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The classical viewpoint is captured well in the 1962 book Representation Theory of Finite Groups and Associative Algebras by Curtis and Reiner (Wiley), Corollary 29.15. This of course doesn't direc …

As Derek Holt comments, cohomology has complications even for fimite general linear groups. Probably you are using the term "reductive" too casually and should replace it by "simple" or perhaps "semis …

It's probably too soon to expect a good historical overview, but for example Steve Kleiman has already written a scholarly article (The development of intersection homology theory) emphasizing the ori …

The answer is yes, for fairly elementary reasons, though it's not easy to give a reference. The point is partly that the polynomials are undefined for two elements of the Weyl group not related by t …

The answer to the question here is yes. (More generallly, if $M$ and $N$ lie in $\mathcal(O)$ and are both infinite dimensional, then $M \otimes N \notin \mathcal{O}$.) The proof is eaiest to writ …

Maybe it would clarify matters if I gave a little more background, in community wiki format. The basic idea of this algebraic proof goes back to a short paper by Richard Brauer (1936) in German in Ma …

What is the earliest published statement and proof of the well-known result: for a simple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0, the convex hull (in the …

EDIT: I misunderstood at first what your basic question is but now understand it better. One cautionary case comes from older work of Richard Block here, which includes the rank 1 simple Lie algeb …

In an irreducible representation (finite or infinite dimensional), every nonzero vector is cyclic. This has nothing to do with Lie algebras as such, as it is true over an arbitrary ring. Thus for e …

It's worthwhile to explain something of the background, since Patrick Delorme's preprint never got published in full. It's a 23 page typed double-spaced document with symbols inserted by hand, dis …

Jantzen first defined translation functors around 1977, using a definite finite dimensional module, in 2.10 of his Habilitationsschrift here. Note that at around the same time, Gregg Zuckerman was …

The problem with your highlighted formulation is that it's wrong as stated, unless for example you require that an "irreducible" representation be finite dimensional or have an integral highest weight …

The books by Onishchik-Vinberg and by Etingof et al. might fit into this category, too.

In the case of the rank 1 simple Lie algebra, your references give a good account of what is known. But in general, it's wise to keep in mind that many of the indecomposable $U_q(\mathfrak{g})$-mod …

Vit's answer is mostly correct but overlooks the extreme case when $I$ is empty: then $\mathcal{O}^{\mathfrak{p}} = \mathcal{O}$. (At the other extreme one gets all finite dimensional modules.) I …