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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8
votes
1answer
159 views

Union of Schubert cells being affine

Let $k$ be a field of characteristic zero, $G$ be a reductive group with a Borel $B$ and $\mathcal{F}:=G/B$ the associated flag variety. Let $W$ be the Weyl-group of G. Then let $S \subset W$ and $Z=\...
2
votes
1answer
123 views

Conjugacy of elements in a parabolic subgroup

Let $G$ be a complex connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. My question is the following: if $g$ and $h$ are elements of $P$ which are conjugate as elements of $G$,...
2
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0answers
52 views

Residual and continuous spectra of $L^2( G(k) \backslash G(\mathbb A) ; \omega)$, and cuspidal automorphic data

Let $G$ be a connected, reductive group over a number field $k$. Let $\mathbb A$ be the ring of adeles of $k$, $\omega$ be unitary character of $Z_G(\mathbb A)/Z_G(k)$, and $V = L^2(G(k) \backslash G(...
10
votes
1answer
188 views

Bialynicki-Birula decompositions and fixed points

I was reading Luna's paper Toute variété magnifique est sphérique and stumbled on a few facts about Bialynicki-Birula decompositions and fixed points that I don't understand. Here is the setup. Let $...
5
votes
1answer
86 views

Element of Weyl chamber contracting $\mathbb{A}^n_k$ to a point

Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0. Fix a Borel subgroup $B$ and a maximal torus $T \subset B$. Let $P \subset G$ be a parabolic subgroup ...
4
votes
1answer
105 views

Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
2
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0answers
91 views

The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
2
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0answers
40 views

$p$-adic Harish-Chandra character of a stable virtual character

Let $F$ be a $p$-adic field and let $G$ be a reductive group over $F$. Associated to an irreducible admissible representation of $\pi$ of $G(F)$, we have a distribution character $\Theta_{\pi}$ ...
5
votes
1answer
97 views

Definition of functions in the induced space from parabolic induction

Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, ...
3
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1answer
141 views

Quotienting $G(\mathbb{Q})_{+}$ by $G^{\text{sc}}(\mathbb{Q})$ and inner forms

Let $G/\mathbb{Q}$ be a connected reductive group, let $G^{\text{ad}}$ be the adjoint group, let $G^{\text{der}}$ be the derived group and let $\rho\colon G^{\text{sc}} \to G^{\text{der}}$ be the ...
3
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1answer
141 views

Reference request - Weyl's integration formula

Is there a reference discussing in an organized way (with a proof) the Weyl integration formula for a reductive group over a local field (Archimedean or not), expressing the Haar integral on the group ...
2
votes
1answer
154 views

Highest weight vector as a global section of an affine scheme

Let $G$ be a connected, reductive quasi-split group over a field $k$, acting on an afffine $k$-variety $X$. Let $B = TU$ be a Borel subgroup of $G$ with maximal torus $T$ and unipotent radical $U$. ...
3
votes
2answers
226 views

Rosenlicht's theorem and fundamental domain for unipotent group acting on $\mathbb A_k^n$

I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very ...
4
votes
1answer
129 views

How to determine a highest weight corresponding to a parabolic subgroup?

Let $G$ be a simply connected, semisimple algebraic group over $\mathbb C$ with maximal torus $T$ and Borel subgroup $B$ containing $T$. If $(V,\pi)$ is an irreducible representation of $G$, then $(V,...
5
votes
1answer
215 views

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 ...
1
vote
1answer
191 views

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 ...
5
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0answers
206 views

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 ...
8
votes
1answer
248 views

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 ...
1
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0answers
64 views

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 ...
2
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0answers
55 views

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 ...
3
votes
1answer
218 views

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 ...
3
votes
1answer
122 views

A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”

I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K_\...
2
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0answers
69 views

Classification of finite dimensional representations of split complex reductive groups

Finite dimensional, irreducible representations of simply connected, complex semisimple algebraic groups can be classified by their highest weight. I was wondering if there is an analogous ...
3
votes
1answer
152 views

split integral model of a reductive group

Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$. Does there exist a ...
4
votes
1answer
98 views

Bruhat order and positive roots made negative

Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s_{\alpha} : \alpha \in \...
6
votes
1answer
118 views

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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0answers
68 views

Connection between global and local notions of a cuspidal representation

Let $k$ be a number field, and $G$ a connected, reductive group over $k$. Let $\omega$ be a unitary character of $Z_G(\mathbb A_k)/Z_G(k)$. An irreducible subspace $(\pi, V)$ of $L^2(G(k) \backslash ...
6
votes
1answer
142 views

Number of irreducible representations of $SO_3(\mathfrak{o}/\mathfrak{p}^l)$

$\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G_l$ denote the finite group $\...
5
votes
0answers
97 views

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 ...
4
votes
1answer
112 views

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 ...
6
votes
2answers
314 views

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 ...
4
votes
0answers
66 views

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 ...
6
votes
1answer
214 views

Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators

I am reading Langlands' pape Euler Products and have a few questions. Let $G$ be a split adjoint semisimple group over $\mathbb Q$. If $p$ is a place of $\mathbb Q$, finite or infinite, let $G_{\...
2
votes
0answers
59 views

Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$

Let $G$ be an adjoint semisimple group over $\mathbb Q$ with parabolic subgroup $P = MN$ in good position relative to a compact subgroup $U= \prod\limits_v K_v$ of $G(\mathbb A)$. Let $L$ be the ...
3
votes
0answers
134 views

Elliptic Maximal Tori and Elliptic Elements

I would be grateful if someone could provide a reference/proof of the following fact (or give a counterexample if I've misunderstood and it's false!) Let $G$ be a reductive group over a field $F$ (in ...
5
votes
2answers
216 views

When is compact induction cuspidal?

Let $G=GL_2(\mathbb{Q}_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$. Question 1: Is $\mathrm{ind}_K^G \rho$ cuspidal? Here ...
11
votes
1answer
363 views

Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits

Let $G$ be a connected reductive group over an algebraically closed field $k$. By the Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $...
4
votes
1answer
188 views

Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
4
votes
1answer
108 views

Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...
1
vote
0answers
56 views

Question on the proof that the Jacquet module preserves admissibility

Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
3
votes
0answers
156 views

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 ...
2
votes
1answer
136 views

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 $\tilde{H} = \operatorname{Res}_{K/k}H$. ...
3
votes
2answers
289 views

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 ...
27
votes
3answers
1k views

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 ...
7
votes
1answer
211 views

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 ...
7
votes
1answer
239 views

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 ...
2
votes
1answer
241 views

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 ...
5
votes
1answer
169 views

$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ 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 ...
1
vote
0answers
81 views

Embedding of discrete series

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 ...
2
votes
0answers
186 views

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 ...

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