# Representation theory seminar 2019-Semester 1

## The University of Melbourne

School of Mathematics and Statistics

Representation theory seminar 2019

**Dates** Feb. 1-May 1, 2019.

**Organisers**: Ting Xue, Yaping Yang

**Time:** Tuesdays 2-3pm or 2-4pm.

**Place:** Peter Hall Building, __Room 107 __weekly

## Schedule

### May. 21, 2-3:30pm, Arik Wilbert (Melbourne)

**Title:** From Kazhdan-Lusztig polynomials to exotic arc algebras

**Abstract: **Kazhdan-Lusztig polynomials can be used to describe multiplicities in many interesting representation theoretic categories (e.g. BGG category O, certain categories of representations of Brauer algebras and Lie superalgebras). Computing these polynomials explicitly is difficult in general. In this talk we will focus on certain parabolic Kazhdan-Lusztig polynomials which can be determined easily using a diagram calculus. The diagrammatics can be used to define different versions of so-called arc algebras whose categories of modules are equivalent to the representation theoretic category of interest we started with.

The main part of this talk will be to explain the construction of a new such family of arc algebras related to the geometry of exotic Springer fibers. These algebraic varieties first appeared in Kato’s work on the Deligne-Langlands correspondence. If time permits, I will explain conjectural connections to categories of perverse sheaves, braid group actions on derived categories and link invariants.

### May. 14, 2-3:30pm, Iva Halacheva (Melbourne)

**Title:** Branching in Schubert calculus and Maulik-Okounkov classes

**Abstract:** I will discuss work in progress with Allen Knutson and Paul Zinn-Justin on computing the restriction of GL(2n) Schubert classes to Sp(2n), both combinatorially via self-dual puzzles and conjecturally also more geometrically on the level of cotangent bundles. I will describe some ideas towards using Lagrangian correspondences to follow the Maulik-Okounkov stable classes in the latter setting.

### May. 7, 2-3:30pm, Peter McNamara (Melbourne)

**Title: ** An introduction to some p-adic representation theory

**Abstract:** This will be an expository lecture giving an introduction to representations of p-adic groups.

### Apr. 30, 2-3:30pm, Gufang Zhao (Melbourne).

**Title:** From generalized geometric Satake to AGT correspondence

**Abstract:** This is a reading report/survey on geometric Satake correspondence of affine Lie algebras and the 4d/2d correspondence of Alday-Gaiotto-Tachikawa. I will follow the sequence of papers of Braverman, Finkelberg, Nakajima on the former, and the work of Schiffmann and Vasserot on the latter. In both correspondences various algebraic structures on (different) doubles of the cohomological Hall algebra of Kontsevich and Soibelman associated to a Calabi-Yau 3-fold arise.

### Apr. 23, Easter holiday, no seminar.

### Apr. 16, 2-3:30pm, Thomas Gobet (the University of Sydney)

**Title:** Positivity properties in Hecke algebras of arbitrary Coxeter groups

**Abstract:** We explain why some generalizations of Kazhdan-Lusztig polynomials, obtained by considering a family of bases of Hecke algebras which generalize both the standard and costandard bases, have nonnegative coefficients. This was conjectured by Dyer, and proven by Dyer and Lehrer for finite Weyl groups. We also explain the nonnegativity of the corresponding "inverse" polynomials, which is obtained by considering a categorical action of the Artin group attached to the Coxeter system, constructed by Rouquier. Both results use the recent proof of Soergel's conjecture by Elias and Williamson.

### Apr. 9, 2-3:30pm, Chenyan Wu (Melbourne)

**Title**: Local Descent Construction and Theta Correspondence

**Abstract**: We will describe the local descent construction which under some conditions give the inverse operation to the Langlands functorial lift. We will derive some properties of this construction and show how it behaves under theta correspondence.

### Apr. 2, 2-4pm, Ian Grojnowski (University of Cambridge)

**Title: **Darboux theorem for derived symplectic varieties.

**Abstract: **I'll start by reminding people what a symplectic vector space is, and work up to explaining local forms of derived symplectic varieties.

No background needed -- this is supposed to be a relaxed Tuesday afternoon talk.

(If we have enough energy, I might also describe some aspects of geometric Langlands on an elliptic curve. The derived geometry is a small ingredient in that, but not really one of the main points).

### March 26, 2-4pm, Yaping Yang (Melbourne)

**Title: **Equivariant (K-) Homology of affine Grassmannian and Toda lattice, after Bezrukavnikov-Finkelberg-Mirković.

**Abstract: **This is a reading report. I will follow the paper of Bezrukavnikov-Finkelberg-Mirković: https://arxiv.org/abs/math/0306413

### March 19, 2-3pm, Robert Coquereaux (CNRS).

**Title: **From orbital measures to Littlewood-Richardson coefficients and hive polytopes

**Abstract: **Several combinatorial models (pictographs) can be used to determine the multiplicities (Littlewood - Richardson coefficients) of those irreducible representations (irreps) that occur in the reduction of a product of two irreps of the Lie group SU(n). Such pictographs can be considered as the integer points of an associated polytope whose volume can be expressed in terms of the Fourier transform of a convolution product of orbital measures.

After describing several variants of this construction, we discuss a few properties of the volume function, whose definition makes sense in a wider context.

### March 12, 2-3pm, Daniel Tubbenhauer (Universität Zürich).

**Title:** Handlebodies, Artin-Tits and Homflypt

**Abstract:** What do handlebodies and Artin–Tits groups have in common? This talk is my attempt to answer this question: I summarize what I know about the connection between handlebodies, Artin–Tits (braid) groups and Soergel bimodules. Joint work in progress with David Rose.

### March 5, 2-3pm, Ivan Marin (Université d'Amiens).

**Title:** New algebras associated to reflection groups

**Abstract:** A number of algebraic constructions have been extended from the symmetric groups to other reflection groups. The Coxeter theory has been the source of most of them, notably for the definition of the Hecke algebras, in the case of real reflection groups. Generalizations of these Hecke algebras to non-real ones have been proposed 20 years ago, using topological means (braid groups and monodromy representations). The properties of these generalizations were partly conjectural, and the main conjecture in this area has been resolved only recently. After having reviewed the corresponding material, we shall describe new extensions of these Hecke algebras which make sense in this general setting, as well as generalizations of the algebra of Brauer diagrams. Altogether, this provides large algebras associated to reflection groups whose structure and combinatorics still has to be determined.

### Feb 26, 2-4pm, Emily Norton (University of Bonn).

**Title: **Rational Cherednik algebras and their representations

**Abstract: **Given a complex reflection group W, we may define a family of noncommutative algebras H_c(W) called rational Cherednik algebras depending on a deformation parameter c. I will discuss the basics of these algebras and their representation theory, with a focus on the finite-dimensional representations.

### Feb 19, 2-3pm, Arun Ram (Melbourne).

**Title:** Combinatorics at level 0.

### Feb 12, 2-4pm, Ting Xue (Melbourne).

**Title:** Global Springer theory.

### Feb 5, 2-4pm, Peter McNamara (Melbourne).

**Title:** Koszul duality.