Moving between weights of weight modules

Abstract

In Lie theory the partial sum property (PSP) says that for a root system in any Kac-Moody algebra, every positive root is an ordered sum of simple roots whose partial sums are all roots. In this paper, we present two generalizations: 1) "Parabolic generalization": if $I$ is a subset of simple roots, every root with positive $I$-height is an ordered sum of roots of $I$-height 1, whose partial sums are all roots. In fact we show this on the Lie algebra level, by showing that every root space is spanned by the Lie words formed from root vectors of $I$-height 1. As an application, we provide a "minimal" description for the set of weights of every (non-integrable) simple highest weight module over any Kac-Moody algebra. This seems to be novel even in finite type. 2) Generalization to weights of weight modules: the PSP gives a chain of roots between 0 (fixed) and any positive root. We generalize this to the weights of weight modules to get a chain of weights between any two comparable weights. This was shown by S. Kumar for any finite-dimensional simple module over a semisimple Lie algebra. In this paper, we extend this result to (i) a large class of highest weight modules over any Kac-Moody algebra $\mathfrak{g}$, which includes all simple highest weight modules over $\mathfrak{g}$; (ii) more generally, for non-highest weight modules such as $\mathfrak{g}$ itself (adjoint representation) and arbitrary submodules of parabolic Verma modules over $\mathfrak{g}$; (iii) arbitrary integrable modules over semisimple $\mathfrak{g}$. Additionally, we also prove the "parabolic" generalizations of this second generalization to the best possible extent. We also find all the highest weight modules which have their sets of weights same as those of parabolic Verma modules and provide a Minkowski difference formula for weights of arbitrary highest weight modules over Kac-Moody $\mathfrak{g}$.