Extended Schur's $Q$-functions and the full Kostant--Toda hierarchy on the Lie algebra of type $D$

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 explicit formulas for the polynomial $\ddot{I}„$-functions of the hierarchy.

24 pages, submitted to a special issue for the memory of Hermann Flaschka