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 HottaKashiwara $D$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 SchurWeyl
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
Collection: Geometric Representation Theory
