Questions tagged [reductive-groups]
A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial
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Quasi-split and inner forms of reductive groups over a curve
Let $X$ be a curve over an algebraically closed field and let $G$ be a split reductive group. Over the generic point $\eta$, Steinberg's theorem implies that we have exact sequence
$$
0 \to H^1(\eta, \...
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Invariants of symplectic group
Let $k,n \in \mathbb{N}$. Consider the conjugation action of $\mathrm{Sp}_{2n}$ on $\mathrm{Sp}_{2n}^k$ and the corresponding invariant algebra $\mathbb{C}[\mathrm{Sp}_{2n}^k]^{\mathrm{Sp}_{2n}}$. Is ...
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The precise structure of the centralizer subgroup of an $\mathfrak{sl}_2$ in a complex simple Lie algebra
I think this question might be folklore to experts (to what extent it can be answered, and to what extent an expected answer is inaccessible), but since I am just a beginner in this direction, please ...
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Almost unipotent characters
Let us consider a split adjoint simple group $G$ over $\overline{\mathbb{F}_q}$. Then we have a Frobenius map $F$, and we can consider a finite group of Lie type, $G^F$. (We can assume that the ...
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Congruent subgroups of $\mathrm{PGL}_2$ vs $\mathrm{SL}_2$
Exercise 4.3 of Mine's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) says:
Show that the image in $\mathrm{PGL}_2(\mathbb{Q})$ of a congruence subgroup in $\mathrm{SL}_2(\...
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Haar measures on reductive groups via top degree differential forms
Let $G$ be a connected reductive group over a local non archimedean field $k$. An invariant differential form is an element of dual of the $\wedge^{top} Lie(G)$. Let $\omega_G$ be an invariant ...
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What changes for reductive groups when the smoothness assumption is dropped for the unipotent radical?
The definition of a reductive group over a field $k$ is that it is smooth (and let us say connected, although not all authors require this, this is the most common definition), and has no non-trivial ...
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Center and Levi subgroups
Let $G$ be a reductive group over $\mathbb{C}$ such that $Z_G
\neq 1$. Can we always find an element $\lambda \in Z_G$ such that
We have that $\lambda \in [G,G]$
For any Levi subgroup $L \subsetneq ...
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$B(\mathcal{O})$-orbits in the affine Grassmannian
Let $G$ be a complex reductive group with Borel subgroup $B$, $\mathcal{O}=\mathbb{C}[\![t]\!]$, $\mathcal{K}=\mathbb{C}(\!(t)\!)$ and $\operatorname{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ the affine ...
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Closed $G$-orbits in an affine variety
Let $k$ be an algebraically closed field (of characteristic $2$ in the case I am interested in). Let $G$ be a connected reductive group over $k$ and $X$ be an affine variety over $k$, on which $G$ ...
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Plücker relations for generalized flag varieties
Let $G$ be a complex simply-connected semisimple group. Then I know that the flag variety of $G$ can be described as the subscheme of $\prod_{\lambda} \mathbb{P}(V_\lambda)$ (with the product over all ...
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Invariants of a maximal unipotent group in GL(n) acting by conjugation on n by n matrices
Let $G = \operatorname{GL}(n, \mathbb{C})$ and let $U \subset G$ be the usual maximal unipotent subgroup (upper triangular matrices with ones on the diagonal). Let $U$ act by conjugation on the space $...
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Compact quotients of $p$-adic reductive groups
Fix a prime $p$ and let $G$ be a connected reductive group over a $p$-adic local field $K$. Let $H \subset G$ be a connected reductive subgroup such that $G(K)/H(K)$ is compact in the $p$-adic ...
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Examples of Harish-Chandra Schwartz functions coming from tempered representations
Let $F$ be a local field and (for simplicity) $G$ a semisimple group split over $F$. Then for in both the archimedean and non-archimedean cases, one has the space $\mathcal{C}(G)$ of Harish-Chandra ...
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Restriction of homogeneous line bundles on G/B to P/B
Let $G$ be a semisimple simply connected algebraic group over $\mathbb C$ and let $B\subseteq G$ be a Borel subgroup. Furthermore let $T\subseteq B$ be a fixed maximal torus, $\lambda \in X^*(T)$ an ...
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