# Lectures

Lectures and Abstracts

**Auguste Hébert:**

Title: Principal series representations of Iwahori-Hecke algebras for Kac-Moody groups over local fields.

Abstract: Let $H$ be the Iwahori-Hecke algebra of a reductive group $G$ over a local field. Mastumoto introduced principal series representations of $H$ at the end of the 70's. They are of particular importance in the representation theory of $H$ and every irreducible representation of $H$ appears as a qoutient of a principal series representation. Matsumoto and then Kato gave criteria of irreducibility for these representations. When $G$ is a Kac-Moody group over a local field, Braverman, Kazhdan and Patnaik and Bardy-Panse Gaussent and Rousseau recently associated an Iwahori-Hecke algebra $H$ to $G$. I will talk about the prinicipal series representations of $H$ and about the generalizations of Matsumoto's and Kato's irreducibility criteria.

**Katerina Hristova:**

Title: Representations of locally compact totally disconnected groups and complete Kac-Moody groups.

Abstract: We study smooth representations of locally compact totally disconnected groups acting continuously on simplicial sets. We discuss some interesting properties of the category of smooth representations of such groups, in particular, its projective dimension and localisation theory. We use our results to obtain information about the representation theory of complete Kac-Moody groups over a finite field. Joint work with Dmitriy Rumynin.

**Victor Kac:**

Title: Cohomology of algebraic structures: from Lie algebras to vertex algebras.

**Axel Kleinschmidt:**

Title: The compact subalgebra of Kac-Moody algebras and its representations.

Abstract: Any (split real) Kac-Moody Algebra has an Chevalley involution invariant subalgebra with definite invariant bilinear form. This infinite-dimensional Algebra has finite-dimensional representations that were discovered using supergravity and these unfaithful representations have interesting properties and quotients that I will review. Based on work with T. Damour, H. Nicolai and A. Vigano.

**Timothée Marquis:**

Title: On the structure of Kac-Moody algebras.

Abstract: In this talk, I will address the following question: given two homogeneous nonzero elements x,y of a symmetrisable Kac-Moody algebra, when is their bracket [x,y] a nonzero element? I will then state some conjectures on the structure of the imaginary subalgebra of a Kac-Moody algebra.

**Dinakar Muthiah:**

Title: Toward double affine flag varieties and Grassmannians.

Abstract: Recently there has been a growing interest in double affine Grassmannians, especially because of their relationship with Coulomb branches of quiver gauge theories. However, not much has been said about double affine flag varieties. I will discuss some results toward unterstanding double affine flag varieties and Grassmannians (and their Schubert subvarieties) from the point of view of $p$-adic Kac-Moody Groups. I will discuss Hecke algebras, Bruhat order, and Kazhdan-Lusztig polynomials in this setting. Ideas originating in the Gaussent-Rousseau theory of masures will play a key role. This includes work joint with Daniel Orr and joint with Manish Patnaik.

**Manish Patnaik:**

Title 1: Metaplectic Kac-Moody groups (construction and basic properties)

Title 2: Representations of p-adic Kac-Moody groups (Satake transforms, Whittaker functions, Jacquet functors, etc.

Title 3: Explicit formulas: Casselman-Shalika formulas, spherical functions, etc.

**Guido Pezzini:**

Title: Geometry and regular functions on symmetric spaces for Kac-Moody groups.

Abstract: I will report on a joint work in progress with Bart Van Steirteghem, on an algebro-geometric theory of symmetric spaces for (minimal) Kac-Moody groups over the complex numbers. Our goal is to study the structure of such spaces as infinite dimensional algebraic varieties (ind-varieties). Inspired by work of Kac and Peterson, we also study the properties of a suitable ring of functions that we consider a good generalization of the ring of regular functions of the finite-dimensional setting. New phenomena, not occurring in the classical setting, will be illustrated, and I will also discuss the possibility of defining equivariant completions of such symmetric spaces.

**Anna Puskás: **

Title: A correction factor for Kac-Moody groups and t-deformed root multiplicities

Abstract: We will discuss a correction factor which arises in the theory of p-adic Kac-Moody groups, for example in formulas for Whittaker functions in the infinite dimensional setting. In affine type, the factor is known by Cherednik's work on Macdonald's constant term conjecture. More generally, it can be represented as a collection of polynomials of t indexed by positive imaginary roots; these are deformations of root multiplicities. The Peterson algorithm and the Berman-Moody formula can be generalized to compute the correction factor for any t. They both reveal some properties of the correction factor and raise further questions and conjectures about its structure. This is joint work with Dinakar Muthiah and Ian Whitehead.

**Bertrand Rémy:**

Title: Topological generation of non-archimedean split simple groups

Abstract: this is joint work with Inna Capdeboscq. We will deal with the problem of counting the minimal number of topological generators for simple algebraic groups over local fields. We have almost complete answers in the split case.

**Guy Rousseau:**

Title: Masures

Abstract: The Bruhat-Tits buildings are very useful to study reductive groups over a non-Archimedean local field K. The masures play a similar part for Kac-Moody groups over such a field K. They enjoy some properties of these buildings, but not some important ones (replaced by new weaker properties). I shall explain the structure of the masures and their construction for an almost split Kac-Moody group over K. I shall also explain quickly their applications (those known up to now).