Research
Analytic Langlands correspondence
Multiplication Kernels for the Analytic Langlands Program in Genus Zero (joint with Sanjay Raman), arXiv:2212.06932
We provide an explicit proof of a recent result of Gaiotto arXiv:2110.02255 which gives an explicit formula for a so-called "multiplication kernel'' $K_3(x, y, z; t)$ intertwining the action of Hecke operators and Gaudin operators in three sets of variables. This function $K_3$ arises naturally in the context of the analytic formulation of the geometric Langlands program in the genus-zero case arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. We also discuss how the kernel $K_3$ relates to other objects typically considered in the analytic Langlands program.
Positive traces
Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras, arXiv:2307.06514
Let $\mathcal{A}$ be a generalized Weyl or $q$-Weyl algebra, $M$ be an $\mathcal{A}-\overline{\mathcal{A}}$-bimodule. Choosing an automorphism $\rho$ of $\mathcal{A}$ we can define the notion of an invariant Hermitian form: $(au,v)=(u,v\rho(a))$ for all $a\in\mathcal{A}$ and $u,v\in M$. In two papers below we obtained a classification of invariant positive definite Hermitian forms in the case when $M=\mathcal{A}=\overline{\mathcal{A}}$, the case of the regular bimodule. In this paper we obtain a classification of invariant positive definite forms on $M$ in the case when $\mathcal{A}$ has no finite-dimensional representations.
Twisted Traces and Positive Forms on Generalized q-Weyl Algebras, arXiv:2105.12652, SIGMA
Let ${\mathcal A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ),$ where $P$ is a Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on ${\mathcal A}$ is called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in {\mathbb C}$ with $|s|=1$ and all $a,b\in {\mathcal A}$. In this paper we classify positive definite invariant Hermitian forms on generalized $q$-Weyl algebras.
Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A (joint with Pavel Etingof, Eric Rains, Douglas Stryker), arXiv:2009.09437, SIGMA
Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies , Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.
On unitarizable Harish-Chandra bimodules for deformations of Kleinian singularities, arXiv:2003.11508, IMRN
The notion of a Harish-Chandra bimodule, i.e. finitely generated $U(\mathfrak{g})$-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [Ivan Losev, Dimensions of irreducible modules over W-algebras and Goldie ranks. arXiv:1209.1083]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, i.e. the algebra itself, for a deformation of Kleinian singularity of type A is unitarizable. We obtain a partial classification of unitarizable regular bimodules.
Other projects
Deformations of pairs of Kleinian singularities, arXiv:1805.08197, accepted to IMRN
Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger
