Related papers: Hikita-Nakajima conjecture for the Gieseker variet…

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…

The Hamiltonian reduction $\mathcal N///T$ of the nilpotent cone in $\mathfrak{sl}_n$ by the torus of diagonal matrices is a Nakajima quiver variety which admits a symplectic resolution $\widetilde{\mathcal N///T}$, and the corresponding…

We give an algebraic description of the equivariant K-theory of Gieseker varieties. Our main result identifies the equivariant K-theory of the Gieseker space with the Jucys--Murphy center of the cyclotomic Hecke algebra, over the…

We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold $X$, such a representation involves an assignment of a complex vector bundle on $X$ to each node of the doubled quiver; to the…

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing…

Convolution in Borel-Moore homology plays an important role in Nakajima's construction of representations of the Heisenberg algebra and of modified enveloping algebras of Kac-Moody algebras. In its most basic form, convolution between two…

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, a reflection functor and a transpose defined by certain bilinear forms. These…

We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with…

The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which…

This paper classifies all 4d Nakajima quiver varieties through a combinatorial approach. For each such variety, we describe the symplectic leaves and minimal degenerations between them. Using the resulting Hasse diagrams and secondary…

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's…

We prove a conjecture of Nakajima (for type A the result was announced by Ginzburg- Vasserot) giving a geometric realization, via quiver varieties, of the Yangian of type ADE (and more in general of the Yangian associated to every symmetric…

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of…

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the…

The goal of this paper is to establish Beilinson-Bernstein type localization theorems for quantizations of some conical symplectic resolutions. We prove the full localization theorems for finite and affine type A Nakajima quiver varieties.…

We relate the representations of the rational Cherednik algebras associated with the complex reflection group G(m,1,n) to sheaves on Nakajima quiver varieties associated with extended Dynkin gaphs via a Z-algebra construction. As the…

Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon…

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic…

In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that…

Let $G$ be a complex reductive algebraic group. In arxiv:2108.03453 Ivan Losev, Lucas mason-Brown and the third-named author suggested a symplectic duality between nilpotent Slodowy slices in $\mathfrak{g}^\vee$ and affinizations of certain…