The "Springer" representation of the DAHA

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 $D(g)$-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.