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Yes. Write $P_\alpha = (SL_2) \ltimes ((\ker \alpha) Rad(P_\alpha))$ where $\ker\alpha$ is the subgroup of $T$. The necessity of your condition is about getting the $SL_2$ to act. Once you impose it, …

There seem to be a bunch of different results of the form "This nice representation $V$ of $G\times H$ breaks up as $\bigoplus_\lambda V_\lambda \otimes W_\lambda$, where the $(V_\lambda)$ are distinc …

Let $\Gamma$ be a finite subgroup of $SL_2({\mathbb C})$, and $Q$ the set of its irreducible representations. McKay makes $Q$ into a directed graph by having $V \to W$ if $W \leq V \otimes {\mathbb C} …

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\mathbb C$. Theorem 1. The highest root is perpendicular to all but one simple root, except in the case ${\mathfrak g}={\mathfrak sl …

Prologue. To $M^n$ a compact real manifold with frame bundle $F$ (a principal $GL_n$ bundle), we associate a line bundle using the representation $M\mapsto \sqrt{|\det M|}$, the bundle of half-densiti …

Yes. In combinatorics this is known as Robinson-Schensted-Knuth vs. just Robinson-Schensted. (Properly speaking the latter is about a yet smaller duality, $\mathbb C[S_n] = \bigoplus_{\lambda\vdash n} …

In my head at least, part of it is this... Let G be reductive. Consider the following algorithm: Extend one component of the Dynkin diagram to its affine diagram, by attaching the lowest root. From …

I've always thought of this formula as an amusing accident. You say "I can go through the proofs", so I don't know if what I'll say helps any more than that, but here goes. If $M$ is any endomorphis …

It's slightly nicer to look at $M_n // U_-$ instead of $GL(n) // U_-$, since then we're looking at a subring of invariants inside a polynomial ring. Namely, the subring generated by all determinants t …

If $\alpha$ is a long root (e.g. if $\mathfrak g$ is simply-laced), then $e_\alpha$ is in the $G$-orbit of the high weight vectors. Projectively, its orbit looks like $G/P_\Theta$ where $P_\theta$ is …

There are a number of better statements (i.e. yes, this is "well-known"). To begin with, wouldn't you rather have the character of this representation, instead of just its dimension? You can get that …

Yes, if $H$ is closed (i.e. not some irrational-flow subgroup inside the closed subgroup $Z(G)$).

Consider the case that $H$ is a maximal torus of $G'$, and your $G = G' \times H$. (Well, you said $G,H$ semisimple, but I'm going to pretend you meant reductive, because really you should have.) Then …

This is not a complete answer, but maybe it will be of use. You're asking which $L$-high weights $\mu$ occur in the $G$-irrep $V_\lambda$. Let me say that $\mu$ occurs classically if for some $N>0$, …

Naively, there can be no reasonable way of distinguishing the red nodes from green in the case $A_{even}$, as the Dynkin diagram automorphism switches them. Less naively, there is indeed a way of dis …