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.

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