Exponents for B-stable ideals

E Sommers, University of Massachusetts - Amherst
J Tymoczko

Article comments

This is the pre-published version harvested from ArXiv. The published version is located at

http://www.ams.org/journals/tran/2006-358-08/S0002-9947-06-04080-3/home.html

http://www.ams.org/journals/tran/2006-358-08/S0002-9947-06-04080-3/S0002-9947-06-04080-3.pdf

Abstract

Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.

When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.