# On the Fibres of Mishchenko-Fomenko Systems

### Peter Crooks

Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA### Markus Roeser

Fachbereich Mathematik, Universität Hamburg, 20146 Hamburg, Germany

## Abstract

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $g$. Their theory associates a maximal Poisson-commutative subalgebra of $C[g]$ to each regular element $a∈g$, and one can assemble free generators of this subalgebra into a moment map $F_{a}:g→C_{b}$. 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 $C_{b}$, and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra $b_{a}⊆g$, defined to be the intersection of all Borel subalgebras of $g$ containing $a$. In the case of a non-nilpotent $a∈g_{reg}$ and an element $x∈b_{a}$, we prove the following: $x+[b_{a},b_{a}]$ lies in the singular locus of $F_{a}(F_{a}(x))$, and the fibres through points in $b_{a}$ form a $rank(g)$-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 $l⊆g$. 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 $F_{a}(0)$, and it generalizes a result of Charbonnel-Moreau. Illustrative examples are included at the end of this paper.

## Cite this article

Peter Crooks, Markus Roeser, On the Fibres of Mishchenko-Fomenko Systems. Doc. Math. 25 (2020), pp. 1195–1239

DOI 10.4171/DM/774