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
173 questions with no upvoted or accepted answers
13
votes
0
answers
509
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 ...
- 22.3k
13
votes
0
answers
405
views
Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient
$\DeclareMathOperator{\GL}{GL}$
$\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
- 6,180
11
votes
0
answers
316
views
Mysterious "raison d'être" of filtrations of congruence subgroups
I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$.
Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
- 5,424
10
votes
0
answers
373
views
Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
- 6,180
9
votes
0
answers
295
views
Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
- 1,666
8
votes
0
answers
228
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 $(\operatorname{Sym}^\bullet V)^W$...
- 3,446
8
votes
0
answers
331
views
$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?
Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...
- 455
8
votes
0
answers
267
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 1,856
8
votes
0
answers
265
views
Chevalley restriction theorem: group vs lie algebra version
Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and ...
- 689
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
7
votes
0
answers
230
views
Field extensions that preserve given cohomology classes
Let $G$ be a connected reductive group over $\mathbb{Q}$ and let $\operatorname{Ker}^1(\mathbb{Q},G) \subset H^1(\mathbb{Q},G)$ be the subset of classes that are trivial at all places. I am trying to ...
- 492
7
votes
0
answers
291
views
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 ...
- 6,180
6
votes
0
answers
86
views
Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
- 709
6
votes
0
answers
110
views
subalgebra of invariants for a reductive subgroup
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
- 887
6
votes
0
answers
468
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 ...
- 927
6
votes
0
answers
236
views
When is an irreducible unramified principal series representation $K$-spherical?
Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$.
Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
- 6,180
6
votes
0
answers
217
views
Dimension of space of K-fixed vectors
If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...
- 91
6
votes
0
answers
308
views
Is there a list of the inner forms of the quasisplit groups over local and global fields of characteristic 0?
From what I've gathered from trying to learn the classification of reductive groups, the classification of semisimple groups over a local or global field $F$ of characteristic 0 proceeds in several ...
6
votes
0
answers
278
views
$G$ is quasisplit at almost all places
Let $G$ be a connected, reductive group over a global field $k$. I am trying to understand why $G_v = G \times_k k_v$ is quasisplit for almost all places $v$ of $k$. There are several equivalent ...
- 6,180
6
votes
0
answers
1k
views
Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180
6
votes
0
answers
239
views
Can we classify reductive group schemes over curves
Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...
- 61
6
votes
0
answers
244
views
Zariski closure of orbits of real groups on complex flag manifolds
Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
- 27.8k
5
votes
0
answers
230
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 ...
- 151
5
votes
0
answers
271
views
Can closure of an orbit under a reductive action contain infinitely many orbits?
I posted this on math.se a week ago, currently it has 23 views and no other feedback.
Here on MO there are several questions about orbit closures but I could not find anything about what I need.
To be ...
- 18.4k
5
votes
0
answers
122
views
Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$
Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
- 71
5
votes
0
answers
1k
views
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_{...
- 6,180
5
votes
0
answers
210
views
rational cohomology of classifying spaces of complex reductive Lie groups
I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
- 2,928
5
votes
0
answers
162
views
Definition of the homomorphism $\textrm{Gal}(k_s/k) \rightarrow \textrm{Aut}(\psi_0(G))$
Let $G$ be a connected, reductive group over a field $k$. Identify $G$ with its $\overline{k}$-points. Let $\Gamma = \textrm{Gal}(k_s/k) = \textrm{Aut}(\overline{k}/k)$. Let $B$ be a Borel subgroup ...
- 6,180
5
votes
0
answers
500
views
How to think about non-connected reductive groups
Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...
- 26k
5
votes
0
answers
223
views
Decomposition of k-split tori of p-adic reductive groups
Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
$$...
- 991
5
votes
0
answers
253
views
Do there exist pseudo-reductive (but not reductive) groups of small dimension?
I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...
- 181
4
votes
0
answers
87
views
Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,622
4
votes
0
answers
237
views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
- 57
4
votes
0
answers
87
views
Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
4
votes
0
answers
213
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 $...
- 43
4
votes
0
answers
168
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 ...
- 413
4
votes
0
answers
118
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 ...
- 63
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
4
votes
0
answers
108
views
When does the null-cone consist entirely of eigenvectors?
Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$.
For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
- 18.4k
4
votes
0
answers
213
views
What are the good maximal compact subgroups in $p$-adic unitary groups?
Let $E/\mathbb Q_{p}$ be a quadratic extension and let $V$ be an $n$-dimensional $E$-hermitian space. Denote the hermitian form by $(\cdot,\cdot):V\times V \rightarrow E$. Let $G := \mathrm{U}(V)$ be ...
- 769
4
votes
0
answers
149
views
Centraliser of a maximal $k$-split torus of a reductive $k$-group
Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...
- 61
4
votes
0
answers
79
views
How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?
I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...
- 6,180
4
votes
0
answers
98
views
Is the union of conic orbits for a reductive group Zariski closed?
Let $G$ be a reductive group over an algebraically closed field $k$ of characteristic $p>0$. If $V$ is a rational $G$-module then we can define the Hilbert nullcone $\mathcal{N}(V)$ to be the zero ...
- 599
4
votes
0
answers
868
views
Definition of Iwahori subgroup independently of the Bruhat-Tits building
Let $G$ be the points of a connected, semisimple algebraic group over a $p$-adic field $k$. To make life easy, let's assume the underlying group scheme is simply connected. The Bruhat-Tits building $...
- 6,180
4
votes
0
answers
211
views
Books on integration on semisimple Lie groups
Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...
- 1,879
4
votes
0
answers
180
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 ...
- 755
4
votes
0
answers
105
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 ...
- 6,180
4
votes
0
answers
313
views
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 ...
- 960
4
votes
0
answers
260
views
Families of Hessenberg varieties for $GL_n$
In short, the question is
What do we know about the sheaf $\pi_*\underline{\bar{\mathbb{Q}}_{\ell}}$ given by the family of (very original, see below) Hessenberg varieties for $GL_n$? As a sum of ...
- 1,856
4
votes
0
answers
288
views
Meaning of a highly ramified character for reductive groups
Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in ...
- 6,180