On the Fibres of Mishchenko-Fomenko Systems

摘要

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra (mathfrak{g}). Their theory associates a maximal Poisson-commutative subalgebra of (mathbb{C}[mathfrak{g}]) to each regular element (ainmathfrak{g}), and one can assemble free generators of this subalgebra into a moment map (F_a:mathfrak{g}ightarrowmathbb{C}^b). This leads one to pose basic structural questions about (F_a) and its fibres, e.g. questions concerning the singular points and irreducible components of such fibres. par We examine the structure of fibres in Mishchenko-Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel-Moreau, Moreau, and others. This includes proving that the critical values of (F_a) have codimension (1) or (2) in (mathbb{C}^b), and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra (mathfrak{b}^asubseteqmathfrak{g} ), defined to be the intersection of all Borel subalgebras of (mathfrak{g}) containing (a). In the case of a non-nilpotent (ainmathfrak{g}_{mathrm{reg}}) and an element (xinmathfrak{b}^a), we prove the following: (x+[mathfrak{b}^a,mathfrak{b}^a]) lies in the singular locus of (F_a^{-1}(F_a(x))), and the fibres through points in (mathfrak{b}^a) form a (ext{rank}(mathfrak{g}))-dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko-Fomenko systems on Levi subalgebras (mathfrak{l}subseteqmathfrak{g}). In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in (F_a^{-1}(0)), and it generalizes a result of Charbonnel-Moreau. Illustrative examples are included at the end of this paper.