几何。对于任何复约群，都存在Kostka-Foulkes多项式的自然推广。然而，只有在特殊线性群的情况下才存在封闭公式。我们相信电荷的几何解释将提供这样一个公式。利特尔曼路径模型是所有复杂约化群的杨氏表的推广。说Littelmann路径在过去二十年中吸引了很多关注，这是一个非常保守的说法。我们的目标是给任何利特曼路径分配一个电荷统计量。最近，Littelmann路径模型也被解释为仿射格拉斯曼的几何形状，使用被称为建筑物的物体。然而，这种解释是不完整的。目的是完成这种解释，这将加深Littelmann路径、仿射格拉斯曼几何和建筑理论之间的现有联系。我们相信这将导致电荷的几何定义。与此相关的是这个项目的第二个目标。在最近与Schumann合作的工作中，我们证明了naito - sagaki的一个猜想，给出了将特殊线性李代数的不可约表示的限制分解为辛李代数的分支规则。这个猜想已经开放了十多年，新规则提供了一种新的方法来研究利特尔曼路径下非利瓦伊子代数的分支规则。我们考虑的子代数是作为有限阶自同构的不动点集得到的子代数。如果自同构是半单无限阶的，则不动点集合是Levi子代数。在这个项目中，我们的目的是描述“李维分支”的建筑理论几何，并在更神秘和困难的非李维情况下提供类似的模拟。我们认为这样的解释不仅提供了分支规则，而且还提供了只存在于Levi子代数的几何Satake等价的建立中的限制函子。

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.