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In addition to the references already mentioned, let me recommend Dennis Snow's excellent notes on homogeneous vector bundles. They cover everything you ask for (and much more): For a description of …

I don't know if you can find the complete list written down anywhere (and I don't know how "nice" such a list would turn out to be), but let me point out that there is a lowbrow approach to this probl …

Lemma 2.1.3 in Rudin's Fourier Analysis on Groups does this for locally compact abelian $G$ and closed $H \leq G$. This might not the best reference (may be a bit too general for a CS readership?), bu …

You can handle the case of $n \leq 3$ one at a time, and so the question really is about $n \geq 5$. Two important names in this regard are Kirby and Siebenmann. The Wikipedia article on the Hauptverm …

You might enjoy this article by Harry Tamvakis.

I believe the reference is to this formula of Euler (see here): If $P(x)/Q(x)$ is a rational function and $ax+b$ is a simple factor of $Q(x)$, then the coefficient of $1/(ax+b)$ in the partial fractio …

I know of two clean approaches to classifying the unitary irreps of the $ax+b$ group. The first is to write the group as a semidirect product $\mathbb R \ltimes \mathbb R_{>0}$. There is a theory (due …

The fact that all $1$-dimensional tori are projective means care is sometimes needed in making analogies with higher dimensional tori. This is one of those times. The natural 'moduli space' of all $d$ …

It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for examp …

Using the maximum modulus principle you can show that $\mathbb{C}^n$ doesn't have any compact complex submanifolds of positive dimension. It follows that lots of complex manifolds, such as complex gra …

If $\theta$ is an involution of a complex linear algebraic group $G$ and if $K=G^\theta$ is its fixed-point set, then $K/K^0$ will always have exponent 2. This follows from a generalized "Cartan deco …

Correct. You can be fairly explicit here. For each root $\alpha$, let $\omega_\alpha \in \mathfrak g^\ast$ be a left-invariant form on $G$ that is dual to $\mathfrak g_\alpha$. Then for $\lambda \in …

I think there's some confusion in the question. For example by "Levi-Civita connection" you must really mean some kind of Laplacian. Anyway, your end result about the cohomology of $G/H$ is essentiall …

You can use some spectral theory to show that the set of unitary operators in $B(H)$ is path-connected. Then the (path-)connectedness of the invertibles follows easily from the polar decomposition. Se …