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For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, …

What I'm writing here seems more like a contribution to a big-list than an "answer", but since you've already chosen one anyway... Say you're interested in which irreps $V_\nu$ occur in $V_\lambda \ …

What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line? I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits …

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ha …

Here's the argument I know that avoids spectral sequences, based on the little-known space $G/N(T)$. In between $T$ and $G$ is $N(T)$. Note that $EG$ "is an" $ET$ and $EN(T)$, since it's contractib …

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V …

I believe the following way (Kostant's, 1970) to be the best way to think about the Hamiltonian condition. First, "why" is there a central extension $H^0(M; {\mathbb R}) \to C^\infty (M) \to symp(M)$ …

The complete list is quite short -- at least, up to changing real form. I'll give the compact group version, up to isogeny. A_1 = B_1 = C_1: $SU(2) \cong Spin(3) \cong U(1,{\mathbb H})$ For the firs …

If $\lambda,\mu,\nu$ are considered as vectors in $\mathbb Z^n$, then the set of triples forms a convex cone, given by a finite (for each $n$) list of inequalities. (The corresponding statement does n …

To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$ (vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching an extra vertex to every old vertex in $Q_0$. Then …

Let's pull back to irreps of $GL_1({\mathbb R}) \times SL_n({\mathbb R})$. By Schur's lemma, the first factor acts by scalars, so the representation is of the form $V \otimes W$ where $V$ is a chara …

The construction you describe appears in Tamvakis' The connection between representation theory and Schubert calculus (Enseign. Math. 50 (2004), 267-2860). Basically, instead of working with represent …

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure o …

Start by deformation-retracting $G$ to its maximal compact, so that $G/T$ will be a flag manifold with a Bruhat decomposition, obtainable by Morse theory as in Agol's answer. If what you want anyway …

There are conjectural ones in the Berenstein-Zelevinsky paper referenced in that one. They have another paper with a general theorem, Tensor product multiplicities, canonical bases and totally positiv …