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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Clarification on the definition of a smooth affine scheme over an integral domain
$\DeclareMathOperator{\Spec}{Spec}$
The following is from Bruhat and Tits article Groupes Reductifs sur un Corps Locale II. $A$ is an integral domain. Here $A$-scheme means "affine $A$-scheme," and $...
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Automorphic quotients for inner forms or $GSp(4)$
For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups:
\begin{equation}
\mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star
\left(
\begin{array}{cc}
& 1 \\
1 &...
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Special fibers of parahorics
Tits' Corvallis article introduces a map on special fibers of group schemes associated to the elements fixing sets pointwise in a building from $\bar{\mathcal{P}_\Omega}$ to $\bar{\mathcal{P}_{\Omega'}...
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Integral smooth model of unramified reductive groups
My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$...
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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-...
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$GSp(4)$ vs $PSp(4)$
After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, ...
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Which groups can have $GSp(4)$ as local component?
In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G_v$ (where $v$ are the places of $F$) are well known. Obviously a group ...
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What condition makes unitary reductive group unramified?
I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...
- 1,759
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Description of the center of a reductive group using absolute and relative roots
Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...
- 6,180
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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 ...
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Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
- 6,180
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Choosing canonical representatives of Weyl group elements, some questions
Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. ...
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Absolute and Relative Coroots
$G$ is a connected reductive group over a field $k$. $T$ is a maximal torus and $S \subset T$ is a maximal $k$-split torus. We have an embedding $X_*(S) \hookrightarrow X_*(T)$. Is it true that if $\...
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Does a reductive group over $K$ always have a torus that becomes maximal split over $L$?
Let $K$ be a field, let $L$ be a field containing $K$, and let $G$ be a reductive group over $K$. Does there always exist a torus $T$ of $G$ so that $T_{/L}$ is a maximal split torus of $G_{/L}$? If ...
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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 ...
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Why is the radical of a reductive group equal to the connected component of the center?
If $G$ is a connected reductive group over a perfect field $k$ (The definition given in Milne's "Algebraic Groups": $G$ is a connected group variety containing no non-trivial connected unipotent ...
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$G$ 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
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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
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Endoscopic group that is not a subgroup
The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he ...
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Centralizers of subtori in reductive groups, derived subgroups
Let $G$ be a split, almost-simple connected reductive group over a field $F$ with split maximal torus $T$. I am trying to understand precisely the groups $[G_{\alpha}, G_{\alpha}]$, where $\alpha$ is ...
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A reductive group has a quasi-split inner form
Let $G$ be a connected, reductive group over a field $k$. Let $\Gamma = \textrm{Gal}(k_s/k)$. I think my question is better suited using the classical language: think of $G$ as an affine $\overline{...
- 6,180
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Central isogenies differ by an element of the maximal torus
Let $G, G'$ be connected, reductive groups over an algebraically closed field $k$, and let $T$ be a maximal torus of $G$. A central isogeny is a surjective morphism of algebraic groups $\phi: G \...
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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
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The unique maximal compact subgroup of a torus
Let $T$ be a torus over a $p$-adic field $F$. Let $q = f(F/\mathbb{Q}_p)$, and normalize the absolute value $| \cdot |$ on $F$ so that a uniformizer has value $\frac{1}{q}$.
Let $X(T)_F$ be the ...
- 6,180
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Bases of a relative root system are parameterized by what?
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus of $G$, and $\Phi = \Phi(T,G)$ the set of roots of $T$ in $G$. The bases $\Delta$ of $\Phi$ ...
- 6,180
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Iwasawa decomposition and compact subgroups
Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context ...
- 6,180
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Simple Proof that a Reductive Group is Unimodular?
Let $G$ be a connected, reductive group over a local field $k$ of characteristic zero. I thought of a simple proof that $G(k)$ is unimodular, but I realize it is almost certainly wrong: $G(k)$ is ...
- 6,180
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There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\...
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Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup
Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$.
My question is about the two ways ...
- 309
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Generalisations of Weyl's construction of irreducible representations
For the moment we work over the complex numbers.
Suppose that $G = \mathrm{SL}(V)$, or $G = \mathrm{Sp}(V)$, or $G = \mathrm{SO}(V)$.
Weyl gave explicit constructions of irreducible representations of ...
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Twisted Levi of a quasi-split group that is not quasi-split
Let $F$ be, say, a non-archimedean local field. Let $G$ be a connected reductive (can be assumed simply connected) quasi-split group $G$ over $F$. Let
$X\in\operatorname{Lie}G$ be semisimple and $G_X:...
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How does a Haar measure on $N$ arise from root subgroups?
Let $G$ be a connected, reductive group, split over a local field $F$. Let $B = TU$ be a Borel subgroup defined over $F$ with maximal torus $T$ and unipotent radical $U$. Let $P$ be a parabolic ...
- 6,180
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How can this argument calculating the Haar measure on a parabolic subgroup be generalized to the non-split case?
Let $\mathbf G$ be a connected, reductive group over a local field $F$. Assume there is a maximal torus $\mathbf T$ which is split over $F$. Let $\mathbf P$ be a parabolic subgroup of $\mathbf G$ ...
- 6,180
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What's the point of a Whittaker model?
Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ ...
- 6,180
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votes
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answer
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Type of place versus type of unitary group
Let $F$ be a totally real number field, $E$ a totally imaginary quadratic extension over $E$, and $V$ an hermitian $n$-dimensional vector space over $F$. I assume $n=2m$ is even. Let $U$ be a unitary ...
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Spherical building at infinity for $SL(n, \mathbb{Q}_p)$
Is there somewhere I can read about the spherical building at infinity for $SL(n, \mathbb{Q}_p)$?
I'm looking for something with lots of explicit examples and computations. (I have books on the ...
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Why are spherical representations subquotients of unramified principal series?
I'm trying to learn the basics of the representation theory of $p$-adic groups and I'm stuck on a few things:
Let $G$ is a connected split reductive group over a non-archimedean local field $F$, and $...
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Understanding the structure of unitary groups
I would like to understand precisely the structure of unitary groups.
Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (i.e. the group of ...
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On existence of a certain irreducible character of $SL(5, q)$
Let $q=p^f$ be a prime power such that $q \equiv 1 \pmod 5$. According to the list of irreducible (complex) character degrees of $SL(5, q)$ in Frank Luebeck's homepage (here), $SL(5, q)$ has 20 ...
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Integral model of a reductive group over a prime field
Let $p$ be a rational prime, $\mathbb{Z}_p$ the ring of $p$-adic integers, and $k$ an algebraic closure of the residue field $\mathbb{F}_p$. Suppose $G$ is an affine smooth group scheme over $\mathrm{...
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Why is Mumford's GIT-quotient so effective?
According to remark 6.14 in Shigeru Mukai's An introduction to invariants and moduli (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine ...
- 1,980
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Unipotent characters of (disconnected) centralizers of semisimple elements: Why these two definitions are equivalent?
Assume that $\mathcal{G}$ is a simple simply-connected algebraic group over $k$, where $k$ is algebraic closure of a finite field of characteristic $p>0$, and $F$ is a Frobenius endomorphism. Let $(...
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Reference Request: (Borel) Iwahori Spherical Representations
I was told that Borel had a result about Iwahori Spherical automorphic representations being upper-triangular (/semistable). Where can I find this?
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Iwahori subalgebra as maximal solvable
I think the following is true, but haven't came up with a proof myself. Thanks in advance!
Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ ...
- 1,856
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Automorphic forms annihilated by $I_1$ but not $I_2$ for finite codim ideals $I_1 \subsetneq I_2$
Suppose $G$ is a connected real reductive Lie group, $\mathfrak{g} = \text{Lie}(G)$, and $\mathcal{Z} = \mathcal{Z}[U(\mathfrak{g})]$ the center of the universal enveloping algebra of $\mathfrak{g}$.
...
- 233
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Correspondence between dual center and linear characters of finite reductive group
Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...
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Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...
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Definition of discrete spectrum and continuous and basic properties
I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...
- 777
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Relation between unipotent cuspidal representations and cuspidal local systems
This could well be a question for reading suggestion. Hope it's not too bad and thanks a lot.
So the question is as in the title. What are the relations between the notion of unipotent cuspidal ...
- 1,856
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votes
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Symmetries of the flag variety
Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma (T\...
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