PIRSA:20060033  ( MP4 Medium Res , MP3 , PDF ) Which Format?
The ``Springer" representation of the DAHA
Speaker(s): Monica Vazirani
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.
Date: 24/06/2020 - 2:00 pm
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