Extended Schur’s [formula omitted]-functions and the full Kostant–Toda hierarchy on the Lie algebra of type [formula omitted

Abstract

The full Kostant–Toda hierarchy on a semisimple Lie algebra is a system of Lax equations, in which the flows are determined by the gradients of the Chevalley invariants. This paper is concerned with the full Kostant–Toda hierarchy on the even orthogonal Lie algebra. By using a Pfaffian of the Lax matrix as one of the Chevalley invariants, we construct an explicit form of the flow associated to this invariant. As a main result, we introduce an extension of the Schur’s Q-functions in the time variables, and use them to give all the polynomial τ-functions of the hierarchy.