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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 $\G\otimes{\bf C}[λ,λ^{-1}]$ are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group ${\bf W}(\G)$ of the simple Lie algebra $\G$. A representative $w\in {\bf W}(\G)$ of a regular conjugacy class can be lifted to an inner automorphism of $\G$ given by $\hat w=\exp\left(2iπ{\rm ad I_0}/m\right)$, where $I_0$ is the defining vector of an $sl_2$ subalgebra of $\G$.The grading is then defined by the operator $d_{m,I_0}=mλ{d\over dλ} + {\rm ad} I_0$ 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 ${\rm ad}I_0$ 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 ${\rm ad} I_0$ 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.