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
263
questions
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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 ...
1
vote
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 ...
