On the Fibres of Mishchenko-Fomenko Systems

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 a\in\mathfrak{g} , and one can assemble free generators of this subalgebra into a moment map F_a:\mathfrak{g}\rightarrow\mathbb{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. 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}^a\subseteq\mathfrak{g} , defined to be the intersection of all Borel subalgebras of \mathfrak{g} containing a . In the case of a non-nilpotent a\in\mathfrak{g}_{\mathrm{reg}} and an element x\in\mathfrak{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 \text{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}\subseteq\mathfrak{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.