2022/2023
Representations of GL(n,F_q)
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Mikhail V. Finkelberg
Language:
English
ECTS credits:
6
Contact hours:
72
Course Syllabus
Abstract
Irreducible characters of GL(n, F_q) were computed by Green in 1955 in elementary combinatorial terms (Kostka polynomials, Hall-Littlewood symmetric functions). Quite mysteriously, there are very deep parallels between this theory and that of (infinite-dimensional) representations of GL(n) over real or p-adic numbers. We will also discuss a few related topics such as A. Weil representation, P. Hall algebra, I. Macdonald polynomials.
Learning Objectives
- Our goal is to study irreducible representations of GL(n, F_q), the related theory of representations of GL(n) over real and p-adic numbers, and related topics such as A. Weil representation, P. Hall algebra, and I. Macdonald polynomials.
Expected Learning Outcomes
- A student will learn a description of the irreducible characters of symmetric groups.
- A student will learn how to prove von Neumann's theorem on the uniqueness of an irreducible representation of a Heisenberg group with a given central character.
- A student will learn how to construct A. Weil representation of the metaplectic central extension of the symplectic group.
- A student will learn a description of the principal series and the cuspidal representations of GL(2, F_q).
- A student will learn how to construct an algebra with a basis given by the isomorphism classes of vector spaces with a nilpotent endomorphism.
- A student will learn a description of the structure constants of the Hall algebra when the ground field is finite.
- A student will learn an identification of the Hall algebra with the ring of symmetric functions and a computation of the image of the natural basis of the Hall algebra.
- A student will learn how to compute the character values of the principal series representations on the unipotent conjugacy classes.
- A student will learn how to organize the class functions on GL(n, F_q) for all n into an algebra.
- A student will learn a description of the irreducible characters of GL(n, F_q).
- A student will learn how to compute the convolution algebra of functions on GL(n, Q_p) biinvariant with respect to the Iwahori subgroup.
- A student will learn various definitions and properties of the Macdonald polynomials.
Course Contents
- Irreducible characters of symmetric groups
- Representations of Heisenberg groups
- A. Weil representation
- Representations of GL(2, F_q)
- P. Hall algebra
- Hall polynomials
- Hall-Littlewood symmetric functions
- Green functions
- Parabolic induction
- Irreducible characters of GL(n, F_q)
- Affine Hecke algebra of GL(n)
- I. Macdonald polynomials
