| Résume | Let g be a finite-dimensional, simple Lie algebra over the field of complex numbers, and U be the quantum, untwisted affine algebra, associated to g. Via Lusztig's q-exponential formulae, it is well known that the affine braid group of g acts on any integrable representation of U. In particular, one obtains an action of the coroot lattice of g on such a representation. In this talk, I will present an explicit formula for these lattice operators. Surprisingly, the answer is given as a normalized limit of a well-known series, which we call Chari-Pressley series, and hinges upon their rationality on integrable representations. Time permitting, I will discuss some applications towards the universal R-matrix of the quantum affine algebra, monodromy of trigonometric Casimir equations and geometry of Nakajima quiver varieties.
This formula and the rationality of CP series were obtained in my recent joint work with V. Toledano Laredo (arxiv:2501.2365). |
