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.

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.

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.

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.

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.

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. ()

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.

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.

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.

We define a notion of slant sum of quiver gauge theories. We relate torus fixed points on the corresponding Higgs branches, which are Nakajima quiver varieties. Then we prove a formula relating the quasimap vertex functions before and after a slant sum, which is a type of "branching rule" for vertex functions. Our construction is motivated by a conjecture, which we make here, for the factorization of the vertex functions of zero-dimensional quiver varieties. The branching rule allows this conjecture to be approached inductively. In special cases, it also provides a formula for the $\hbar=q$ specialization of vertex functions for quiver varieties not necessarily of Dynkin type as a sum over reverse plane partitions. When passed through the quantum Hikita conjecture, such expressions provide conjectural formulas for graded traces of Verma modules on the 3d mirror dual side. We also consider the Coulomb side. We make some conjectures reflecting what can be seen on the Higgs side and prove them in ADE type. We study slant sums of Coulomb branches and their quantizations. We prove that for one-dimensional framing, slant sum operation on the Coulomb branch side corresponds to taking products.

We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest a variant of the conjecture, with K-theoretic Coulomb branches involved. We show that this version follows from the usual (homological) one for a set of theories. We apply this result to prove the conjecture in finite ADE types. In the course of the proof, we show that appropriate completions of K-theoretic and homological (quantized) Coulomb branches are isomorphic.

Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453 authors suggested a symplectic duality between nilpotent Slodowy slices in $\mathfrak{g}^{\vee}$ and affinizations of certain $G$-equivariant covers of special nilpotent orbits. In this paper, we study the various versions of Hikita conjecture for this pair. We show that the original statement of the conjecture does not hold for the pairs in question and propose a refined version. We discuss the general approach towards the proof of the refined Hikita conjecture and prove this refined version for the parabolic Slodowy varieties, which includes many of the cases considered in arXiv:2108.03453 and more. Applied to the setting of arXiv:2108.03453, the refined Hikita conjecture explains the importance of special unipotent ideals from the symplectic duality point of view. We also discuss applications of our results. In the appendices, we discuss some classical questions in Lie theory that relate the refined version and the original version. We also explain how one can use our results to simplify some proofs of known results in the literature. As a combinatorial application of our results we observe an interesting relation between the geometry of Springer fibers and left Kazhdan-Lusztig cells in the corresponding Weyl group.

We calculate explicit special values of parabolic affine inverse Kazhdan-Lusztig polynomials for subregular nilpotent orbits. We thus obtain explicit character formulas for certain irreducible representations of affine Lie algebras. To do so, using the geometry of the Springer resolution, we identify the cell quotient of the anti-spherical module over $\hat{W}$ corresponding to the subregular cell with a certain one-dimensional extension of a module defined by Lusztig. We describe the canonical basis in this module geometrically and 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 that we describe using an equivariant version of the derived McKay correspondence.

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.

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.