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Questions tagged [reductive-groups]

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Dominant Weyl chamber and co-adjoint orbit for Riemann symmetric pair

Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}_0$. Suppose that $\sigma$ is an involutive automorphism of $K$ which defines a symmetric pair $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
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1answer
48 views

Does every simply-connected reductive group have trivial Galois cohomology?

Let $G$ be a linear algebraic group over a field $k$ with separable closure $k^s$ and absolute Galois group $\Gamma\!_k$. Consider the Galois cohomology group $$ H^1(\Gamma\!_k,G(k^s)). $$ This group ...
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18 views

Basic properties of the adelic matrix norm

This is from Moeglin and Waldspurger's book Spectral Decomposition and Eisenstein Series. Here $G$ is a linear algebraic group over a number field $k$. One fixes a closed embedding $i'$ of $G$ into ...
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1answer
27 views

Finite volume of $G(k) \backslash G(\mathbb A)$ implies that a real-valued character trivial on $G(k)$ is unramified

Let $G$ be a connected, reductive group over a number field $k$, and $X(G)_k$ the group of $k$-rational characters of $G$. We define $G(\mathbb A)^1$ to be the subgroup of $g \in G(\mathbb A)$ such ...
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20 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 ideles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
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1answer
34 views

centralizers of nilpotent element in simple Lie algebra and associated Levi subalgebra

Let $\{e,h,f\}$ be a $sl_2$ triples in simple Lie algebra $\mathfrak g$ with usual relations $[h,e]=2e,~ [h,f]=-2f,~[e,f]=h$. Then the centralizer of $e$ is $\mathfrak g_e=\{b:[b,e]=0\}$ and the ...
2
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1answer
61 views

When does the Borel subgroups of affine (linear) algebraic groups come from Borel subgroups of general linear group?

Let $G=SL_n(\mathbb C), $ or $SO_n(\mathbb C)$ or $Sp_{2n}(\mathbb C)$. Then it is known that $B\cap G$ is a Borel subgroup of $G$ where $B$ is the Borel subgroup of $GL_n$ (for the right $n$) of ...
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1answer
41 views

Intersection of Levi subgroups is a Levi

Let $P_0 =M_0N_0 \subset G$ be a minimal parabolic subgroup of a connected, reductive group with maximal split torus $A_0$. Let $\Phi^+ = \Phi(A_0,N_0)$ and $\Delta \subset \Phi^+$ be a set of simple ...
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1answer
56 views

A couple of questions about admissible representations and $K$-finite vectors

Let $G$ be the points of a connected, reductive group over $\mathbb R$, and let $K$ be a maximal compact subgroup of $G$. Let $(\pi,V)$ be a continuous representation of $G$ on a Hilbert space for ...
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2answers
60 views

Computing central isogenies of reductive groups

Is there any easy way to figure out what the groups $GL_n/ \mu_k$ ($\mu_k$ central) are depending on $n$ modulo $k$, or something? For example $GL_3/ \{ \pm \mathrm{Id}_3 \}$ I suspect it is $\cong ...
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24 views

Comparing the relative and absolute Bruhat decompositions for quasi-split reductive groups

Let $G$ be a connected, reductive group over a field $k$. Assume $G$ is quasi-split. Let $A_0$ be a maximal split torus of $G$ with centralizer $T$, and let $B$ be a minimal parabolic (Borel) ...
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28 views

Technical question on adelic quotient of centralizers

Let $G$ be a connected reductive group over $k =\mathbb Q$ with split component $A_G$. Let $M$ be a $k$-Levi subgroup of $G$ with split component $A_M$. Let $\gamma$ be a semisimple element of $M(k)...
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71 views

Semisimple elements of a parabolic subgroup are contained in some Levi

Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $...
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28 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 ...
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29 views

Weil group and conjugacy classes of cocharacters

We have the following setup: $G$ a compact reductive group over $\mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$ In Variétiés de Shimura Lemma 1.2.4 p....
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37 views

Elementary question in reductive group

I am very sorry for asking some elementary question. Let $G$ be a reductive group over a number field $F$ and $N$ a unipotent radical of some parabolic subgroup of $G$. Then I am wondering whether $...
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42 views

The definition of a coroot for non-split reductive groups?

Let $G$ be a connected, reductive group over a field $k$, and $A_0$ a maximal split torus of $G$. Let $\Phi = \Phi(G,A_0)$ be the set of roots of $A_0$ in $G$. Then the $\mathbb R$-linear span $\...
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1answer
23 views

Principal series for $\operatorname{GL}_2$, question about an exact sequence

I have a question about Proposition 7.2 of these notes by Gordon Savin. Here $G = \operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, and $\...
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1answer
29 views

Incorrect modulus character computation in Casselman's notes?

In Proposition 6.3.3 in Casselman's notes on representation theory of $p$-adic groups, I believe there is an error in the statement of the Proposition. Let $G$ be the points of a connected, reductive ...
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1answer
28 views

Union of Bruhat cells is open in $G$

Let $G$ be (the points of) a connected, reductive group over a local field $k$ with maximal split torus $S$, minimal parabolic $P_0$, and Weyl group $W = N_G(S)/Z_G(S)$. Let $\Delta$ be the ...
4
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1answer
82 views

Brauer-Severi varieties as quotients of forms of $\text{GL}_2$

Let $L/F$ be a finite galois extension of fields, with galois group $\Gamma$. Let $X$ be a variety over $F$ such that $X_L \cong \mathbb{P}^1_L$ over $L$, corresponding to a cohomology class $\alpha \...
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16 views

Adjacent parabolic subgroups and proportionality of $H_P(n')$ 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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52 views

Are permutation matrices over reals reductive?

Question Let $H$ denote the subgroup of permutation matrices of $\operatorname{GL}(n,\mathbb{R})$. Is $H$ a reductive group? My Work I know that $H$ is a linear algebraic group being a finite ...
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1answer
34 views

$W(G,T)$-invariant form induces a $W(G,A_0)$-invariant form

Let $G$ be a connected, reductive group over a $p$-adic field $k$, with maximal split torus $A_0$ contained in a maximal torus $T$. Let $X = X(T)$, and $V = X \otimes \mathbb R$. Let $W = W(G,T)$ ...
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1answer
72 views

Parabolic Induction in Stages

$\DeclareMathOperator{\Ind}{Ind}$Let $G$ be a connected, reductive group over a local field $k$. Let $P_{\ast} = M_{\ast}N_{\ast} \subseteq P = MN$ be parabolic subgroups, standard with respect to ...
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1answer
42 views

Split component is the identity component of the intersection of the kernels of the roots

Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$, and $T$ a maximal torus of $G$ which contains $S$ and which is defined over $k$. Let $A$ be the ...
0
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1answer
29 views

Image of the Harish-Chandra map is of finite index?

Let $G$ be a connected, reductive group over a global or local field $k$ with absolute value $| \cdot |$. Let $X(G)_k$ be the group of rational characters of $G$ which are defined over $k$, and let $...
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45 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 ...
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43 views

Definition of the Hilbert space $L^2(G/Z,\nu)$

Let $G = \operatorname{GL}_n(\mathbb Q_p)$, and let $Z$ be the center of $G$. Let $\nu: Z \rightarrow \mathbb C^{\ast}$ be a continuous, unitary character of $Z$. Then $L^2(G/Z,\nu)$ is defined to ...
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60 views

Irreducible admissible representation of a minimal parabolic is finite dimensional

Let $k$ be a nonarchimedean locally compact field, $G$ a connected reductive group over $k$ with minimal parabolic subgroup $P = MN$. Let $\sigma$ be a smooth representation of $M$. In chapter 3 of ...
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1answer
40 views

Unramified principal series and basis for $I(\chi)^B$

I am reading The unramified principal series of $p$-adic groups by W. Casselman and am stuck on some basic details. $G$ is a connected, reductive group over a $p$-adic field, $P = MN$ is a minimal ...
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38 views

Proving that for a reductive group $G$, $GV_r$ is closed

Let $G$ be a reductive group over an algebrically closed field $\mathbb{K}$ acting on a finite dimensional vector space $V$. Let $\lambda$ be a one-parameter subgroup of $G$. Let's consider for $r \in ...
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0answers
16 views

Exactness of the functor $(\quad)^G$ for $G=k^\times$

Suppose $G=k^\times$ where $k$ is an algebraic closed field of characteristic $0$(I'm not sure whether this assumption is necessary). Then show that $(\quad)^G$ is an exact functor i.e. given an exact ...
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1answer
168 views

The derived group of a reductive connected group is semisimple

I am actually having some trouble in demonstrating the following result in the book Lie algebras and Algebraic groups of P. Tauvel and R. W. T. Yu: Let $G$ be a connected reductive algebraic group. ...
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64 views

On the right-invariance of the Reynolds Operator

Let $K$ be a field of characteristic $0$ (and maybe algebraically closed...) and $G$ be a linear algebraic group, that is a group $G$ which is an affine variety where the multiplication and inversion ...
4
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1answer
111 views

$\operatorname{dim}V^G = \operatorname{dim}(V^\ast)^G$, or $G$ linearly reductive implies $V^G$ dual to $(V^\ast)^G$

NOTE: The beginning part, which might sound like a lot of complicated theory to some, is just for completeness. I might just need $\operatorname{dim}V^G = \operatorname{dim}(V^\ast)^G$, which anyone ...
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86 views

Contragredient of a cuspidal representation

Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal. A ...
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1answer
41 views

Integral converges “$UZg$ is closed in $G$”

This is a followup to my previous question here. Let $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $Z$ be the center of $G$, and let $f: G \rightarrow \mathbb{C}$ be a function which is ...
2
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1answer
77 views

Kirillov model: integral vanishes for some $n$, or for all large $n$?

Let $G = \operatorname{GL}_2(F)$ for a $p$-adic field $F$, and $(\pi,V)$ an irreducible, admissible representation of $G$. Let $\tau$ be a continuous character of $F$. For $n \geq 1$ and $\xi \in V$,...
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1answer
57 views

Compactly supported modulo the center implies convergence over the unipotent radical

Let $G$ be a quasisplit reductive group over a $p$-adic field $F$ with center $Z$, and let $\omega: Z \rightarrow \mathbb{C}^{\ast}$ be a quasicharacter. Let $f: G \rightarrow \mathbb{C}$ be a ...
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0answers
32 views

Does a one-parameter subgroup of a reductive group 'act reductively'?

For the present question I am only interested in the case where the underlying field is $\mathbb{C}$. Hilbert fourteenth can be stated as follows: Let $V$ be an algebraic variety and $G$ a linear ...
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1answer
90 views

Definition of the unitary group via a cocycle on $\operatorname{GL}_n$

$\DeclareMathOperator{\GL}{GL}$ I thought I understood something, but now I don't think I do. Let $E/F$ be a separable quadratic extension with nontrivial automorphism $\tau$. Let $\Gamma_E = \...
3
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0answers
65 views

Relating height to length in a root system

I have a question about a proposition from Casselman's notes on representation theory. Let $\mathfrak a$ be a finite dimensional vector space, and $\Sigma$ a reduced root system in the dual $\...
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1answer
26 views

The structure of $\textrm{c-Ind}_H^G(\sigma)$

Let $G$ be a group of td type, and $H$ a closed subgroup. Let $(\sigma,W)$ be a a smooth representation of $H$. I have always had trouble thinking about the compactly induced representation $\textrm{...
1
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1answer
37 views

Equivalent definitions of compactly supported in the case of induced representation

Let $G$ be a topological group of totally disconnected (td) type, meaning every neighborhood of $1_G$ contains a compact open subgroup. Let $H$ be a closed subgroup of $G$. Then $H$ is also of td ...
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1answer
116 views

$C_c^{\infty}(G) \rightarrow \textrm{c-Ind}_H^G(\delta_H)$ is surjective

Let $G$ be a unimodular topological group of totally disconnected type, and $H$ a closed subgroup of $G$. Let $\delta_H$ be the modular character of $H$, and let $dh$ be a right Haar measure on $H$. ...
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1answer
38 views

Underlying space of the contragredient representation of a Hilbert space

Let $G$ be a topological group of totally disconnected type, and let $(\pi,V)$ be a smooth representation of $G$, for $V$ a complex vector space. Then $\pi$ induces a representation on the dual $\...
2
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0answers
32 views

Absolute roots restricting to a given root form a Galois orbit

Let $G$ be a quasisplit, connected reductive group over a field $k$. Let $S$ be a maximal split torus of $G$, and $T = Z_G(S)$, which is a maximal torus of $G$ which is defined over $k$. Let $B$ be ...
2
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1answer
79 views

Some questions on the Hecke algebra in Casselman's notes

Let $G$ be a connected, reductive group over a $p$-adic field $F$, which is unramified in the sense that $G$ is quasisplit to split over an unramified extension of $F$. Then $G$ is necessarily ...
3
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4answers
114 views

Significance of $G$ being reductive

Let $G$ be an algebraic group. What is the significance of $G$ being reductive? By reductive, I mean that: Let $R(G)$ be the largest subgroup such that $R(G)$ is a connected, solvable, normal ...

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