REGULAR CONJUGACY CLASSES IN THE WEYL GROUP AND INTEGRABLE HIERARCHIES

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction w ith grade 1 regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G x C[lambda, lambda(-1)] are studied. The graded Heisenberg subalgebras containing such elements are labelle d by the regular conjugacy classes in the Weyl group W(G) of the simpl e Lie algebra G. A representative w epsilon W(G) of a regular conjugac y class can be lifted to an inner automorphism of G given by (w) over cap = exp (2i pi ad I-0/m), where I-0 is the defining vector of an sl( 2) subalgebra of G. The grading is then defined by the operator d(m),( I0) = m lambda(d/d lambda) + ad I-0 and any grade 1 regular element La mbda from the Heisenberg subalgebra associated with [w] takes the form Lambda = (C-+ + lambda C--), where [I-0, C--] = -(m - 1)C-- and C-+ i s included in an sl(2) subalgebra containing I-0. The largest eigenval ue of adI(o) is (m - 1) except for some cases in F-4, E(6),(7,8). We e xplain how these Lie algebraic results follow from known results and a pply them to construct integrable systems. If the largest ad I-0 eigen value is (m - 1), then using any grade 1 regular element from the Heis enberg subalgebra associated with [w] we can construct a KdV system po ssessing the standard W-algebra defined by I-0 as its second Poisson b racket algebra. For G a classical Lie algebra, we derive pseudo-differ ential Lax operators for those non-principal KdV systems that can be o btained as discrete reductions of KdV systems related to gl(n). Non-Ab elian Toda systems are also considered.