Let $G$ be a connected semisimple Lie group with Lie algebra $\mathfrak{g}$. There are two natural duality constructions that assign to it the Langlands dual group $G^\lor$ (associated to the dual root system) and the Poisson-Lie dual group $G^∗$. Cartan subalgebras of $\mathfrak{g}^\lor$ and $\mathfrak{g}^∗$ are isomorphic to each other, but $G^\lor$ is semisimple while $G^∗$ is solvable.
In this talk, we explain the following non-trivial relation between these two dualities: the integral cone defined by the Berenstein-Kazhdan potential on the Borel subgroup $B^\lor \subset G^\lor$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on $K^∗ \subset G^∗$ (the Poisson-Lie dual of the compact form $K \subset G$). The first cone parametrizes canonical bases of irreducible $G$-modules. The corresponding points in the second cone belong to integral symplectic leaves of $K^∗$.
The talk is based on a joint work with A. Berenstein, B. Hoffman and Y. Li.[-]

The theory of cluster algebras has proved useful in proving theorems about the characters of graded tensor products or Demazure modules, via the $Q$-system. Upon quantization, the algebra associated with this system is shown to be related to a quantum affine algebra. Graded characters are related to a polynomial representation of the quantum cluster variables. This immediately suggests a further deformation to the spherical DAHA, quantum toroidal algebras and elliptic Hall algebras.[-]

We study invariant differential operators on representations of supergroups associated with simple Jordan superalgebras, in the classical case this problem goes back to Kostant. Eigenvalues of Capelli differential operators give interesting families of polynomials such as super Jack polynomials of Sergeev and Veselov and factorial Schur polynomials of Okounkov and Ivanov. We also discuss connection with deformed Calogero-Moser systems in the super case.[-]

The algebra $U(gl_n)$ contains a famous and beautiful commutative subalgebra, called the Gelfand-Tsetlin subalgebra. One problem which has attracted great attention over the recent decades is to classify the simple modules on which this subalgebra acts locally finitely (the Gelfand-Tsetlin modules). In investigating this question, Futorny and Ovsienko expanded attention to a generalization of these algebras, saddled with the unfortunate name of “principal Galois orders”. I'll explain how all interesting known examples of these (and some unknown ones, such as the rational Cherednik algebras of $G(l,p,n)!)$ are the Coulomb branches of N = 4 3D gauge theories, and how this perspective allows us to classify the simple Gelfand-Tsetlin modules for $U(gl_n)$ and Cherednik algebras and explain the Koszul duality between Higgs and Coulomb categories O.[-]

In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras an their truncations, in the spirit of the shifted Yangians of Brundan-Kleshev, Braverman-Finkelberg Nakajima, Kamnitzer-Webster-Weekes-Yacobi... We discuss the following points in representation theory of (truncated) shifted quantum affine algebras that we relate to representations of Borel quantum affine algebras by induction and restriction functors. We establish that the Grothendieck ring of the category of finite-dimensional representations has a natural cluster algebra structure. We propose a conjectural parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual quantum affine Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.[-]