GAAG seminar - Baxter polynomials and representations of shifted quantum groups

We explain the application of polynomiality of Q-operators to representations of truncated shifted quantum affine algebras (and quantized K-theoretical Coulomb branches). The Q-operators are transfer matrices associated to prefundamental representations of the Borel subalgebra of a quantum affine algebra, via the standard R-matrix construction. In a joint work with E. Frenkel, we have proved that, up to a scalar multiple, they act polynomialy on simple finite-dimensional representations of a quantum affine algebra. This establishes the existence of Baxter polynomial in a general setting (Baxter polynomiality). In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras and their truncations. We propose a conjectural parameterization of simple modules of a non simply-laced truncation in terms of the Langlands dual quantum affine Lie algebra (this has various motivations, including the symplectic duality relating Coulomb branches and quiver varieties). We prove that a simple finite-dimensional representation of a shifted quantum affine algebra descends to a truncation as predicted by this conjecture. This is derived from Baxter polynomiality.