Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies

Abstract

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 $\hat{w}=\exp \left(2i\pi \mathrm{ad}{\mathrm{I}}_0/m\right)$, where $I_0$ is the defining vector of an $s{l}_2$ subalgebra of $\G$.The grading is then defined by the operator $d_{m,I_0}=m\lambda \frac{d}{d\lambda }+\mathrm{ad}{I}_0$ and any grade one regular element $\Lambda$ from the Heisenberg subalgebra associated to $[w]$ takes the form $\Lambda =(C_{+}+\lambda C_{-})$, where $[I_0,C_{-}]=-(m-1)C_{-}$ and $C_{+}$ is included in an $s{l}_2$ subalgebra containing $I_0$. The largest eigenvalue of $\mathrm{ad}{I}_0$ 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 $\mathrm{ad}{I}_0$ 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 $g{l}_n$. Non-abelian Toda systems are also considered.

44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use OLATEX (preferred) or LATEX)

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