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Near the bottom of Jacob Lurie's homepage, you can find an exposition of the Borel-Weil-Bott theorem from an algebro-geometric standpoint. It is "easily readable" if you're familiar with the things l …

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a …

I think that the best reference for the infinitesimal site, especially if one is motivated to learn crystalline cohomology, is found in Grothendieck's lectures "Crystals and the De Rham Cohomology of …

Brian's comment does what you want, and describes the almost direct product caveat. A standard and excellent reference for all things of this nature is Demazure's "Sous-groupes Paraboliques des group …

A no-nonsense construction, over $Z$, following work of Kostant and Chevalley, is given in Lusztig's paper "Twelve bridges from a reductive group to its Langlands dual". The heart of the construction …

Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$. So that I stop worrying, …

I don't know anything about quantum groups, but the Peter-Weyl theorem for compact groups generalizes nicely to Type I second-countable locally compact topological groups, a result of Segal and Mautne …

Though I'm not an expert on motives, by any measure, I think that an answer to your question can be given by considering periods. As Kontsevich and Zagier recall in their paper "Periods", publ. IHES, …

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid. Suppose moreover that $X$ a …

Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$. Then I'm pr …

I think that going all the way to barcodes and persistent homology is a big leap, and probably not one where there will be something interesting. But maybe there are interesting things if you just go …

One suggestion: "Period mappings and Period Domains", by Carlson, Muller-Stach, and Peters, in the Cambridge studies in advanced mathematics series. It's a very nice read, and each chapter comes wit …

I'll try to answer both questions, though I will change the first question somewhat. Let's work in the setting of a real reductive algebraic group $G$ and a closed subgroup $H \subset G$. Your fir …

The places to look are: Steinberg, "Endomorphisms of linear algebraic groups." Memoir AMS 80, (1968), and Gross, "The motive of a reductive group" Invent. math. 130, 287 ± 313 (1997). (I learne …

Here are some more details. As John Voight said, the quaternion algebra is kind of irrelevant here. If $\gamma$ is a regular semisimple element, then its centralizer is a torus ${\mathbf T}$ over ${ …