arXiv subjects
Publications and source records attributed to Ekaterina Bogdanova.
We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks $\operatorname{Ge}_{Z_G}$ and $\operatorname{Ge}_{π_1(\check{G})}$, which classify gerbes on $X$ with respect to the center $Z_G$ of $G$ and algebraic fundamental group $π_1(\check{G})$ of $\check{G}$.
We prove that after inverting the Planck constant $h$ the Bezrukavnikov-Kaledin quantization $(X, \mathcal{O}_h)$ of symplectic variety $X$ in characteristic $p$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.
We prove that for any reductive group $G$ of adjoint type cuspidal automorphic twisted D-modules have non-vanishing quantum Whittaker coefficients. The argument provides a microlocal interpretation of quantum Whittaker coefficients for any $\checkΛ^+$-valued divisor under some hypothesis on singular support.
We define the stack of $G$-local systems with restricted variation on the formal puntured disc and study its properties. We embed sheaves of categories over this stack into the category of factorization module categories over $\operatorname{Rep}(G)$. Along the way we develop a theory of factorization structures in families and study functorialities of such under changes of the base curve.
Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[η] \in H^0(X, Ω^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In this paper we construct a functorial in $(X, [η])$ formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves on $X$. We also construct its natural extension to a quasi-coherent sheaf of categories $\mathrm{QCoh}_h$ on the product $X^{(1)} \times {\mathbb S}$ of the Frobenius twist of $X$ and the projective line ${\mathbb S}=\mathbb P^1$, viewed as the one-point compactification of $\mathrm{Spec}\ \! k[h]$. Its global sections over $X^{(1)} \times \{0\}$ is the category of quasi-coherent sheaves on $X$. If $X$ is affine, $\mathrm{QCoh}_h$, restricted to $X^{(1)}\times \mathrm{Spf} \ \! k[[h]]$, is equivalent to the category of modules over the distinguished "Frobenius-constant" quantization of $(X,[η])$ defined by Bezrukavnikov and Kaledin.