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08/06/2026, 11:00
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.
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08/06/2026, 11:50
Spectral triples provide a noncommutative analogue of spin manifolds and play a central role in noncommutative geometry. While the compact Riemannian case is well understood, far less is known in the noncompact pseudo-Riemannian setting. In this talk, I will present the construction of an indefinite spectral triple for the Lie group SU(1,1), highlighting the role of representation theory in...
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08/06/2026, 14:00
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...
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08/06/2026, 14:50
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...
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08/06/2026, 16:10
In this talk I will revisit the spin-Ruijsenaars–Macdonald system, given by a matrix-valued generalisation of Macdonald operators that arises in the context of induced modules of double affine Hecke algebra and fits in a quantum-affine version of Schur–Weyl duality. After reviewing the construction of this system, I will outline recent developments, including elliptic...
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09/06/2026, 09:00
Short star products are filtered deformations of graded algebras satisfying a truncation condition first considered by Beem-Peelaers-Rastelli and further developed by Etingof and Stryker. In this talk I will explain that quantum symmetric pair coideal subalgebras are realized as short star products on quantum horospherical subalgebras. The shortness property allows for immediate conceptual...
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09/06/2026, 09:50
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...
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09/06/2026, 10:15
Kostant's branching law for a symmetric pair $(\mathfrak{g},\mathfrak{k})$ determines the multiplicity of a irreducible $\mathfrak{k}$-module inside irreducible $\mathfrak{g}$-modules. In this talk I will explain how to derive this branching law using the language of Watanabe's integrable modules for quantum symmetric pairs. This is joint work in progress with Stefan Kolb.
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09/06/2026, 11:20
It is known that the quantum coordinate algebra of a Kac--Moody algebra and its crystal basis admit Peter--Weyl type decompositions.
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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... -
09/06/2026, 14:00
The celebrated work of De Concini–Kac–Procesi established that the representation theory of quantum groups at roots of unity is closely linked to the conjugacy classes of the underlying group. In this talk, we extend this approach to quantum symmetric pairs.
For an i-quantum group $U^\imath$ associated with an involution θ, we consider a De Concini–Kac type integral form and study its...
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09/06/2026, 14:50
We study finite-dimensional representations of the quantum group analogue of $\mathfrak{so}_{2N}$ inside $\mathfrak{so}_{2N+1}$ appearing in the theory of quantum symmetric pairs. Using a Verma module approach, we classify finite-dimensional simple modules in terms of highest weights. These highest weights are joint eigenvalues of the Letzter-Cartan subalgebra. Using a modified...
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09/06/2026, 15:15
Given a quantum symmetric pair $(U,B)$, it is a well-known fact that (with some caveats) the (elementary) zonal spherical functions (ZSF) restrict to symmetric Macdonald polynomials.
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We use the Haar functional on $U$'s dual to construct an inner product for the matrix spherical functions (MSF). Since the MSF form a free module over the ZSF, we can interpret this inner product to be related to... -
09/06/2026, 16:10
Spin graph functions of Reshetikhin-Stokman are generalizations of vector-valued spherical functions that arise in many contexts in harmonic analysis and physics. In this talk I will consider a special class of spin graph functions for the (identity connected component of the) Lorentz group SO$(n,1)_e$, which is relevant for the analysis of Euclidean four-point local correlation functions in...
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10/06/2026, 09:00
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...
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10/06/2026, 09:50
Let $G$ be a semisimple Lie group and $K$ the maximal compact subgroup. Principal series representations of $G$ induced from a parabolic subgroup $P = MAN$ whose unipotent radical $N$ is abelian have been well studied. The analysis of these representations exploits the fact that $(K,L)$ has the structure a symmetric pair, where $L = K \cap M$. The next easiest case is when $N$ has the...
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10/06/2026, 10:15
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...
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10/06/2026, 11:20
Aiming for a revival of the dormant theory of crystallographic complex reflection groups, we give Coxeter-like reflection presentations for the top family of such groups. These new presentations behave à la Coxeter—encoding many of the group's properties at a glance—and further achieve the braid theorem, allowing to deform into the generic Hecke algebra. As part of a final showcase...
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10/06/2026, 14:00
The Riemann--Liouville fractional integral interpolates iterated integration to non-integer orders and turns a hypergeometric solution of rank $p$ into one of rank $p+1$. Katz recast this into the so-called \emph{middle convolution}, an operation on local systems on the punctured Riemann sphere that preserves the rigidity index. Dettweiler and Reiter subsequently made the operation explicit at...
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10/06/2026, 14:50
Consider the rank 1 DAHA of type $\check C_1 C_1$ with generators $T_1$, $T_0$, $Z$, $Z^{-1}$ and depending on the Askey--Wilson parameters $a,b,c,d,q$. Its basic representation is a faithful representation on the space of Laurent polynomials in $z$. We extend the generators $Z,Z^{-1}$ to all rational functions in $Z$. Then the basic representation extends to a faithful representation on the...
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10/06/2026, 16:10
Dirac-type operators and Howe dualities provide a uniform approach to decomposition problems and symmetry algebras in the orthogonal and symplectic settings. Quantizing these structures is subtle and in the orthogonal Dirac setting competing frameworks do not currently agree.
In this talk I focus on the symplectic (metaplectic) Howe duality in the first nontrivial case, rank one, and...
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11/06/2026, 09:00
In this talk, we discuss a conceptual perspective on when and how branching multiplicities change in restrictions of representations. We develop a framework for the stability of branching multiplicities arising in the restriction of finite- and infinite-dimensional representations of real reductive Lie groups.
Focusing on pairs whose complexifications are of type (gl(n+1), gl(n)) and...
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11/06/2026, 09:50
Controlling multiplicities in branching problems is a central question
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in representation theory. In the setting of infinite-dimensional representations of real reductive Lie groups, analytic methods based on symmetry breaking operators provide one explanation of a surprising blow-up phenomena mentioned in the talk by T. Kobayashi.
We shall present this approach considering the basic case of... -
11/06/2026, 11:20
Harish-Chandra's c-function on a real semisimple Lie group gives the leading coefficient of the zonal spherical function and determines the Plancherel measure for the spherical transform. Gindikin and Karpelevič gave an explicit formula for the c-function. Moreover, Heckman and Opdam developed a theory of hypergeometric functions associated with root systems, which are generalizations of...
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11/06/2026, 14:00
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.
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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... -
11/06/2026, 14:50
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...
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11/06/2026, 15:15
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...
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11/06/2026, 16:10
Bivariate Jacobi polynomials is a family of orthogonal polynomials on the triangle that are solutions to a second order differential equation. This talk will present an algebraic interpretation for this family of polynomials based on the representation theory of the so-called rank 2 Jacobi algebra.
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Bob Yuncken12/06/2026, 09:00
The structure theory of the finite dimensional representations of a quantized enveloping algebra undergoes a massive simplification when the deformation parameter q goes to 0. This is the fundamental observation in the theory of crystal bases of Kashiwara and Lusztig. On might ask if there is a similar simplification of the Pontrjagin dual, meaning a crystallisation of the quantized algebra...
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12/06/2026, 09:50
(Joint work with Valerio Toledano Laredo)
The well known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these...
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12/06/2026, 11:20
Braided monoidal categories are governed by braid groups of type A and the associated Yang-Baxter equation. Given a semisimple compact Lie group G, the Kazhdan-Lusztig theorem gives a non-trivial equivalence between two particular braided monoidal unitary categories: one constructed from a non-trivial associator on the category of unitary G-representations, using solutions to the so-called...
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