On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

TL;DR

This paper investigates the structure of Borel stable abelian subalgebras in Z_2-graded Lie algebras, providing parametrizations and dimension formulas, and unifying several existing results in the field.

Contribution

It introduces a natural parametrization of maximal Borel stable abelian subalgebras and derives their dimension formulas, extending previous work by Kostant, Panyushev, and Suter.

Findings

Parametrization of maximal Borel stable abelian subalgebras

Dimension formulas for these subalgebras

Unified framework recovering previous results

Abstract

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.