MIT Infinite Dimensional Algebra Seminar (Spring 2022)

Many interesting algebras in (geometric) representation theory arise as convolution algebras. Based on these examples we develop a general framework using Chow rings and Chow motives. Chow motives are objects in a weights structure of the triangulated derived category of motives. I will explain weight structure and weight complex functors and try to explain why it might be interesting for representation theorists.  We finally indicate formality results using motives instead of perverse sheaves.

The notion of chiral homology of a chiral algebra was introduced by Beilinson and Drinfeld, generalising conformal blocks. The construction of a chiral algebra from a conformal vertex algebra and a smooth complex curve provides a large supply of interesting examples, but in general the chiral homology of these examples seems not to be well understood. Motivated by questions in the representation theory of vertex algebras, we study the behaviour of the chiral homology of families of elliptic curves degenerating to a nodal curve. After introducing chiral homology in general, I will explain how to develop explicit complexes to compute it in the case of interest, relate it to the Hochschild homology of the corresponding Zhu algebra, and establish links with identities of Rogers-Ramanujan type and their generalisations. (Joint work with R. Heluani)

I will describe a program that uses shuffle algebras to yield generators-and-relations presentations for quantum loop groups. The main idea is that the necessary relations are dual to the so-called wheel conditions that describe the shuffle algebras in question, and we will use this to get a complete presentation of two interesting algebras that arise in geometric representation theory: K-theoretic Hall algebras of quivers, and Hall algebras of coherent sheaves on curves over finite fields (the latter project joint work with Francesco Sala and Olivier Schiffmann).

A representation of $gl(V)$ is a map $V \otimes V^* \otimes M \to M$ satisfying some conditions, or via currying, it is a map $V \otimes M \to V \otimes M$ satisfying different conditions. The latter formulation can be used in more general symmetric tensor categories where duals may not exist, such as the category of sequences of symmetric group representations under the induction product. Several other families of Lie algebras have such "curried" descriptions and their categories of representations have nice compact descriptions as representations of diagram categories, such as the (walled) Brauer category, partition category, and variants. I will explain a few examples in detail and how we came to this definition. This is joint work with Andrew Snowden.

We present a new approach to the problem of quantising integrable systems of differential-difference equations. The main idea is to lift these systems to systems defined on free associative algebras and look for the ideals there that are stabilized by the new dynamics. In a reasonable class of candidate ideals, there are typically very few that are invariant for the first equation in the hierarchy. Once these ideals are picked the challenge is to prove that the whole hierarchy of equations stabilizes them. We will discuss these ideas using as a key example the hierarchy of the Bogoyavlensky equation.

This is a joint work with A. Mikhailov (Leeds) and J. P. Wang (U. of Kent). To be published soon.

I will present joint work with Penghui Li on our theory of graded sheaves on Artin stacks. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. The theory of graded sheaves has a natural interpretation in terms of mixed geometry à la Beilinson--Ginzburg--Soergel and provides a uniform construction thereof. In particular, it provides a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Historically, constructions of graded lifts were done on a case-by-case basis and were technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. As an application, I will conclude the talk by showing that the category of constructible B-equivariant graded sheaves on the flag variety G/B is a geometrization of (and is equivalent to) the DG-category of bounded chain complexes of Soergel bimodules.

In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R.Frassek, I. Karpov, and V.Pestun.

This topology induces the Lie topology on the Levi factors of parabolics of spherical type, in the indefinite cases it provides new examples for Kramer's theory of topological twin buildings (which he developed in 2002 for loop groups), and in the Archimedian case it is possible to determine their fundamental groups, actually providing a structural explanation for the (known by classification) fundamental groups of semisimple split real Lie groups.

Moreover, in the 2-spherical situation these topological groups turn out to satisfy Kazhdan's Property (T), and allow Mostow-Margulis-type rigidity results for (S-)arithmetic subgroups.

Kac-Moody groups also admit symmetric spaces. In the non-spherical situation, these symmetric spaces have a causal structure with the two halves of the twin building visible at infinity in the future, resp. past directions. One can prove that either time-travel is impossible on Kac-Moody symmetric spaces (i.e., there are no non-trivial closed causal piecewise geodesic curves) or all points of the Kac-Moody symmetric space are causally equivalent. It turns out that the question which of the two cases occurs is equivalent to the question whether (global) Kostant convexity holds for Kac-Moody groups; it also seems, by an observation that Hartnick and Damour pointed out to me, that this question is closely related to what physicists call "cosmological billards", so from a physical point of view one should expect Kostant convexity to hold and, thus, time travel to be impossible on Kac-Moody symmetric spaces.

(Global) Kostant convexity is the question how the A-part in the Iwasawa decomposition KAN changes if one multiplies with an element in K from the wrong side. There is a local version concerning the adjoint action on the Kac-Moody algebra, and this holds by a result by Kac and Peterson from 1984.

I started thinking about Kac-Moody groups as a postdoc in 2005 (when together with three peers at TU Darmstadt we founded what we then called the "anonymous Kac-Moody theorists"), and this is a report on various things we encountered along the way; two of the three other anonymous Kac-Moody theorists became key collaborators of mine over the years. My collaborators and students during various stages of my work will be mentioned explicitly as we make our way through the various observations.

We’ll revisit the theorem of Feigin and Frenkel that says that screening operators that act between Wakimoto modules satisfy quantum Serre relations. We’ll use this to giving an alternative construction of (the Iwahori) variant of Kazhdan-Lusztig equivalence.

The Weyl group and the nilpotent orbits are two basic objects attached to a semisimple Lie group. In this talk, I will describe two very different geometric constructions relating these two objects, due to Kazhdan-Lusztig, Lusztig, and myself.

The main result is that these two constructions give the same maps between conjugacy classes in the Weyl group and the set of nilpotent orbits. It confirms a conjecture of Kazhdan and Lusztig in the 80s. The proof uses affine Springer fibers, and leads to more open questions about them.

The Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogs of classical objects provide insights into their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in the Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of the Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in the Deligne categories, it is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of the Deligne categories.​

Coulomb branches are a new construction of symplectic singularities based in 3-dimensional N=4 supersymmetric quantum field theory. Even in the case where these recover well-known singularities such as nilcones and symmetric powers of C^2, they still shed new light on these varieties. In particular, they are a presentation which is much better adapted to the construction of tilting generators using the approach of Bezrukavnikov and Kaledin. I'll discuss how this leads to interesting noncommutative symplectic resolutions of Coulomb branches, and a description of the wall-crossing functors for the categories of coherent sheaves on resolutions.

I will talk about relations between modules over the small quantum group $u_q$ at a root of unity and geometry of a specific affine Springer fiber F. Cohomology of F is related to the center of $u_q$, while (Koszul dual of) the category of $u_q$ modules is related to microlocal sheaves on F.Based on joint project with Pablo Boixeda Alvarez, Peng Shan and Eric Vasserot, and with Boixeda Alvarez, Michael McBreen and Zhiwei Yun.

I will discuss a new construction of pre-Tannakian categories associated to oligomorphic groups -- a class of groups arising in model theory. This gives a new concrete realization of Deligne's interpolation categories Rep $(S_t)$, as well as new examples of pre-Tannakian categories in characteristic zero which are not interpolations or ultraproducts of Tannakian categories.