Stokman: Quasi-polynomial analogs of Askey-Wilson polynomials I will first introduce quasi-polynomial analogs of the nonsymmetric and symmetric Askey-Wilson polynomials using an explicit representation of a rank one double affine Hecke algebra. In the second part of the talk, I will explain how they are related to Mizhan-Rahman's associated Askey-Wilson polynomials. The second part of the talk is joint work with Mikhail Isachenkov.

Vlaar: On quantum affine symmetric pairs (part 1) The Yang–Baxter equation and the reflection equation, or boundary Yang–Baxter equation, are fundamental identities in quantum integrable systems, governing factorizable particle interactions on a line and on a half-line, respectively. While the Yang–Baxter equation is deeply connected to quantum groups, solutions of the reflection equation, called K-matrices, arise naturally from quantum symmetric pairs. In this two-part talk, we will survey recent joint work and open problems on quantum symmetric pairs, with special emphasis on affine type.

Appel: On quantum affine symmetric pairs (part 2) The Yang–Baxter equation and the reflection equation, or boundary Yang–Baxter equation, are fundamental identities in quantum integrable systems, governing factorizable particle interactions on a line and on a half-line, respectively. While the Yang–Baxter equation is deeply connected to quantum groups, solutions of the reflection equation, called K-matrices, arise naturally from quantum symmetric pairs. In this two-part talk, we will survey recent joint work and open problems on quantum symmetric pairs, with special emphasis on affine type.

Wang: Weight modules for gl_2 \times \gl_2 We develop a theory of weights for the quantum symmetric pair $(\mathfrak{gl}_4,\mathfrak{gl}_2\times\mathfrak{gl}_2)$ of type AIII. We define ``magical'' operators that are compatible with weight spaces (wrt. Letzter's Cartan subalgebra) and use them to study Verma modules and irreducible quotients. We then prove the existence of weight bases in tensor products by explicitly constructing some highest weight vectors. These constructions allow us to mimic the important aspects of the classical finite dimensional representation theory. Applications include a definition of rational representations, the BGG resolution, a Clebsch--Gordan formula, the Harish-Chandra isomorphism and central characters, as well as a classification and description of all irreducible polynomial representations.

Meereboer: Kostant's branching law for quantum symmetric pairs It is known that the category of finite dimensional modules for quantized enveloping algebras is semisimple and that its simple objects are uniquely classified by their dominant integral weights. For quantum symmetric pairs such a classification remains open. In this talk I will present Watanabe's category of integrable modules and explain that its Grotendieck group is isomorphic to its classical counterpart using a generalization of a branching law of Kostant. This is part of joint work with Stefan Kolb.

Watanabe: Quantizations of coordinate algebras of symmetric pair subalgebras It is known that the quantum coordinate algebra of a Kac--Moody algebra and its crystal basis admit Peter--Weyl type decompositions. Also, Kashiwara proved that, for finite type, the crystal basis is isomorphic to the crystal basis of the modified quantum group as bicrystals. The main topic of this talk is quantum symmetric pair analogues of these results. In particular, I will show you some examples of "bi-icrystals" of type A. This talk is partly based on a joint work with Mao Hoshino.

Langen: Uniform bounds on the Dunkl kernel In Dunkl theory the Dunkl kernel replaces the classical exponential function in its predominant role in harmonic analysis. For regular spectral parameters, we present upper bounds for the Dunkl kernel and its derivatives which are uniform in the spatial variable. These estimates generalize sharp uniform upper bounds for spherical functions of Cartan motion groups and classical Bessel functions. The proof is based on an asymptotic study of the differential system satisfied by the Dunkl kernel with respect to parabolic subgroups of the Weyl group. As a consequence we obtain results regarding the Lebesgue density of the representing measure of Dunkl’s intertwining operator.

Terwilliger: The q-Onsager algebra and its finite-dimensional irreducible modules This talk is about the $q$-Onsager algebra $O_q$. The algebra $O_q$ is defined by two generators and two relations called the $q$-Dolan/Grady relations. We will describe the finite-dimensional irreducible $O_q$-modules $V$ that satisfy a mild assumption. We will show that the $O_q$-generators act on $V$ as a tridiagonal pair. We will describe the tridiagonal pairs and the related tridiagonal systems, using the concept of a tetrahedron diagram. We will classify up to isomorphism the tridiagonal systems, and explain which ones come from an $O_q$-module.

Rösler: Limits of Bessel functions for root systems as the rank tends to infinity In this talk, we consider the asymptotic behaviour of Dunkl-type Bessel functions associated with root systems of type A and type B with positive multiplicities as the rank tends to infinity. To obtain limits, one has to take sequences of spectral parameters which are of Vershik-Kerov type, i.e. tend to infinity in a suitable way. The situation is similar to the case of Heckman-Opdam polynomials whose limits were studied by Okounkov and Olshanski more than 20 years ago. Nowadays, there is renewed interest in such topics within the area of integrable probability. We characterize both the possible limit functions as well as the spectral sequences for which limits of Bessel functions can be obtained, both in the cases of type A and type B. For multiplicities related to group cases, these results have an interpretation in the context of asymptotic harmonic analysis in the sense of Olshanski. The talk is based on joint work with Dominik Brennecken

van Horssen: Shift operators for non-symmetric Heckman-Opdam polynomials and non-symmetric Macdonald-Koornwinder polynomials We present an algebraic construction of shift operators for the non-symmetric Heckman-Opdam polynomials and the non-symmetric Macdonald-Koornwinder polynomials. To each linear character of the finite Weyl group, we associate forward and backward shift operators, which are differential-reflection and difference-reflection operators that satisfy certain transmutation relations with the Dunkl-Cherednik operators. In the Heckman-Opdam case, the construction recovers the non-symmetric shift operators of Opdam and Toledano Laredo for the sign character. This talk is based on joint work with Maarten van Pruijssen (arXiv:2602.06784).

Gaudillot-Estrada: A Mackey Analogy for real semisimple quantum groups For a given real semisimple group $G$, the Mackey analogy consists of a collection of explicit relationships between the groups algebra of $G$ and that of its Cartan motion group $G_0$. The weakest of these relationships is the Connes-Kasparov isomorphism $K_*(C^*(G_0)) \cong K_*(C^*_r(G))$. In this talk, on the basis of small dimensional examples, I will explain why this analogy may also hold for real semisimple quantum groups, which have been introduced by De Commer.