Institute for Mathematics and its Applications

The Grothendieck-Springer resolution of a semisimple complex group G carries a canonical Poisson structure whose symplectic leaves are affine bundles over the flag variety. This structure is obtained by Hamiltonian reduction of the cotangent bundle T*G with respect to the action of a Borel. We construct a multiplicative counterpart to this setting by introducing a general method for quasi-Poisson reduction relative to the action of a subgroup and applying it to the quasi-Hamiltonian internal fusion double. The resulting structure has a natural foliation by quasi-Hamiltonian fiber bundles over the flag variety.

Let G be a complex, connected, reductive, algebraic group, and $\chi:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of G. Let $G_0$ be the centralizer of $\chi(\mathbb{C}^\times)$. Here I will talk about $G_0$-equivariant parity sheaves on the $n$-graded piece, $\mathfrak{g}_n$. For the first half we will spend on the derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role.

I will define parabolic induction and restriction in graded settings.

We will dive into the results of Lusztig in characteristic 0 in the graded setting.

Under some assumptions on the field $\Bbbk$ and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role.

The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on $\mathbb{Z}$-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

I will present my recent result on homological mirror symmetry for the universal centralizer (a.k.a Toda space) associated with a complex semisimple Lie group. The A-side is a partially wrapped Fukaya category on the universal centralizer, and the B-side is the category of coherent sheaves on the categorical quotient of the dual maximal torus by the Weyl group (with some modifications if the group has nontrivial center). I will illustrate many of the geometry and ideas of the proof using the example of SL_2 or PGL_2.

The motivation behind this work in progress is to unify different approaches to Khovanov-Rozansky triply graded link homology, HHH. By an old theorem of Markov links up to isotopy is equivalent to braids up to the so called Markov moves. Hence, braid group actions, which already play a central role in geometric representation theory, provides a natural starting point for algebro-geometric realizations of link homologies. Many such braid group actions are know on e.g. Soergel bimodules, mixed Hodge D-modules, Coherent sheaves, and Matrix Factorizations. There exist several different (some of them conjectural) algebro-geometric approaches to calculating HHH utilizing the above categories. The goal of this project is to unite some of them by relating the geometry of the categories used in the different approaches.

An important topic in representation theory is understanding the relationship between different bases for the same representation. In this talk, we discuss our work showing upper triangularity of the change-of-basis matrix between the sl_3 web basis and Specht basis for irreducible symmetric group representations of three-row rectangular shape. Specht bases are classical in combinatorial representation theory. Web bases arise in the context of studying Lie algebra representations diagrammatically. We will focus on our combinatorial analysis of the expansion of Specht vectors in the sl_3 web basis. This is joint work with Julianna Tymoczko at Smith College.

Closed webs in Kuperberg's B_2 spider evaluate to Laurent polynomials in q with integer coefficients. We will discuss ongoing work on the categorification of the graphical calculus for so_5 webs using decorated web cobordisms or foams.

The Springer fiber of a matrix $X$ is a subvariety of the flag variety consisting of the flags fixed by $X$. One of the classical examples of geometric representation theory shows that their cohomology admits a representation of the symmetric group (or the Weyl group, in general Lie type). Like Schubert varieties, the geometry of Springer fibers is deeply entwined with combinatorics. Unlike Schubert varieties, very little is known about even straightforward questions about this geometry. In this talk, we study the combinatorics and geometry of a particular family of Springer fibers that arise in combinatorics, representation theory, and knot theory. We give some results about how to partition these Springer fibers into cells that are encoded by a kind of graph called a web.

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