Talks

I will explain a generalized Albanese property for smooth curves, which implies Deligne's geometric class field theory with arbitrary ramification. The proof essentially reduces to some well-known Cartier duality statements. This is joint work with Andreas Hayash.

The Springer resolution and resulting Springer sheaf are key players in geometric representation theory. While one can construct the Springer sheaf geometrically, Hotta and Kashiwara gave it a purely algebraic reincarnation in the language of equivariant $D(\mathfrak{g})$-modules. For $G = GL_N$, the endomorphism algebra of the Springer sheaf, or equivalently of the associated $D$-module, is isomorphic to $\mathbb{C}[\mathcal{S}_n]$ the group algebra of the symmetric group. In this talk, I'll discuss a quantum analogue of this. In joint work with Sam Gunningham and David Jordan, we define quantum Hotta-Kashiwara $D$-modules $\mathrm{HK}_\chi$, and compute their endomorphism algebras. In particular $\mathrm{End}_{\mathcal{D}_q(G)}(\mathrm{HK}_0) \simeq \mathbb{C}[\mathcal{S}_n]$. This is part of a larger program to understand the category of strongly equivariant quantum $D$-modules. Our main tool to study this category is Jordan's elliptic Schur-Weyl duality functor to representations of the double affine Hecke algebra (DAHA). When we input $\mathrm{HK}_0$ into Jordan's functor, the endomorphism algebra over the DAHA of the output is $\mathbb{C}[\mathcal{S}_n]$ from which we deduce the result above. From studying the output of all the $\mathrm{HK}_\chi$, we are able to compute that for input a distinguished projective generator of the category the output is the DAHA module generated by the sign idempotent. This is joint work with Sam Gunningham and David Jordan.

I will give an overview of a joint project with Simon Riche and Laura Rider and another one with Dima Arinkin aimed at a modular version of the equivalence between two geometric realization of the affine Hecke algebra and derived Satake equivalence respectively. As a byproduct we obtain a proof of the Finkelberg-Mirkovic conjecture and a possible approach to understanding cohomology of higher Frobenius kernels with coefficients in a G-module.

The Beilinson-Drinfeld Grassmannian of a simple complex algebraic group admits a natural stratification into "global spherical Schubert varieties". In the case when the underlying curve is the affine line, we determine algebraically the global sections of the determinant line bundle over these global Schubert varieties as modules over the corresponding Lie algebra of currents. The resulting modules are the global Weyl modules (in the simply laced case) and generalizations thereof. This is a joint work with Ilya Dumanski and Evgeny Feigin.

Elliptic stable envelopes, introduced by Aganagic and Okounkov, are a key ingredient in the study of quantum integrable systems attached to a symplectic resolution. I will describe a relation between elliptic stable envelopes on a hypertoric variety and a certain 'loop space' of that variety. Joint with Artan Sheshmani and Shing-Tung Yau.

Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers positive parts of Yangians as defined by Maulik-Okounkov. For general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem. One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are positive parts of quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.