Related papers: Equivariant Satake category and Kostant-Whittaker …

In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group $G$ acting on a variety $X$ are in fact equivalent. In the first part of this…

Let $G_\mathbb R$ be a connected real reductive group and let $X$ be the corresponding complex symmetric variety under the Cartan bijection. We construct a canonical equivalence between the relative Satake category of $G(\mathcal…

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…

Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove…

We construct a new affine Grassmannian which connects an equal characteristic affine Grassmannian and Zhu's Witt vector affine Grassmannian. As a result, we deduce the mixed characteristic version of the Bezrukavnikov-Finkelberg's derived…

Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in Arkhipov and {\O}rsted (2018a, 2018b), we define a dg-model for the derived category of…

We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis,…

We consider the semi-direct products $G=\mathbb Z^2\rtimes GL_2(\mathbb Z), \mathbb Z^2\rtimes SL_2(\mathbb Z)$ and $\mathbb Z^2\rtimes\Gamma(2)$ (where $\Gamma(2)$ is the congruence subgroup of level 2). For each of them, we compute both…

We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the…

Given a simple, simply connected compact Lie group G, let M be a G-space. We describe the quantization of the category of positive energy representations of the loop group of G at a given level and parametrized over the loop space LM. This…

Let $G$ be a connected, simply-laced, almost simple algebraic group over $\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, and let $T_c$ be a maximal torus therein. Let $\mathrm{Gr}_G$ denote the affine Grassmannian…

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant $K$-class is given by the partition function of an integrable loop model,…

Let $\text{X}$ denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands'…

We calculate the category of D-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham…

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the…

The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures…

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra…

We provide a factorization model for the continuous internal Hom, in the homotopy category of $k$-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we…

This article establishes a geometric Satake equivalence for affine Kac-Moody groups as an equivalence of abelian semisimple categories over algebraically closed fields. We define a well-behaved category of equivariant sheaves on the double…

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. We relate it to a certain t-structure on the derived category of…