Relative modality of elements in generalized Takiff Lie algebras

TL;DR

This paper investigates the (m, n)-modality of adjoint orbits in generalized Takiff Lie algebras, establishing bounds, properties, and conditions for equality, especially in the context of quadratic and reductive Lie algebras.

Contribution

It introduces the (m, n)-modality invariant for generalized Takiff Lie algebras and proves key properties and bounds, including conditions for equality in reductive cases.

Findings

(n - m) * chi(g) is a lower bound for the modality

Equality holds for a dense set of orbits and in certain cases when g is reductive

Conjecture: equality holds for all m when g is reductive

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.