On Sevostyanov’s Construction of Quantum Difference Toda Lattices

Abstract

Abstract We propose a natural generalization of the construction of the quantum difference Toda lattice [6, 22] associated with a simple Lie algebra $\mathfrak{g}$. Our construction depends on two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data (which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we provide an alternative construction via Lax matrix formalism, cf. [15]. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of [7]. We give a geometric interpretation of all Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, generalizing the construction of [4].