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Let $G$ be an affine algebraic group defined over $\mathbf Z$. The kernel of the natural homomorphism $G(\mathbf Z/p^2\mathbf Z)\to G(\mathbf Z/p\mathbf Z)$, if abelian, is a group which comes along w …

Here is just one example (I know there are others too): Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are i …

$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}0 & 1\\0&0\end{pmatrix}$

I address mainly Question C in the simplest special case where $R$ is $\mathbb Q$: In this case you are looking at locally nilpotent endomorphisms of a vector space. Similarity classes of such endomo …

My answer starts off just like Emerton's answer above; you want the $G$-orbits on $G/B\times G/B$. But now, I diverge from Emerton to say that $G/B$ is the space of full flags $F_0\subset F_1\subset \ …

Bruhat decomposition over $\mathbf Z/p^r\mathbf Z$ is precisely the problem we looked at in this paper. We defined several invariants of double cosets, and classified the pairs $(n,k)$ for which, when …

$K\pi^\lambda K$ has a transitive right action of $K$. The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$. Thus, $K\pi^\lambda K = \coprod_x K\pi^\lambda x$ as $x$ …