We organise an informal seminar on
HARMONIC ANALYSIS ON QUANTUM GROUPS AND HECKE ALGEBRAS
where we study and discuss various topics in this area with staff, postdocs and PhD-students.
The informal RU-UvA seminar is organised by Erik Koelink, Maarten van Pruijssen (RU) and Mikhail Isachenkov, Jasper Stokman (UvA).
Please send an email to Erik Koelink (e.koelink@math.ru.nl) if you want to be on the email list for further announcements.

Time schedule (2023-2024)


2024, spring session
Thursday, April 18, 2024, 13:00-17:00 at UvA (location: F3.20).
13:00-14:00 Kenny De Commer (VU Brussel).
14:30-15:00 Kenny De Commer (VU Brussel).
Title: Unitary Doi-Koppinen modules and quantized real semisimple Lie groups
Abstract: In the first part of this talk, I will explain the algebraic formalism of Doi-Koppinen data. A Doi-Koppinen datum consists of a bialgebra A together with a right A-comodule algebra B and a left A-module coalgebra C. A Doi-Koppinen module is then an A-module which is at the same time a C-comodule, such that the module and comodule structure are compatible in a natural way. I will revisit the theory of Doi-Koppinen modules in the setting of right coideal *-subalgebras of compact quantum group Hopf *-algebras. I then introduce the notion of a unitary Doi-Koppinen module, and explain how the collection of unitary Doi-Koppinen modules forms a tensor C*-category. In the second part of this talk, I will explain how this framework, when applied to Letzter's quantum symmetric pair coideal subalgebras, allows to quantize the representation category of a semisimple real Lie group. As an example, I will consider in detail the case of quantum SL(2,R). This is based on joint work with J.R. Dzokou Talla.

16:00-17:00 Stefan Kolb (Newcastle U).
Title: The center of quantum symmetric pair coideal subalgebras
Abstract: In finite type, the center of Drinfeld-Jimbo quantized enveloping algebras can be described in terms of their universal R-matrix. In this talk I will explain how the universal K-matrix for quantum symmetric pairs (QSP) can be employed in a similar fashion to describe a distinguished basis of the center of QSP coideal subalgebras. This simplifies joint work with G. Letzter from 2008. I will discuss the multiplicative behavior of the distinguished basis elements.

Thursday, March 14, 2024, 13:00-17:00 at RU (location: HG00.307).
13:00-14:00 Jort de Groot (UvA).
Title: Making SUq(1,1) into a von Neumann algebraic quantum group
Abstract: This is a follow-up on my previous talk ''an introduction to von Neumann algebraic quantum groups''. In the talk, I will explain the construction of quantum SU(1,1) in the von Neumann algebraic setting. By a theorem of Woronowicz and a result of Korogodsky, one should consider the normalizer of SU(1,1) in SL(2,C). I will explain the construction of the von Neumann algebra, the comultiplication, and the Haar weight, following the work of Koelink and Kustermans. If time permits, we will also have a look at the dual von Neumann algebraic quantum group.
14:30-15:30 Martijn Caspers (TUD).
Title: The structure of right-angled Hecke von Neumann algebras
Abstract: Right-angled Hecke von Neumann algebras are q-deformations of Coxeter groups, which are groups freely generated by elements whose square is the identity and where the generators are either free or they commute. In this talk we review the structural properties of operator algebras of right-angled Hecke algebras.
16:00-17:00 Stein Meereboer (RU).
Title: Introduction to braid group action for quantum symmetric pairs and applications to spherical functions
Abstract: In the framework of symmetric pairs (G,K) there is a natural action of the relative Weyl group on K. However finding an analog of this action on quantum symmetric pairs has presented an ongoing challenge. Initial progress had been made by Kolb and Pelegrinni, later by Dobson. Wang and Zang recently provided these actions, notably showing that the operators are closely related to the Lusztig braid group operators. In the talk I would like to introduce these operators and explain how one may apply these operators to the study of spherical functions.

Thursday, February 22, 2024, 13:00-17:00 at UvA (location: F3.20).
13:00-14:00 Jort de Groot (UvA).
Title: An introduction to von Neumann algebraic quantum groups.
Abstract: In this talk I will introduce von Neumann algebraic quantum groups as defined by Kustermans and Vaes, the early motivation for which was to generalize Pontryagin duality for locally compact abelian groups to a larger category. In the talk, I will explain the definition of locally compact quantum groups and show that this indeed leads to a generalized Pontryagin duality theorem, carrying around the example of a locally compact group.
14:30-15:30 Martijn Caspers (TUD).
Title: An overview of what is (not) known about Hecke von Neumann algebras.
Abstract: Hecke von Neumann algebras are von Neumann algebras generated by representations of Coxeter groups and their Hecke deformations. Thus they are topological closures of images of representations and they have first been considered in 2007 by Davis, Dymara, Januszkiewicz and Okun in the study of cohomology of Coxeter groups. After that surprising links with various objects in von Neumann algebra theory have been found. In this talk we present a short introduction to the subject and an overview of what is known about Hecke von Neumann algebras as well as some of the most important open questions.
16:00-17:00 Carel Wagenaar (TUD). (change in programme!)
Title: An algebraic approach to stochastic duality
Abstract: A useful tool in studying stochastic processes is stochastic duality: expectations of one process can be computed using expectations of another (often much simpler) process with the help of duality functions. In this talk I will introduce some interacting particle systems and show how Lie- and quantum algebras can be used to find dualities. In particular, we will look at the dynamic asymmetric exclusion process and show that Askey-Wilson polynomials appear as duality functions.

2023, Tuesday, October 24, 13:00-17:00 at Radboud Universiteit (location HG00.308)
October 24, 2023
13:00-14:00: Philip Schlösser (RU) ``Intermediate Macdonald Polynomials and their connections to Jasper and Maarten''
Abstract: As a continuation of AH talks 2 and 3 I will discuss a way to construct Macdonald polynomials that are invariant under a parabolic subgroup W_J of the finite Weyl group (in the extreme cases, we obtain the symmetric and nonsymmetric Macdonald polynomials). The space of polynomials thus symmetric turns out to be in bijection both with the spherical subspace of a DAHA-module from Jasper's 23 May talk, and with a power of the ring of Weyl-symmetric polynomials using Steinberg's theorem. We thus end up with two methods of making these polynomials vector-valued. The constructions discussed are essentially the q-version of Maarten's talk from 16 February.
14:30-15:30: Mikhail Isachenkov (UvA) ``A glimpse into mathematics of the conformal bootstrap''
Abstract: I will give a short review of the conformal bootstrap program, with the emphasis on the aspects of the theory of symmetric spaces and harmonic analysis that are of use there. In particular, we will discuss conformal field theories, conformal crossing equation, conformal blocks and a simple example of their relation with the (generalized) spherical functions.
16:00-17:00: Ilija Burić (Università di Pisa, Italy) ``Spherical weight-shifting algebra''
Abstract: Next to their importance in harmonic analysis, spherical functions appear in partial wave decompositions of QFT amplitudes and thus form the basis for the ‘bootstrap' approach to field theory. I will describe a systematic way to construct spinning spherical functions by applying certain ‘weight-shifting' operators to scalar ones. The talk is based on joint works with Francesco Russo, Volker Schomerus and Alessandro Vichi.

Time schedule (2022-2023)


Please consult the overview and more detailed information on the programme here. It will be updated whenever necessary.

May 23, 2023
AH talk: Jasper Stokman, ``Vector-valued Macdonald polynomials using DAHA''
Abstract: In this talk I will realize a particular class of induced DAHA modules on spaces of vector-valued Laurent polynomials and show how this gives rise to vector-valued Macdonald polynomials. I will relate the vector-valued Macdonald polynomials to the scalar-valued quasi-polynomial extensions of Macdonald polynomials, which were recently introduced by myself, Sahi and Venkateswaran.
Research talk: Valentin Buciumas (UvA)
Representation of p-adic groups and special functions
Abstract: I will introduce Hecke algebras on p-adic groups and certain representations of them related to Whittaker functions and explain how to understand these objects from a special functions point of view (this is equivalent to understanding Macdonald's formula for the spherical function and the Casselman-Shalika problem). I will also explain what is known about generalizing these results to other settings of interest (for example to metaplectic covers of p-adic groups or loop groups). Towards the end of the talk (and time permitting), I will try to explain the connections to other topics discussed in this seminar, like the polynomial representation of the DAHA, quantum symmetric pairs, and SSV polynomials.
Research talk: Stefan Kolb (Newcastle University, UK)
Representations of very non-standard quantum so(2N-1)
Abstract: Letzter's theory of quantum symmetric pairs provides new quantum deformations of the Lie algebra so(n-1) considered as a Lie subalgebra of so(n). These deformations are realized as coideal subalgebras B of the Drinfeld-Jimbo quantum enveloping algebra U=Uq(so(n)). For even n=2N the algebra B has an obvious Cartan subalgebra which makes it possible to mimic quantum group constructions. In this talk I will discuss this example as an illustration of Letzter's theory. I will outline a Poincare-Birkhoff-Witt Theorem and the classification of finite-dimensional irreducible representations of B in this case. The talk is based on joint work with Jake Stephens.
April 25, 2023
QG talk: Erik Koelink, ``Harmonic Analysis on Quantum Groups III. Introduction to Crystal Basis'' notes available
QG talk: Stein Meereboer, ``Harmonic Analysis on Quantum Groups IV. Introduction to iota-Crystal Basis'' notes available
AH talk: Edward Berengoltz, ``Harmonic Analysis on Hecke algebras III'' notes available and also Edward's personal summary of Macdonald's book is available.

March 30, 2023
QG talk: Jasper Stokman notes available
AH talk: Philip Schlösser
Research talk on February 16
Carel Wagenaar (TU Delft)
Title: A rank 2 Askey-Wilson algebra and bivariate Askey-Wilson polynomials
Abstract: The rank one Askey-Wilson algebra AW(3), introduced by Zhedanov in 1991, is a quantum algebra encoding properties of Askey-Wilson polynomials. In recent years, an extension to a higher rank version of this algebra has been studied. In this talk, I will present recent work with Wolter Groenevelt where we introduce a rank 2 Askey-Wilson algebra. This algebra encodes properties of bivariate Askey-Wilson polynomials. Important tools we use are the Hopf algebra Uq(sl(2)) and Koornwinders' twisted primitive elements.
February 16, 2023
QG talk: Valentin Buciumas
AH talk: Stein Meereboer notes available
Research talk on February 16
Maarten van Pruijssen (Radboud Universiteit)
Title: Heckman-Opdam polynomials -- a Steinberg variation.
Abstract: The symmetric and non-symmetric Jacobi polynomials associated to a root system with a multiplicity function have been studied by Heckman and Opdam. In this talk I will report on my investigations of the space of polynomials that are invariant for a parabolic subgroup of the Weyl group. We obtain families of orthogonal polynomials that are separated by a commutative algebra of differential-reflection operators. For the extreme cases we recover the results by Heckman and Opdam. A result of Steinberg tells us that the function space under consideration is actually freely and finitely generated as a module over the (polynomial) algebra of Weyl group invariant polynomials. In this way we can interpret our orthogonal polynomials as orthogonal vector-valued polynomials. They are now separated by a commutative algebra of differential operators. In Dynkin types BC1 and A2 we can recognize the new families of polynomials obtained in this fashion as spherical functions on a compact symmetric space (U,K), now related to a "higher K-type".