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Maybe I should try to defend myself, or at least the self I was four decades ago when I improvised my graduate text. But first I should disclaim any originality in the proof of Weyl's theorem, which …

While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into …

I don't think there is a one-line answer to this question, since it depends a lot on the direction from which you approach semi-simple Lie theory. For one thing, it's probably best at first to empha …

In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031), Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corr …

It's always a good idea to ask (as students typically do) why one is studying a particular subject or theorem. Here are some of my views, from the algebraic side of representation theory: 1) The or …

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras: 37602. Like some others who started graduate study in …

One implicit aspect of the question is the intrinsic interest of studying monoid representations. This is addressed by Qiaochu's answer and comments on it. But I'd emphasize more the role of appli …

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of …

The questions raised here will probably need some substantial research papers to answer, inventing new approaches and methods. In any case, the question of what to do about "small" primes has be …

The statement is false. The standard definition of "reductive" for a finite dimensional Lie algebra $\mathfrak{g}$ over an arbitrary field of characteristic 0 is given in a number of equivalent ways …

Borel's lengthy 1953 Annals paper is essentially his 1952 Paris thesis. It was followed by work of Bott, Samelson, Kostant, and others, which eventually answers your side question affirmatively. …

I'd add to what Ben says the observation that people have found a number of different module categories valuable for different purposes within this same Lie algebra context. (And some Lie theory pe …

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary, the result of historical accidents. In order to avoid repetitions, the four infinite families $A_\ell, B_\ell, C_ …

Motivation in mathematics is always a tricky question, but I'd call attention to one name you've omitted from your list: Serre. It's definitely worthwhile to look at his paper Cohomologie des groupe …

This is a reasonable question, and there has been some relevant literature over many decades. But my impression is that no definitive answer has been given, even for the classical types. I've int …