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 $F_a:\mathfrak{g}\rightarrow\mathbb{C}^b$. 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 $a\in\mathfrak{g}_{\text{reg}}$ 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 $\mathrm{rank}(\mathfrak{g})$-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.