Relative modality of elements in generalized Takiff Lie algebras

Abstract

Given a natural number m and a finite dimensional complex Lie algebra $g$, the $m^{th}$ generalized Takiff Lie algebra of $g$ is the Lie algebra $g_m:=g\otimes C\left[T\right]/T^{m+1}$. For $n\ge m$, we define the $(m,n)$-modality of an adjoint orbit ${\Omega }_m$ in $g_m$ to be the minimum codimension of an adjoint orbit in the pullback of ${\Omega }_m$ in $g_n$.

In this paper, we study these invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra $g$. We show that these invariants satisfy some concavity and hereditary properties from which we deduce that $(n-m)\chi \left(g\right)$ is a lower bound, where $\chi \left(g\right)$ is the index of $g$. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if $g$ is reductive, it is always an equality when $m=0$ (and also some special orbits). We conjecture that equality holds for all m, n when $g$ is reductive.