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October 2019

Andrey Smirnov, UNC-Chapel Hill – Geometric Methods in Representation Theory Seminar

October 25, 2019 @ 4:00 pm - 5:00 pm

Title: Elliptic stable envelope for Hilbert scheme of points on C^2. Abstract: In this talk I describe an explicit formula for elliptic stable envelope of  torus fixed points on the Hilbert scheme of points in C^2. In K-theoretic limit we obtain new combinatorial formulas for Schur, rational Schur and Macdonald polynomials. In particular, we obtain explicit combinatorial formula for the coefficients of the Kostka matrix.

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November 2019

Cris Negron, UNC-Chapel Hill Mathematics Department – Geometric Methods of Representation Theory Seminar

November 1, 2019 @ 4:00 pm - 5:00 pm

Title: Modularization of quantum groups and some conformal field theory Abstract: I will discuss recent work on constructing small quantum groups at even order roots of unity. (Recall that the small quantum group for a given simple Lie algebra is a characteristic 0, q-analog of its corresponding restricted enveloping algebra.) Our investigations are inspired by a conjectured equivalence of categories between representations for small quantum sl_2, at a certain even order parameter q, and representations for the so-called triplet conformal field theory.…

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Alex Yong, University of Illinois at Urbana-Champaign – Geometric Methods in Representation Theory Seminar

November 15, 2019 @ 4:00 pm - 5:00 pm

Title: The A.B.C.D’s of Schubert calculus Abstract: We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory. This is joint…

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Prof. V. Balaji – Geometric Methods in Representation Theory Seminar

November 22, 2019 @ 3:00 pm - 4:00 pm

Title: Torsors on semistable curves and the problem of degenerations. Abstract: Let G be an almost simple, simply connected algebraic group G over the field of complex numbers. In this talk I answer two basic questions in the classification of G-torsors on curves. The first one is to construct a at degeneration of the moduli stack G-torsors on a smooth projective curve when the curve degenerates to an irreducible nodal curve. Torsors for a generalization of the classical Bruhat-Tits group schemes to…

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Gurbir Dhillon – Geometric Methods in Representation Theory Seminar

November 22, 2019 @ 4:15 pm - 5:15 pm

Title: The tamely ramified Fundamental Local Equivalence Abstract: Let G be an almost simple algebraic group with Langlands dual G’. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker D-modules on the affine flag variety of G’ at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell,…

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February 2020

Rekha Biswal (MPIM) – Geometric Methods in Representation Theory Seminar

February 14, 2020 @ 4:00 pm - 5:00 pm

Title:  Macdonald polynomials and level two Demazure modules for affine sl_{n+1}. Abstract: An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials.In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights of sl_{n+1} which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher…

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January 2021

Yau Wing Li, MIT – Geometric Methods in Representation Theory Seminar

January 29, 2021 @ 4:00 pm - 5:00 pm

Yau Wing Li, MIT Zoom Meeting ID: 975 6220 3148 Title: Endoscopy for affine Hecke categories Abstract: Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral…

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February 2021

Charlotte Chan, MIT – Geometric Methods in Representation Theory Seminar

February 19, 2021 @ 4:00 pm - 5:00 pm

Charlotte Chan (MIT) Meeting ID: 975 6220 3148 Title: Flag varieties and representations of p-adic groups Abstract: Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called “depth-zero” supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint…

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March 2021

Anne Dranowski, IAS – Geometric Methods in Representation Theory Seminar

March 5, 2021 @ 4:00 pm - 5:00 pm

Anne Dranowski, IAS Title: How to compute the fusion product of MV cycles in type A Abstract: In their recent paper on the MV basis and DH measures, Baumann, Kamnitzer and Knutson showed that Mirkovic and Vilonen’s geometric Satake basis of singular algebraic cycles yields a biperfect basis of the coordinate ring of a unipotent subgroup. Moreover, they showed that the structure constants of multiplication of basis vectors in this ring are given by intersection forms. We present joint work…

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Tsao-Hsien Chen, University of Minnesota – Geometric Methods in Representation Theory Seminar

March 19, 2021 @ 4:00 pm - 5:00 pm

Tsao-Hsien Chen, University of Minnesota Time: 4:00 pm, 03/19/21 Title: Towards derived Satake equivalence for symmetric varieties Abstract: In an ongoing project of  D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh, the authors propose a conjectural generalization of the derived Satake equivalence for complex reductive groups to spherical varieties. I will describe a program aimed at establishing their conjecture in the case of symmetric varieties (an important class of spherical varieties). A key ingredient is the relation between the Satake category for…

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