## University of Sydney Algebra Seminar

# Arun Ram (University of Melbourne)

Full Schedule of Lectures:

1) Tuesday June 14, 12-1pm, Carslaw 375

2) Thursday June 16, 12-1pm, Carslaw 375

3) Friday June 17, 12-1pm, Carslaw 375

## Tuesday 14 June, 12-1pm, Place: Carslaw 375

## Combinatorics of the Loop Grassmannian

I will explain what the loop Grassmannian and the affine flag variety are and how to
label their points. This labelling is a refinement of the labelling of crystal bases by
Littelmann paths. I’ll show the picture which summarises the connection to the affine
Hecke algebra and the spherical affine Hecke algebra. I’ll give a summary of the
relationship between Mirkovic-Vilonen cycles and the crystal bases and explain how this
is reflected in the path model indexing.

## Thursday 16 June, 12-1pm, Place: Carslaw 375

## Combinatorics of representations of affine Lie algebras

This will be a survey of my current understanding of the combinatorial representation
theory of affine Lie algebras. For category O at negative level, Verma modules have
finite composition series with decomposition numbers determined by Kazhdan-Lusztig
polynomials. The structure of affine Weyl group orbits controls the pretty patterns.
For category O at positive level, Verma modules have infinite compositions with
decomposition numbers given by inverse Kazhdan-Lusztig polynomials, and at critical
level, the patterns correspond to the periodic Kazhdan-Lusztig polynomials. I’ll also
discuss parabolic category O. Finite dimensional modules (which are level 0) are
indexed by Drinfeld polynomials and then there are various collections of smooth
representations where our combinatorial understanding has increased greatly in recent
years.

## Friday 17 June, 12-1pm, Place: Carslaw 375

## Combinatorics of affine Springer fibres

This talk will be a survey of the relation between affine Springer fibres and representations of the double affine Hecke algebra. I will likely focus on a favourite example of the elliptic homogeneous case where I can draw a nice picture illustrating how the affine Springer fibre is decomposed into cells indexed by connected components of complement of a hyperplane arrangement called the Shi arrangement (the same one that appears in the K-theory version of the Chevalley-Shephard-Todd theorem for reflection groups). These regions then correspond to a Macdonald polynomial basis of the corresponding representation of the double affine Hecke algebra.