The weights of simple modules in Category O for Kac�Moody algebras

Dhillon, G and Khare, A (2022) The weights of simple modules in Category O for Kac�Moody algebras. In: Journal of Algebra, 603 . pp. 164-200.

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Abstract

We give the first positive formulas for the weights of every simple highest weight module L(λ) over an arbitrary Kac�Moody algebra. Under a mild condition on the highest weight, we also express the weights of L(λ) as an alternating sum similar to the Weyl�Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding D-module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group. © 2022 Elsevier Inc.

Item Type:	Journal Article
Publication:	Journal of Algebra
Publisher:	Academic Press Inc.
Additional Information:	The copyright for this article belongs to Elsevier
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	10 May 2022 15:59
Last Modified:	10 May 2022 15:59
URI:	https://eprints.iisc.ac.in/id/eprint/71731

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