# 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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### Properness of reductive group actions on smooth varieties

Suppose that $G$ is a reductive algebraic group acting on a smooth variety $X$, and that the action has finite stabilizers. When is the action of $G$ on $X$ proper? What is an example where the action ...
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### Property of representations of reductive group schemes over characteristic 0 field

I originally posted this on Maths SE, but then I thought it MO might be more fitting. Let $k$ be a characteristic $0$ field and let $G$ be a linear algebraic group scheme over $k$. Then is it true ...
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### Is there a scheme-theoretic description of spherical varieties?

I haven't found much expository literature, and in particular, I'd like some references on how spherical varieties could be viewed schematically, especially functorially (i.e. viewed from the functor ...
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### Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$

Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines ...
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### Reference request: Commutator relations for the exceptional group F4

Is there any standard reference for the commutator relations for the exceptional group of type $F_4$? If this question is not appropriate here, please let me know and I will delete it. Thanks in ...
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### rational representants of sigma-conjugacy classes

Let $G$ be a connected reductive group over a local non-archimedean field $K$. Let $\widehat{K}^{nr}$ be the completion of the maximal unramified extension of $K$ and let $\sigma$ denote the Frobenius ...
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### branching laws for $p$-adic representations of reductive groups

There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations? For ...
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### Are all cuspidals induced?

This is a follow-up to this question by Marc Palm asked 7 years ago: Let $K$ be a finite extension of $\mathbb{Q}_p$, and $G$ a reductive group over $K$. Is every irreducible cuspidal ...
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### How to compute (unipotent) radicals?

My question follows some previous one, essentially this one. I want to understand, given an algebraic group $G$ (say linear), how to compute its radical and unipotent radical. The (unipotent) radical ...
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### The reductive $p$-adic group $^{2}\!A_3''$ via Galois decent

I am running into some confusion when trying to explicitly describe the group $^{2}\!A_3''$ (using the naming convention that Tits gives in his Corvallis notes). If anyone can give me any advice, I ...
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### Global integral model for unitary groups

I'm a bit puzzled about the following considerations, and am looking for some explanations or maybe some references about it. Setting: Let $E/F$ be a CM extension of number fields ($F$ being totally ...
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### Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$

Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
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### What is the meaning of the “constant term of Eisenstein series” in terms of Fourier analysis

Let $G$ be a connected, reductive group over $\mathbb Q$, with parabolic subgroup $P = MN$. Let $\pi$ be a cuspidal automorphic representation of $M(\mathbb A)$. For a smooth, right $K$-finite ...
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### If $H \subset \operatorname{GL}(n)$, can we realize $\operatorname{Res}_{K/k} H$ inside $\operatorname{GL}([K : k]n)$?

Let $K/k$ be a finite separable extension. If necessary, we can assume $[K : k] = 2$. Let $H$ be a $K$-closed subgroup of $\operatorname{GL}_n$, and let $\tilde{H} = \operatorname{Res}_{K/k}H$. ...
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### Variety of conjugacy classes

Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
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### What is the defining property of reductive groups and why are they important?

Having read (skimmed more like) many surveys of the Langlands Program and similar, it seems the related ideas apply exclusively to groups that are "reductive". But nowhere, either in these surveys or ...
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### L-packets in the local Langlands correspondence: why finite sets?

Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
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### How should the local Langlands correspondence for general reductive groups take into account different inner forms?

Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
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### The centralizer of a semisimple element which is not contained in any proper parabolic subgroups

Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the ...
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### $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
### Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$
My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus ...