Timeline for Simultaneous decomposition into generalized eigenvectors

6 events

when toggle format	what		by	license	comment
Dec 14, 2011 at 12:47	vote	accept	Sh4pe
Dec 12, 2011 at 19:41	comment	added	Ben Webster♦		I meant the endomorphism space. This is finite dimensional since any $\mathfrak{g}$-module map between objects in category $\mathcal{O}$ is determined by what it does on finitely many weight spaces (those which appear as highest weights in composition factors) and the space of vector space maps between those weight spaces is finite dimensional. For example, any endomorphism of a Verma module sends the highest weight vector to a highest weight vector, and thus is scalar multiplication.
Dec 12, 2011 at 17:33	history	edited	Ben Webster♦	CC BY-SA 3.0	edited body
Dec 12, 2011 at 16:20	comment	added	Sh4pe		First of all: thank you very much! That helps me a lot! But do you mean that my module itself is finite dimensional? One typical example would be a verma module, and they are in generally not finite dimensional. Or do you mean that the endomorphism spaces are finite dimensional? If true, why are they finite dimensional? (The merely linear endomorphisms of a verma would also form an infinite dimensional space). I'm not at all trying to argue with you, just trying to learn. Thank you!
Dec 12, 2011 at 16:14	vote	accept	Sh4pe
Dec 14, 2011 at 12:47
Dec 12, 2011 at 15:35	history	answered	Ben Webster♦	CC BY-SA 3.0