# RESEARCH

## On a geometric realization of the asymptotic affine Hecke algebra: **arXiv:2312.10582**

### Roman Bezrukavnikov, Ivan Karpov, Vasily Krylov

A key tool for the study of an affine Hecke algebra $\mathcal{H}$ is provided by Springer theory of the Langlands dual group via the realization of $\mathcal{H}$ as equivariant $K$-theory of the Steinberg variety. We prove a similar geometric description for Lusztig's asymptotic affine Hecke algebra $J$ identifying it with the sum of equivariant $K$-groups of the squares of fixed points of Springer fibers, as conjectured by Qiu and Xi. As an application, we give a new geometric proof of Lusztig's parametrization of irreducible representations of $J$. We also reprove Braverman-Kazhdan's spectral description of $J$. As another application, we prove a description of the cocenters of $\mathcal{H}$ and $J$ conjectured by the first author with Braverman, Kazhdan and Varshavsky. The proof is based on a new algebraic description of $J$, which may be of independent interest.

## Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals: **arXiv:2212.11995**

### Vasily Krylov, Inna Mashanova-Golikova, Leonid Rybnikov

In this paper, we construct a natural structure of affine crystals on the spectra of Bethe subalgebras acting on the tensor product of Kirillov-Reshetikhin modules for the Yangian in type A. We conjecture that such a construction exists for arbitrary g and gives Kirillov-Reshetikhin crystals. As an application of our results, one can explicitly describe the monodromy of solutions of Bethe equations. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to certain commutative subalgebras in the universal enveloping of the current Lie algebra. These commutative subalgebras are of independent interest, we show that they come from the Feigin-Frenkel center on the critical level and can be considered as "universal" versions of shifted Gaudin subalgebras.

## Subregular nilpotent orbits and explicit character formulas for modules over affine Lie algebras: **arXiv:2209.08865**

### Roman Bezrukavnikov, Victor Kac, Vasily Krylov

Let g be a simple finite dimensional Lie algebra of type $A$, $D$, $E$, and let $\hat{\mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a simple highest weight $\hat{\mathfrak{g}}$-module are either bounded or are given by a linear function of the weight. We explain and generalize this observation by means of Kazhdan-Lusztig theory, namely, by computing values at $q=1$ of certain (parabolic) affine inverse Kazhdan-Lusztig polynomials. The calculation relies on the explicit description of the canonical basis in the cell quotient of the anti-spherical module over the affine Hecke algebra corresponding to the subregular cell. We also present an explicit description of the corresponding objects in the derived category of equivariant coherent sheaves on the Springer resolution, they correspond to irreducible objects in the heart of a certain t-structure related to the so called non-commutative Springer resolution. (**accepted to the special volume of PAMQ dedicated
to Corrado De Concini’s birthday**)

## Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications: **arXiv:2209.01481**

### Merrick Cai, Vasily Krylov

We study the Frobenius pushforwards of invertible sheaves on the wonderful compactifications, and in particular its decomposition into locally free subsheaves. We give necessary and sufficient conditions for a specific line bundle to be a direct summand of the Frobenius pushforward of another line bundle. In the case of $G=\operatorname{PSL}_n$, we offer lower bounds on the multiplicities (as direct summands) for those line bundles satisfying the sufficient conditions. We also decompose Frobenius pushforwards of line bundles into a direct sum of vector subbundles, whose ranks are determined by invariants on the weight lattice of G. We study a particular block which decomposes as a direct sum of line bundles, and identify the line bundles that appear in this block. Finally, we present two approaches to compute the class of the Frobenius pushforward of line bundles on wonderful compactifications in the rational Grothendieck group and in the rational Chow group. (**submitted to Communications in Algebra**)

## Bethe subalgebras in antidominantly shifted Yangians: **arXiv:2205.04700**

### Vasily Krylov, Leonid Rybnikov

The loop group of a simple complex Lie group $G$ has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras in functions on the loop group depending on the parameter $C \in G$ called classical universal Bethe subalgebras. To every antidominant cocharacter $\mu$ of the maximal torus T⊂G one can associate the closed Poisson subspace $\mathcal{W}_\mu$ of $G((z^{-1}))$ (the Poisson algebra $\mathcal{O}(\mathcal{W}_\mu)$ is the classical limit of so-called shifted Yangian $Y_\mu(\mathfrak{g})$). We consider the images of universal Bethe algebras in $\mathcal{O}(\mathcal{W}_\mu)$, that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular $C$ centralizing $\mu$, we compute the Poincaré series of these subalgebras. For $\mathfrak{g}=\mathfrak{gl}_n$, we define the natural quantization of functions on the loop group of $\mathfrak{gl}_n$ and universal Bethe subalgebras there. Taking the images of $B(C)$ in $Y_\mu(\mathfrak{gl}_n)$ we recover Bethe subalgebras $B_\mu(C)$ proposed by Frassek, Pestun and Tsymbaliuk and prove their conjecture about the Poincaré series of these subalgebras. (**submitted to IMRN**)

## Hikita-Nakajima conjecture for the Gieseker variety: **arXiv:2202.09934**

### Vasily Krylov, Pavel Shlykov

Hikita-Nakajima conjecture is a general conjecture about the relation between the geometry of symplectically dual varieties. In this paper, we prove Hikita-Nakajima conjecture for the ADHM spaces (also known as Gieseker varieties). We also develop a general approach towards the proof of Hikita-Nakajima conjecure and formulate a very explicit "numerical" conjecture relating Higgs and Coulomb branches, this conjecture should imply Hikita-Nakajima conjecture in the case of Nakajima quiver varieties. (**submitted to Selecta**)

## Representations with minimal support for quantized Gieseker varieties: **arXiv:2002.06741**

### Pavel Etingof, Vasily Krylov, Ivan Losev, José Simental

We study the minimally supported representations of quantizations of Gieseker moduli spaces. We relate them to $\operatorname{SL}_n$-equivariant D-modules on the nilpotent cone of $\mathfrak{sl}_n$ and to minimally supported representations of type $A$ rational Cherednik algebras. Our main result is character formulas for minimally supported representations of quantized Gieseker moduli spaces. (**Mathematische Zeitschrift**)

## Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A: **arXiv:1905.01810**

### Ivan Mirkovic, Maxim Vybornov (with an appendix by Vasily Krylov)

This paper contains an appendix written by myself where the explicit formula for the isomorphism between type $A$ quiver varieties and slices in affine Grassmanian is given. (**Advances in Mathematics**)

## Almost dominant generalized slices and convolution diagrams over them: **arXiv:1903.08277**

### Vasily Krylov, Ivan Perunov

We study generalized slices in affine Grassmannians $\operatorname{Gr}_G$ corresponding to $\lambda$ and $\mu$ (for simply-laced $G$ these varieties are known to be isomorphic to Coulomb branches of the corresponding quiver gauge theories). We apply our results to compute characters of tangent spaces at torus fixed points of convolution diagrams over slices (symplectic resolutions of the corresponding Coulomb branches), construct open affine coverings of these convolution diagrams for quasi-dominant $\mu$ (confirming one of the predictions of $3D$-mirror symmetry), and to compute their Poincaré polynomials. (**Advances in Mathematics**)

## Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory): **arXiv:1805.07721**

### Michael Finkelberg, Vasily Krylov, Ivan Mirković

We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with the universal enveloping algebra of the positive nilpotent subalgebra of the Lie algebra $\mathfrak{g}$. To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory. (**Journal of Topology**)

## Integrable crystals and restriction to Levi via generalized slices in the affine Grassmannian: **arXiv:1709.00391**

### Vasily Krylov

We construct the integrable crystals $B(\lambda)$ (corresponding to finite dimensional irreducible representations of simple Lie algebra $\mathfrak{g}$), using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $B(\lambda_1) \otimes B(\lambda_2) \rightarrow B(\lambda_1+\lambda_2) \cup \{0\}$ in terms of multiplication of generalized transversal slices. Let $L \subset G$ be a Levi subgroup of $G$. We describe the restriction to Levi $\operatorname{Res}\colon \operatorname{Rep}(G) \rightarrow \operatorname{Rep}(L)$ in terms of the hyperbolic localization functors for the generalized transversal slices. (**Functional Analysis and Its Applications**)