# Representation theory seminar 2021 semester 1

## Representation theory seminar 2021, Semester 1

**Organisers**: Kari Vilonen, Ting Xue, Yaping Yang

**Topics: **Affine Lie algebras, Kazhdan-Lusztig conjectures, localisation

**Time:** Tuesdays 3:15pm-5pm.

15 June 2:15pm **Arun Ram** (Melbourne)** Representations of affine Hecke algebras IV: Where do Hecke algebras come from? **Notes

**Abstract: **This will be a talk about the origins of the game:

(a) double centralisers and correspondence between some representations of A and representations of the centralizer

(b) Convolution, Inducing from the trivial representation, double coset algebras, and correspondence between some representations of the group and the representations of the double coset algebra

(c) Generators and relations for p-adic groups, cosets and double cosets

(d) Computation of the Iwahori-Hecke algebras and correspondence between some representations of the p-adic groups and representations of the Hecke algebra.

8 June 2:15pm **Arun Ram** (Melbourne)** Representations of affine Hecke algebras III: Standard modules **Notes

**Abstract:** Most standard modules are built by induction. I’ll describe principal series modules and the basic theorems about their simplicity and their composition factors. Then I’ll define tempered and square integrable modules, and explain how the standard modules correspond to generalized Springer fibers and the square integrable modules correspond to cuspidal nilpotent elements. This provides an indexing of irreducible representations of affine Hecke algebras by Deligne-Langlands parameters.

1 June **2:15pm** (note time change!) **Arun Ram** (Melbourne) **Representations of affine Hecke algebras II Central characters, weight spaces and intertwiners **Notes

**Abstract:** I will try to explain how the representation theory of the affine Hecke algebra is coded by local regions in a hyperplane arrangement, sometimes called the Shi arrangement. I’ll introduce the favourite induced modules (standard modules) and explain how to study their structure by weight spaces and intertwiners.

25 May **Arun Ram** (Melbourne) **Representations of affine Hecke algebras I **Notes

**Abstract: **In this first talk, I’ll define the isogeneous affine Hecke algebras and explain how to compare their representations. Then I’ll explain how to get the representations of finite Hecke algebras of the finite complex reflection groups G(r,p,n) from the affine Hecke algebra of type GL_n.

18 May **Anna Romanov** (Sydney) **The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules, continued **Notes

11 May **Arun Ram** (Melbourne) **Decomposition numbers for standard objects in categories O **Notes

**Abstract: **The Verma modules are indexed by their highest weight. They have a simple quotient and so the irreducibles are indexed by their

highest weight. The composition factors of a Verma module must all lie in the same orbit of the Weyl group. In the affine case the orbits take three different shapes depending on whether it is positive level, negative level, or critical level. In each case there is a different family of Kazhdan-Lusztig type polynomials that describes the multiplicity of the irreducible in the layers of the Jantzen filtration of the Verma module. I’ll try to explain what these affine Weyl group orbits and Kazhdan-Lusztig polynomials are.

4 May **Anna Romanov** (Sydney) **The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules, continued **Notes

27 April **Anna Romanov** (Sydney) **The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules** Notes

**Abstract: **In these talks, I will describe some geometric objects that play a role in the representation theory of Kac-Moody Lie algebras. In the Kac-Moody setting, we have no Beilinson—Bernstein localisation theorem. However, as Yaping told us last week, D-modules on certain infinite-dimensional varieties can still be used to prove results about representations of affine Lie algebras. This series of talks has two goals: (1) to clarify the state of affairs on Beilinson—Bernstein-type equivalences of categories for affine Lie algebras, and (2) sketch the proof (due to Kashiwara-Tanisaki) that negative level blocks of category O for an affine Lie algebra are equivalent to certain categories of equivariant D-modules on the affine flag variety.

In the first week, I will define Kac-Moody groups and their flag varieties carefully, describe their ind-variety structure, and do an SL2 example.

In the second week, I will introduce D-modules, state the known results about equivalences of categories between D-modules and representations of affine Lie algebras, and sketch the Kashiwara-Tanisaki negative-level proof.

20 April **Yaping Yang** (Melbourne) **Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras: continued **Notes

13 April **Yaping Yang** (Melbourne) **Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras**

**Abstract: **In my talk, I will review the Kac Moody Lie algebras, the affine root systems, and the affine Weyl groups. I will state the Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras. In particular, at the non-critical level, the transition matrix of the characters of the irreducible highest weight modules and the characters of the Verma modules is given by the affine Kazhdan-Lusztig polynomial.

30 Mar **Kari Vilonen **(Melbourne) **Summary and conclusion of the Jantzen conjecture**

23 Mar **Kari Vilonen **(Melbourne) **the Jantzen conjecture**