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
448 questions
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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,...
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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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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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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_\...
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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 ...
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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 ...
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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 \...
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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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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 ...
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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 $\...
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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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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_{\...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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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What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?
Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
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Some questions about cuspidal representations and automorphic representations
My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{...
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Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is "uniform" across $P \overline{N}$
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
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Do we have $K \cap P = (K \cap M)(K \cap N)$?
Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
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Some confusion about weights and roots in parabolic root systems
I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...
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How to determine the unramified character corresponding to an unramified Langlands parameter?
Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
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Volume of a double class of a parahoric subgroup
Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...
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If $f \in \operatorname{c-Ind}_Q^P \sigma$, then $f|_N$ is compactly supported modulo $Q \cap N$?
There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true.
The thing I can't figure ...
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$G(k)/H(k)$ as a submanifold of $G/H(k)$
Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an ...
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The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$
Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P_0$ containing a maximal split torus $A_0$. Let $W = N_G(A_0)(k)/Z_G(A_0)(k)$ be the Weyl group, and $S \...
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If $\Pi$ and $\Sigma$ agree at almost all places, then the central character of $\Pi$ corresponds to $\operatorname{Det} \Sigma$
Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at ...
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Reference for parabolic root systems
Let $G$ be a connected reductive group with maximal split torus $A_0$, and $P = MN$ a parabolic subgroup with Levi $M$ containing $A_0$. Let $A_M$ be the split component of $\mathfrak a_M^{\ast} = X(...
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Adjacent parabolic subgroups and proportionality to $\alpha^{\vee}$
Let $P = MN$ be a parabolic subgroup of a $p$-adic reductive group $G$ with split component $A_M$. There is bijection from the set of parabolic subgroups of $G$ with Levi $M$ and the chambers of $\...
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The convergence of the factor $\gamma(P)$ in the Iwasawa decomposition
Let $G$ be a connected, reductive group over a $p$-adic field $k$, $A_0$ a maximal split torus of $G$, and $P = MU$ a parabolic subgroup with $M$ containing $A_0$. Let $\overline{P} = M \overline{U}$ ...
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