Weights of highest weight modules over Kac-Moody algebras

Abstract

In this dissertation, broadly, we treat the weight-sets of arbitrary highest weight modules (uniformly) over all general complex Kac-Moody Lie algebras $\mathfrak{g}$, achieving the below.

We obtain a uniform, explicit, cancellation-free and positive formula for the weight-sets of all highest weight modules $V$ (of all highest weights) over all Kac-Moody $\mathfrak{g}$. Interestingly, our formula for the weights of $V$ involves basic ingredients, namely the independent subsets in the Dynkin diagram of $\mathfrak{g}$ determined by $V$, and nothing else! Prior to our work, it seems that even for all (non-integrable) simple highest weight $\mathfrak{g}$-modules - despite these being treated for over a half century - their weight-formulas were not known until a few years ago. Namely, these were written-up in Khare [J. Alg. 2016] and Dhillon-Khare [Adv. Math. 2017 and J. Alg. 2022]; and our general formula recovers the formulas for simples in these papers. Leaving out general Kac-Moody or even general semisimple settings, given an arbitrary highest weight module $V$ over $\mathfrak{sl}_n(\mathbb{C})$ or in particular $\mathfrak{sl}_4(\mathbb{C})$, even in this case the weights of $V$ were not written down in the literature to the best of our knowledge.

More importantly, our weight-formula naturally drives us to introduce and study a finite family of ``higher order Verma modules" $\mathbb{M}(\lambda,\mathcal{H})$ over Kac-Moody $\mathfrak{g}$, for every highest weight $\lambda\in \mathfrak{h}^*$ and (any) collection $\mathcal{H}$ of independent subsets in the Dynkin diagram of $\mathfrak{g}$. This family generalizes and includes: all Verma modules $M(\lambda)=\mathbb{M}(\lambda,\emptyset)$ at zeroth order level, and the parabolic Verma modules $M(\lambda,J)=\mathbb{M}(\lambda, \{ \{j\} \ |\ j\in J \})$ (which were introduced and studied by Lepowsky, Kumar, Mathieu,... to name but a few) at first order level. Importantly, our higher order Verma modules are crucial and universal for weight considerations: their weight-sets (which are finitely many when we fix their highest weight) are pairwise disjoint and exhaust the weight-sets of all highest weight $\mathfrak{g}$-modules. In this thesis, we also initiate the study of the characters of these universal modules, by computing them via BGG-type resolutions in certain cases.

Next, a phenomenon in root systems which rewarded us with many applications on the weights side. First, recall the partial sum property for Kac-Moody root systems: every root of $\mathfrak{g}$ is an ordered sum of simple roots such that each partial sum is also a root. The course/journey to finding and proving our weight-formula mentioned above begins from proving a parabolic-generalization of this property, which we call as the parabolic partial sum property. Given a subset $S$ of simple roots, it allows one to write any (positive) root $\beta$ involving some simple roots from $S$ as: an ordered sum of roots, in which each root involves exactly one simple root from $S$ (unit $S$-height roots) and with each partial sum also being a root. The parabolic partial sum property has been devised in order to obtain a ``minimal" description for the weights of all (non-integrable) highest weight simples, which was posed by Khare. This thesis exhibits such minimal descriptions as an immediate application of the parabolic partial sum property; in fact we will more strongly show this property at the level of Lie words for any general Lie algebra graded over any free abelian semigroup. There is also another generalization of the partial sum property due to Khare and Kumar for highest weight simples in finite type, which we extend to the Kac-Moody setting in this thesis.

There is another notable application of the parabolic partial sum property shown in this thesis. Chari and her coauthors [Adv. Math. 2009 and J. Geom. Phys. 2011] introduced and studied certain combinatorial subsets called weak faces and $(\{2\};\{1,2\})$-closed subsets of finite root systems. This is in order to construct Koszul algebras, study Kirillov-Reshetikhin modules over specializations of quantum affine algebras, and also to classify nilpotent ideals in the parabolic Lie subalgebras of finite type $\mathfrak{g}$, etc. These subsets (say subsets of a set $X$) are some discrete analogues/ generalizations of the faces for convex sets (the faces of the convex hull of $X$). In this thesis, we are concerned with such subsets of $X=$ a weight-set, generalizing the faces for convex hulls of weight-sets. Using the weak faces for weight-sets of finite-dimensional simples (in finite type) Khare extended the aforementioned results of Chari et al. Motivated by the applications of these two modern notions to representation theory, we completely classify the weak faces and $(\{2\};\{1,2\})$-closed subsets of weights of all highest weight modules (again uniformly) over all Kac-Moody $\mathfrak{g}$; extending and completing the partial classification results of Chari, Khare, and their co-authors from finite type. We more strongly show that both these notions are the same as the weights falling on the faces, for the convex hulls of these weight-sets. This shows the equivalence of these two notions, to the classical faces for the convex hulls of weights (which have been pursued from the 1960s).