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 $F_a:\mathfrak{g}\to {\mathbb{C}}^b$. 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 $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\mathfrak{g})$-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.