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
On the Fibres of Mishchenko-Fomenko Systems cover
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This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra g\mathfrak{g}. Their theory associates a maximal Poisson-commutative subalgebra of C[g]\mathbb{C}[\mathfrak{g}] to each regular element aga\in\mathfrak{g}, and one can assemble free generators of this subalgebra into a moment map Fa:gCbF_a:\mathfrak{g}\rightarrow\mathbb{C}^b. This leads one to pose basic structural questions about FaF_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 FaF_a have codimension 11 or 22 in Cb\mathbb{C}^b, and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra bag\mathfrak{b}^a\subseteq\mathfrak{g}, defined to be the intersection of all Borel subalgebras of g\mathfrak{g} containing aa. In the case of a non-nilpotent agrega\in\mathfrak{g}_{\mathrm{reg}} and an element xbax\in\mathfrak{b}^a, we prove the following: x+[ba,ba]x+[\mathfrak{b}^a,\mathfrak{b}^a] lies in the singular locus of Fa1(Fa(x))F_a^{-1}(F_a(x)), and the fibres through points in ba\mathfrak{b}^a form a rank(g)\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 lg\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 Fa1(0)F_a^{-1}(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