Quantum Character Theory

TL;DR

This paper introduces a quantum analogue of conjugation equivariant D-modules on complex reductive groups, developing new tools and computations in quantum algebra, with applications to skein theory and quantum groups.

Contribution

It defines quantum Hotta-Kashiwara modules and computes their endomorphism algebras, extending classical theory into the quantum setting with new algebraic tools.

Findings

Computed endomorphism algebras of quantum modules

Developed tools from double affine Hecke algebra

Calculated skein algebra of the 2-torus for quantum groups

Abstract

We develop a $\mathtt{q}$-analogue of the theory of conjugation equivariant $\mathcal D$-modules on a complex reductive group $G$. In particular, we define quantum Hotta-Kashiwara modules and compute their endomorphism algebras. We use the Schur-Weyl functor of the second author, and develop tools from the corresponding double affine Hecke algebra to study this category in the cases $G=GL_N$ and $SL_N$. Our results also have an interpretation in skein theory (explored further in a sequel paper), namely a computation of the $GL_N$ and $SL_N$-skein algebra of the 2-torus.