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There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ i …

As stated, there is a non-semisimple counterexample: take $G$ to be the additive group $\mathbb{F}_q$ with a unipotent representation $\rho(a)=\left(\begin{matrix}1 & a\\ 0 & 1\end{matrix}\right)$. It …

I am far from being expert in this subject, but I will try to present my understading of there this multiplicativity comes from. I wiil refer to authors you mention but only to the parts which I hope …

See Theorem 4.2.2 in https://www-fourier.ujf-grenoble.fr/~mbrion/lin.pdf In particular, in your example properness of $X=G/B$ simplifies the left part of the sequence, turning it into $$0\to \hat{G} …

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt isomorphi …

In the previous lines it is proven that $U(T)^W$ is integral over $\gamma(Z^{\mathcal G})$. Since $U(T)^W\subset Frac(U(T)^W)=Frac(\gamma(Z^{\mathcal G}))$ the equality follows from $\gamma(Z^{\mathca …

Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in t …