Relative modality of elements in generalized Takiff Lie algebras

Abstract

Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm\,:= g $\otimes$ C[T ]/T m+1 . For n $\ge$ m, we define the (m, n)-modality of an adjoint orbit $\Omega$m in gm to be the minimum codimension of an adjoint orbit in the pullback of $\Omega$m in gn. In this paper, we study this family of invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra g. We show that this family of invariants satisfies some concavity and hereditary properties. From which we deduce that (n -m)$\chi$(g) is a lower bound, where $\chi$(g) 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 when g is reductive.