  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.
