The ``Springer" representation of the DAHA

          @misc{ scivideos_PIRSA:20060033,
            doi = {10.48660/20060033},
            url = {https://pirsa.org/20060033},
            author = {Vazirani, Monica},
            keywords = {Mathematical physics},
            language = {en},
            title = {The {\textquoteleft}{\textquoteleft}Springer" representation of  the DAHA},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2020},
            month = {jun},
            note = {PIRSA:20060033 see, \url{https://scivideos.org/pirsa/20060033}}
          }
          

Abstract

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 $D(\mathfrak{g})$-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. This is joint work with Sam Gunningham and David Jordan.