The Littelmann path model via the affine Grassmannian

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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.