On the Brylinski-Kostant filtration

Abstract:

Let $\mathfrak g$ be a semisimple Lie algebra and $V$ a finite dimensional simple $\mathfrak g$ module. The Brylinski-Kostant (simply, BK) filtration on weight spaces of $V$ is defined by applying powers of a principle nilpotent element. It leads to a $q$-character of $V$. Through a result of B. Kostant the BK filtration of the zero weight space is determined by the so-called generalized exponents of $\mathfrak g$. Later R. K. Brylinski calculated the BK filtration on dominant weights of $V$ assuming a vanishing result for cohomology later established by B. Broer. The result could be expressed in terms of $q$ polynomials introduced by G. Lusztig. In the present work, Verma module maps are used to determine the BK filtration for all weights. To do this several filtrations are introduced and compared, a key point being the graded injectivity of the ring of differential operators on the open Bruhat cell viewed as a $\mathfrak g$ module under diagonal action. This replaces cohomological vanishing and thereby Brylinski’s result is given a new proof. The calculation for non-dominant weights uses the fact that the corresponding graded ring is a domain as well as a positivity result of G. Lusztig which ensures that there are no accidental cancellations. This method allows one to compare the BK filtrations in adjacent chambers.