Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G ⊗ C[λ, λ −1 ] are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group W(G) of the simple Lie algebra G. A representative w ∈ W(G) of a regular conjugacy class can be lifted to an inner automorphism of G given byˆw = exp (2iπadI 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,I 0 = mλ d dλ + adI 0 and any grade one regular element Λ from the Heisenberg subalgebra associated to [w] takes the form Λ = (C + + λC −), where [I 0 , C − ] = −(m − 1)C − and C + is included in an sl 2 subalgebra containing I 0. The largest eigenvalue of adI 0 is (m − 1) except for some cases in F 4 , E 6,7,8. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems. If the largest adI 0 eigenvalue is (m − 1), then using any grade one regular element from the Heisenberg subalgebra associated to [w] we can construct a KdV system possessing the standard W-algebra defined by I 0 as its second Poisson bracket algebra. For G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to gl n. Non-abelian Toda systems are also considered.