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SUMMARY:Monica Vazirani (UC Davis)
DTSTART:20200624T180000Z
DTEND:20200624T190000Z
DTSTAMP:20260424T135123Z
UID:GRT-2020/19
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GRT-2020/19/
 ">The "Springer" representation of  the DAHA</a>\nby Monica Vazirani (UC D
 avis) as part of Geometric Representation Theory conference\n\n\nAbstract\
 nThe Springer resolution and resulting Springer sheaf\nare key players in 
 geometric representation theory.\nWhile one can construct the Springer she
 af geometrically\,\nHotta and Kashiwara gave it a purely algebraic reincar
 nation in\nthe language of equivariant $D(\\mathfrak{g})$-modules.\n\nFor 
 $G = GL_N$\, the endomorphism algebra of the Springer sheaf\,\nor equivale
 ntly of the associated $D$-module\,\nis isomorphic to $\\mathbb{C}[\\mathc
 al{S}_n]$ the group algebra of\nthe symmetric group.\nIn this talk\, I'll 
 discuss a quantum analogue of this.\nIn joint work with Sam Gunningham and
  David Jordan\, we define\nquantum Hotta-Kashiwara $D$-modules $\\mathrm{H
 K}_\\chi$\,\nand compute their endomorphism algebras.\nIn particular $\\ma
 thrm{End}_{\\mathcal{D}_q(G)}(\\mathrm{HK}_0)\n\\simeq \\mathbb{C}[\\mathc
 al{S}_n]$.\n\nThis is part of a larger program to understand the category\
 nof strongly equivariant quantum $D$-modules.\nOur main tool to study this
  category is Jordan's elliptic  Schur-Weyl\nduality functor to representat
 ions of the double affine Hecke algebra\n(DAHA).\nWhen we input $\\mathrm{
 HK}_0$ into Jordan's functor\,\nthe endomorphism algebra over the DAHA  of
  the output is\n$\\mathbb{C}[\\mathcal{S}_n]$ from which we deduce the res
 ult above.\n\nFrom studying  the output of all the $\\mathrm{HK}_\\chi$\, 
 we are\nable to compute that for input  a distinguished projective\ngenera
 tor of the category\nthe  output is the DAHA module generated by the sign 
 idempotent.\n\nThis is joint work with Sam Gunningham and David Jordan.\n
LOCATION:https://researchseminars.org/talk/GRT-2020/19/
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