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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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Asymptotic behavior of Shalika germs near non-regular elements

Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
youknowwho's user avatar
3 votes
1 answer
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Factoring out an element of a root subgroup to make a conjugation integral

Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix $$\begin{pmatrix} a & \varpi b \\ c & d \...
Ashwin Iyengar's user avatar
2 votes
0 answers
69 views

Number of rational points of a connected reductive group in a compact subset

Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
Sentem's user avatar
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4 votes
1 answer
136 views

Centralizer of conjugacy classes

Let $\mathrm{G}$ be a complex reductive group and let $\mathrm{O}_g$ be the adjoint orbit of $g\in \mathrm{G}$. I wonder is the centralizer $\mathrm{C}_{\mathrm{G}}(\mathrm{O}_g)$ still a reductive ...
TaiatLyu's user avatar
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4 votes
1 answer
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Do parabolic inductions share a composition factor if and only if the inducing data are associate?

Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
user449595's user avatar
2 votes
1 answer
211 views

Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$. Associated with this ...
a_g's user avatar
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7 votes
1 answer
307 views

Nilpotent orbits of a parabolic subgroup

Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm{...
Alexander's user avatar
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4 votes
2 answers
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Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
Antoine Labelle's user avatar
1 vote
0 answers
91 views

Injection of $G(k)/Z(k)$ into $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
Μάρκος Καραμέρης's user avatar
3 votes
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Representations of a reductive Lie group vie central character and K-types

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
Nandor's user avatar
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$\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group

Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...
Otto's user avatar
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4 votes
3 answers
268 views

Does every nilpotent lie in the span of simple root vectors?

Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span ...
Alexander's user avatar
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5 votes
1 answer
268 views

Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
a_g's user avatar
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3 votes
1 answer
93 views

Are isomorphic maximal tori stably conjugate?

Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
David Schwein's user avatar
7 votes
0 answers
203 views

Chevalley-Solomon formula and Weyl character formula

Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
Antoine Labelle's user avatar
4 votes
2 answers
189 views

References for $K$-orbits in $G/B$

Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
Hadi's user avatar
  • 731
6 votes
2 answers
254 views

Reference for Langlands dual homomorphisms

I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article ...
user449595's user avatar
13 votes
0 answers
470 views

Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?

Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
John Baez's user avatar
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3 votes
0 answers
51 views

One parameter subgroups of reductive algebraic groups

If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
Ekta's user avatar
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2 votes
0 answers
64 views

Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$

Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
youknowwho's user avatar
4 votes
0 answers
190 views

Gelfand-Kazhdan criterion, exposition by Paul Garrett

Here is Paul Garrett's exposition on the Gelfand-Kazhdan Criterion. In page 4 of the exposition, he showed the following lemma. Lemma (Page 4): Let $B, t, S$ be as above and for $\alpha, \beta$ in $...
L-JS's user avatar
  • 151
2 votes
1 answer
172 views

Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma

$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
youknowwho's user avatar
0 votes
1 answer
196 views

Variants of the classical Satake classfication

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] describes as a consequence of the Satake ...
Coherent Sheaf's user avatar
5 votes
0 answers
177 views

Question on the unramified local Langlands conjecture

I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...
Giulio Ricci's user avatar
1 vote
0 answers
62 views

A variation of the dual group of the adjoint group

Let $\mathbf{G}$ be connected reductive group over a $p$-adic field $F$. Denote by $\mathbf{Z}$ the center of $\mathbf{G}$, and $\mathbf{A}$ the maximal split torus of $\mathbf{Z}$ (also called the ...
youknowwho's user avatar
5 votes
1 answer
274 views

Local triviality of torsors for relative reductive groups

Let $X \to S$ be a relative (smooth proper) curve, and $G \to X$ a reductive group scheme. The following two results are well-known: (Drinfeld-Simpson) For arbitrary $S$, if $G$ is defined over $S$, ...
C.D.'s user avatar
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0 answers
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What does it mean for a linear algebraic group to act reductively

I was reading this paper by Baues and on page 918 he mention that $S$ acts reductively on the cochain complex and on page 919 again he mention the word "Since $T$ acts reductively on the complex.....
Uncool's user avatar
  • 181
0 votes
0 answers
77 views

Character of principal series representations of $\mathrm{GL}(n,\mathbb{R})$

I am looking for an explicit form of the character of principal series representations of $\mathrm{GL}(n,\mathbb{R})$. At the moment I am particularly interested in the case $n=2$. A reference would ...
asv's user avatar
  • 21.2k
2 votes
0 answers
109 views

On the character of a representation of $\mathrm{GL}(n,\mathbb{R})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G=\GL(n,\mathbb{R})$. Given a continuous admissible irreducible representation of $G$ in a Frechet (or a Banach) space. Then its character ...
asv's user avatar
  • 21.2k
0 votes
0 answers
70 views

Embeddings of unitary groups over $\mathbb{Q}$

$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation: suppose we have an Hermitian vector space $V=K^3$ of matrix $$ J=\begin{pmatrix}& & \...
Fra's user avatar
  • 91
3 votes
1 answer
239 views

Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)

I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled. So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
youknowwho's user avatar
7 votes
2 answers
256 views

Holomorphic discrete series vs. discrete series

(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
youknowwho's user avatar
2 votes
0 answers
122 views

Classification of restricted Lie algebras of reductive groups

$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
Martin Ortiz's user avatar
3 votes
0 answers
80 views

Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?

I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof). Suppose $G$ is a ...
David Schwein's user avatar
1 vote
0 answers
44 views

Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?

In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
C.D.'s user avatar
  • 545
3 votes
1 answer
119 views

Connected components of a spherical subgroup from spherical data?

This question is in a similar spirit to this one by Mikhail Borovoi. Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety. Losev proved that the spherical $...
Spencer Leslie's user avatar
3 votes
1 answer
312 views

Does a quasi-split reductive group scheme admit a maximal torus?

Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
C.D.'s user avatar
  • 545
2 votes
1 answer
213 views

Stabilizer of a Levi subgroup in the Weyl group and its quotient

(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.) For simplicity, let $G$ be a connected reductive ...
youknowwho's user avatar
4 votes
0 answers
145 views

Representation rings of disconnected reductive groups

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of ...
Stefan  Dawydiak's user avatar
1 vote
0 answers
102 views

Does $\tilde{\mathrm E}_{6,3}^{(2)6}$ exist over a p-adic field?

Does a form of $\tilde{\mathrm E}_6^{(2)}$ with this Satake-Tits diagram exist over a p-adic field?
Daniel Sebald's user avatar
4 votes
1 answer
176 views

Canonicality of group of integers for reductive groups over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ and $\mathcal{O}_K$ be ...
user267839's user avatar
  • 6,034
1 vote
0 answers
101 views

Structure theory of Schubert varieties (extend results from semisimple groups to reductive)

The lecture notes Borel–Weil–Bott theorem and geometry of Schubert varieties by Shrawan Kumar present a concise summary of major results on cohomology of flag varieties $G/B$ for $G$ semisimple, ...
user267839's user avatar
  • 6,034
3 votes
0 answers
90 views

Langlands parameters and Weyl group actions

Let $F$ be a $p$-adic field and $\mathbf{G}$ a connected reductive group over $F$, assumed to be quasi-split. Let $\mathbf{T}$ be a maximal split torus of $\mathbf{G}$ and $\mathbf{P}=\mathbf{M}\...
youknowwho's user avatar
0 votes
1 answer
128 views

Calculating relative root systems

Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...
Ann's user avatar
  • 43
4 votes
0 answers
107 views

Reference Request: Classification of spherical varieties by "Weyl group invariant fans"

Apologies in advance for the vague question. Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...
Dcoles's user avatar
  • 51
3 votes
2 answers
205 views

Reductive groups over arbitrary fields with disconnected relative root systems

Let $\mathbf{G}$ be a connected reductive group over a field $k$, not necessarily algebraically closed. Let $\Phi$ be the relative root system for $\mathbf{G}$ with respect to $k$, and assume that $\...
Gina's user avatar
  • 131
4 votes
1 answer
111 views

Cohomology of Deligne-Lusztig variety associated to Coxeter element

Determining the individual ($l$-adic) cohomology groups of Deligne-Lusztig varieties has only been done for the general linear group and for some other very specific cases (as far as I know). However, ...
EJB's user avatar
  • 153
2 votes
0 answers
88 views

Torsion equivariant cohomology of reductive groups

Let $G$ be a reductive group with maximal torus $T$. One knows that the equivariant cohomology ring of a point with rational coefficients is $\mathbb{Q}[X^*(T)]^W$, and also there is an equivariant ...
user333154's user avatar
0 votes
0 answers
99 views

Tempered representations and unramified principal series

For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...
InteresetingStuff's user avatar
2 votes
1 answer
82 views

Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$

I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...
D_S's user avatar
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