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I assume, since you haven't explicitly stated it, that you're taking these Lie algebras in characteristic 0 -- the question is much harder in positive characteristic (and in particular, the word "comp …

All of these recommendations are very good, and I'd like to add that the book D-Modules, Perverse Sheaves, and Representation Theory (which you can download at the provided link if you have institutio …

For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alph …

This statement is false in general for algebraic groups. It's true in characteristic 0, but it is not in general true in positive characteristic. Instead, one has a weaker statement in positive charac …

As Vladimir mentions, the statement of your theorem is unclear. However, since you have written the $G$- and $H$-actions as tensor product actions where each acts nontrivially on one tensor factor and …

The reason your argument doesn't work is because it's not true that $\text{Sym}^l \mathfrak n^\vee$ has a filtration with 1-dimensional graded pieces where $T$ acts with anti-dominant weights. In fact …

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of th …

Background Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be …

This question is perhaps somewhat soft, but I'm hoping that someone could provide a useful heuristic. My interest in this question mainly concerns various derived equivalences arising in geometric rep …

Setup Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the positi …

For your first question, let $i : T \hookrightarrow X$ be the inclusion; then $\mathcal O_T$ becomes a $\mathcal O_X$-module by identifying $\mathcal O_T$ with $i_* \mathcal O_T$, an $\mathcal O_X$-mo …

There is a nice answer to question 5 for algebraic groups in arbitrary characteristic over algebraically closed fields, although one needs to consider a larger category. Given an algebraic subgroup H …

Chris Brav's answer gives a nice description of the cohomology in the $D$-module context. Just to expand a bit on that, I'd like to give a direct description, and also say a word about positive charac …

I don't have a complete answer, but let me just note that your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diag …

Let $G$ be an algebraic group over an algebraically closed field $k$ of arbitrary characteristic and let $U$ be the hyperalgebra of $G$. Recall that $U$ is defined as the subspace of the full linear d …