Séminaires : Séminaire d'Algèbre

Let $G$ be a reductive algebraic group over a field $k$. When $k=\mathbb{C}$, R. K. Brylinski constructed a filtration of weight spaces of a $G$-module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's $q$-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. Recently, we managed to generalise this result to the case where $k$ is a field of good positive characteristics. I will give a brief introduction to both historical results and our new result in the talk.