# Questions tagged [reductive-groups]

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### 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. ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 $k$. ...
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### 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.
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### 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, ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
### 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 ...