q,t-characters and the structure of the $\ell$-weight spaces of standard modules over simply laced quantum affine algebras

Abstract: We establish, for all simply laced types, a q,t-character formula, first conjectured by Nakajima. It relates, on one hand, the structure of the $\ell$-weight spaces of standard modules regarded as modules over the Heisenberg subalgebra of some quantum affine algebra and, on the other hand, the t-dependence of their q,t-characters, as originally defined in terms of the Poincar\'e polynomials of certain Lagrangian subvarieties in quiver varieties of the corresponding simply laced type. Our proof is essentially geometric and generalizes to arbitrary simply laced types earlier representation theoretical results for standard $\mathrm{U}_q (\widehat{\mathfrak{sl}}_2)$-modules.