On the singularities of Mishchenko-Fomenko systems

2021/03/25 by Crooks, Peter, Röser, Markus
#FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2103.14168

Abstract

To each complex semisimple Lie algebra \mathfrakg and regular element a∈\mathfrakg_reg, one associates a Mishchenko-Fomenko subalgebra F_a⊆ℂ[\mathfrakg]. This subalgebra amounts to a completely integrable system on the Poisson variety \mathfrakg, and as such has a bifurcation diagram Σ_a\subseteqSpec(F_a). We prove that Σ_a has codimension one in Spec(F_a) if a∈\mathfrakg_reg is not nilpotent, and that it has codimension one or two if a∈\mathfrakg_reg is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of Σ_a.

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