# Questions tagged [reductive-groups]

Reductive groups are almost semisimple, they have nice representation theory, and they are classified by root data. This class of groups is the natural setting of a wide variety of representation theoretic problems in algebraic geometry. Use this tag for questions about algebraic groups of types A, B, C, D, E, F, and G—for Lie groups of the same type use [lie-groups]. Consider using with the [group-theory] and/or [representation-theory] tags.

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### Reference request: multiplicative group of a central simple algebra, their reductivity and parabolic subgroups

During my study of the theory of automorphic forms and $L$-functions, I never found any literature dealing with the following: Suppose $D$ is a central division algebra over a local or global field $F$...
1 vote
41 views

### Why is the multiplicative group of a central division algebra anisotropic?

Let $k$ be any field. Suppose $D$ is a central division algebra over $k$ of degree $n^2$, then we can understand its multiplicative group $D^{\times}$ as an algebraic group (defined over $k$). I ...
20 views

### How to show that two Whittaker functionals are equal

I'm checking a calculation with induced Whittaker functionals and have confused myself a little bit. Let $P_{\ast} = M_{\ast} N_{\ast} \subset P = MN \subset G$ be parabolic subgroups of a $p$-adic ...
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1 vote
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### Is the highest weight space of $V^{\otimes k}$ an irreducible representation of $S_k$?

Let $V$ be $n$-dimensional vector space over $\mathbb{C}$ with given action of $GL_n$. Let $W_\lambda$ be the highest weight space of weight $\lambda$ in $V^{\otimes k}$. Is $W_\lambda$ an irreducible ...
44 views

### Iwasawa decomposition of the general symplectic groups over the ring of adeles

This post is going to be quite a silly one, as its subject is something I believe to be the case, but which I am unable to prove to be the case (due to my lack of experience with algebraic and ...
1 vote
58 views

### $G$-composition length of representation

I do not understand what is meant by $G$-composition length. The textbook I'm using - The Local Langlands for GL(2) - makes the following statement. Let $\chi=\chi_1 \otimes \chi_2$ be a character of ...
30 views

### Where do the "classical lattices" come from?

I am trying to understand sphere packing problems, and it seems well-known that lattices can be only of certain kinds: the classical types (e.g. $A_n, B_n$, etc.) or the exceptional types (e.g. $E_8$)....
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1 vote
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### Quotient of $GL_n$ by unipotent group and maximal torus

Assume we are working over a field $k$ and $G= GL_n$. Then we get the (standard) maximal torus $T$, the unipotent group $U$ and the Borel $B$ of upper triangular matrices. More generally we have we ...
70 views

### Steinberg representation for finite groups VS for $p$-adic groups and parahorics

The term "Steinberg representation" comes out in both the context of finite groups of Lie type (ie. connected reductive groups over a finite field) and of $p$-adic groups (ie. connected ...
121 views

### Defining Levi subgroups: roots versus centralizer

I want to unify two points of view that remain distinct for me, even though they should match. Let $G$ be a reductive group (I will take examples in $GL(3)$). Let $P_0$ a minimal parabolic, $M_0$ its ...
172 views

### Reductive group terminology questions [closed]

$\DeclareMathOperator{\Hom}{Hom}$ $\newcommand{\g}{\mathfrak{g}}$ I am a beginner in the subject of reductive groups and I am hoping someone might be able to walk me through some basic terminology. ...
76 views

### What is the algebraic group whose $\mathbb Q_p$-rational points are the group $\mathrm{GU}_n(\mathbb Q_p)$?

For any field $K$, I denote by $\Gamma_K = \mathrm{Gal}(\overline K/K)$ its absolute Galois group. Let $p$ be a prime number and let $\mathbb Q_{p^2}$ denote a quadratic unramified extension of the ...
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### algebraic complex group and reductive algebraic group

A complex algebraic simply-connected group is always reductive? I saw that you con defined a langlands dual group for a reductive algebraic group in general but then I've found out a text that use the ...
### Flat and general holomorphic sections of the induced representation $\operatorname{Ind}^{\operatorname{GL}_2}$
Let $G = \operatorname{GL}_2(\mathbb Q_p)$, $K = \operatorname{GL}_2(\mathbb Z_p)$, and $B = TU$ the usual parabolic subgroup of $G$. For a pair of characters $\chi_1, \chi_2$ of $\mathbb Q_p^{\ast}$ ...