Questions tagged [reductive-groups]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
0answers
75 views

Describing an action on the Weyl group $W(T)=N_G(T)/T$ for different maximal tori $T$

While trying to understand more the structure of reductive group, I came upon the situation that I describe below. I can't find a mistake in my dissertation, however I arrive to an absurd conclusion. ...
2
votes
0answers
42 views

Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
2
votes
0answers
25 views

Bruhat-Tits theory: how does the normalizer act on an apartment?

Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...
0
votes
0answers
47 views

When does an element lie in a Levi subgroup?

Let $G$ be a complex reductive group, and let $P$ be parabolic, with unipotent radical $U$ and reductive quotient $L=P/U$. A Levi subgroup is a lift of $L$ into $P$. My question is how do you tell if ...
1
vote
0answers
34 views

Group extension of $G$ by $\mathbb{G}_m$ and the Picard group.

I was reading about algebraic groups and I ran into a proof that I don't understand. The setup is as follows: $k$ is a separably closed field, and $G$ is a simply connected semi-simple group over $k$. ...
0
votes
0answers
12 views

relation between weyl group of reductive group $W(G,T)$ and weyl group of the dual group $W(\hat{G},\hat{T})$

what is the relation between Weyl group of reductive group $W(G,T)$ and Weyl group of the dual group $W(\hat{G},\hat{T})$? I think they must be isomorphic but I'm not sure.
2
votes
1answer
47 views

Definition of functions in the induced space from parabolic induction

Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, ...
2
votes
1answer
29 views

The relative weights form a basis of $\mathfrak a_P^{G \ast}$

Let $G$ be a connected, reductive group over $\mathbb Q$ with minimal parabolic $P_0 = M_0 N_0$. For $P = MN$ a standard parabolic subgroup of $G$, let $A_P$ be the split component of $M$ and let $\...
2
votes
0answers
34 views

Examples of reductive p-adic groups representations

I am currently reading D. Renard's book "Représentations de groupes réductifs p-adiques". He provides no concrete examples (which would be of great help). I found some notes by Fiona Murnaghan on the ...
0
votes
0answers
21 views

Is the product group of finitely many copies of the multiplicative group of positive real numbers a reductive group?

Let G be the multiplicative group of positive real numbers. Is the finite product group $G \times \cdots \times G$ reductive? I am trying to construct the moduli space for some quiver representations ...
0
votes
1answer
24 views

Double Coset Space Hausdorff?

Let $G$ be a locally compact, totally disconnected Hausdorff group, and let $H,N \subseteq G$ be closed subgroups. It is known that $G/N$ is Hausdorff (in the quotient topology, and has the other two ...
3
votes
0answers
41 views

Are semisimple Lie groups always reductive Lie groups?

The example I am looking at is the universal cover of $SL_2(\mathbb{R})$ which is a semisimple Lie group with infinite center. Using Knapp's definition of reductive Lie groups, we need to find a ...
2
votes
0answers
105 views

simply-connected compact Lie group v.s. semi-simple Lie group

If G is a simply-connected compact Lie group, is it true that its Lie algebra must be semi-simple? What I found is that https://en.wikipedia.org/wiki/Compact_group#Classification: The ...
3
votes
0answers
67 views

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 $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
2
votes
1answer
81 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 ...
0
votes
1answer
44 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 ...
0
votes
1answer
28 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 ...
2
votes
0answers
22 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 ...
0
votes
1answer
68 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
votes
1answer
84 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 ...
2
votes
1answer
122 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 ...
1
vote
1answer
86 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 ...
2
votes
2answers
63 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 ...
0
votes
0answers
34 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) ...
1
vote
0answers
41 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....
1
vote
0answers
39 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 $...
0
votes
0answers
103 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 $\...
0
votes
1answer
27 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 $\...
0
votes
1answer
61 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 ...
0
votes
1answer
45 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
votes
1answer
97 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 \...
0
votes
0answers
64 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 ...
0
votes
1answer
36 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 $W = W(G,T)$ be ...
2
votes
1answer
100 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 ...
0
votes
1answer
71 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
votes
1answer
39 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 $...
0
votes
0answers
58 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 ...
1
vote
0answers
56 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 ...
3
votes
0answers
82 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 ...
1
vote
1answer
54 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 ...
4
votes
0answers
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 ...
1
vote
0answers
20 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 ...
0
votes
1answer
371 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. ...
1
vote
0answers
76 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
votes
1answer
119 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 ...
3
votes
0answers
91 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 ...
2
votes
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
votes
1answer
84 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 $\xi \in V$,...
1
vote
1answer
58 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 ...
1
vote
0answers
45 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 ...