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 $F$...
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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 ...
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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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Is the unitary group of a $p$-adic anisotropic hermitian space commutative?

Let $E/\mathbb Q_p$ be a quadratic extension where $p \not = 2$. Let $V$ be an $E$-vector space equipped with a non-degenerate hermitian form, and assume that $V$ is anisotropic. In particular, $\dim(...
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5 votes
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Points stabilised by conjugate of a group

Let $G$ be a reductive group acting on an affine variety $X$. Let $x\in X$ such that the stabiliser of $x$ in $G$ is a subgroup $H$ which is itself a reductive group. Then, it is easy to see that any ...
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6 votes
1 answer
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Computing $\chi(1)$ and $\chi(s)$ for $\chi\in\widehat{\mathrm{GL}_2(\mathbb{F}_q)}$ and semisimple non-regular $s$ using formulas of Deligne-Lusztig

Let $G=\mathrm{GL}_2$ and $s=\left(\begin{smallmatrix} a & \\ & b \end{smallmatrix}\right)$ be semisimple and non-regular in $G(\mathbb{F}_q)=\mathrm{GL}_2(\mathbb{F}_q)$ (i.e. $a\neq b$ and $...
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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 ...
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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 ...
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1 vote
2 answers
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$G$-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 ...
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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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Showing two representations of the unipotent radical of Borel subgroup are isomorphic.

We consider the unipotent radical $N$ of the Borel subgroup $B$ of $\operatorname{GL}_2(F)$ where $F$ is a local field. Let $\phi$ be a character of the maximal split torus $T$. We inflate $\phi$ to $...
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Parabolic induction of p-adic groups independent of the choice of parabolic.

I noticed many papers concerning the theory of smooth representations of connected reductive p-adic groups, omit the mention of the specific parabolic subgroup $P\subseteq G$ used in defining the ...
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Classification of Reductive groups.

Let $K$ be a field with a non trivial discrete valuation and with $K$ complete and $\bar{K}$ perfect. What role do the Buildings play in the classification of reductive groups over $K$ ? Also there ...
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Is the natural projection of flag varieties a flat morphism?

Let $G$ be a reductive group over a field k (of characteristic zero) with parabolic subgroup $P$ and Borel $B \subset P$. Is the natural projection $ \pi: G/B \rightarrow G/P $ of flag varieties then ...
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Cohomology of $G/T$ for complex reductive group $G$

Okay, I will try this again, last time people didn't like how I asked it :) So let's say $G$ is a reductive complex algebraic group; it could be $GL_n(\mathbb{C})$ if that makes you happy (and in ...
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What is $\dim(M_1wP_2)$ for parabolics $P_i = M_iN_i$ of a reductive group?

Let $B$ be a Borel subgroup of a connected reductive algebraic group $G$ with maximal torus $T$. Let $\Phi = \Phi(G,T)$ be the root system, $\Delta$ be the basis of $\Phi$ defined by $B$, and $W$ be ...
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Representations of reductive groups and stabilisers.

Let $k$ be an algebraically closed characteristic zero field. Let $G$ be a reductive group. Let $G\to\mathrm{GL}_n$ be a representation of $G$, inducing an action of $G$ on $\mathbb A^n$. In proving ...
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Reductive group is infinitesimally flat and its distribution of algebra is isomorphic to the distribution of algebra of any big cell

I'm reading the book Representations of Algebraic Groups of Jantzen, chapter 1, part II. When discussing big cells of a split, connected reductive group $G$ over an integral domain $k$, the author ...
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6 votes
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Algebraic Peter-Weyl in positive characteristic

To my understanding there is an algebraic version of Peter-Weyl that holds in characteristic $0$ that says for any reductive group $G$ one has that: $$k[G]=\bigoplus V\otimes V^*$$ as a $G\times G$-...
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Dominance of $w\mu$ for dominant cocharacter $\mu$

NOTE: The question has now been posted on MathOverflow: Dominance of $w\mu$ for dominant cocharacter Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ ...
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Dominance order on cocharacter group $X_*(T)$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. By $X_*(T)$ we denote ...
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references on irreducible $(\mathfrak{g},K)$-modules for small rank groups

Classification of irreducible $(\mathfrak{g},K)$-modules for $SL_2(\mathbb{R})$ and $SL_2(\mathbb{\mathbb{C}})$ can be found in many standard textbooks, like Wallach's "real reductive groups"...
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packets for motives

I want to know if there exists a set of motives that corresponds to automorphic representations of a reductive group. Such a set must be divided into sets of the same L(or ε)function and satisfy the ...
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3 votes
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Maximal split tori in parabolic subgroups [answered]

Let $G$ be a reductive group over a field $k$. Let $P$ be a parabolic $k$-subgroup of $G$ and $P = L \ltimes U$ be a Levi decomposition. Is it true that every maximal split torus of $P$ lies in $L$ ? ...
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Representations of an algebraic group $G$ versus representations of the group $G(k)$

The following question stems from the discussion here. Let $G$ be a group scheme, $G$ acting on a vector space $V$ over a field $k$ is morphism of group valued functors $G \rightarrow GL_V$, where $...
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1 vote
1 answer
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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 ...
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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 ...
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2 votes
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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 ...
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6 votes
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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. ...
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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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0 votes
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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 ...
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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 $\mathbb Q_p^{\ast}$ ...
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2 votes
1 answer
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Trace of an induced representation on $G(\mathbb A)$

Let $G = \operatorname{GL}_2$, with center $Z$, diagonal matrices $T$, and upper triangular unipotents $N$. Let $K$ be the standard maximal compact subgroup of $G(\mathbb A) = G(\mathbb A_{\mathbb Q})...
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1 vote
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Describing the depth zero compact induction of a representation which is not cuspidal

Let $G$ be a reductive $p$-adic group. I would like to describe a representation of the type $$\rho = \mathrm{c-Ind}_K^G\,\sigma$$ where $K$ is a maximal parahoric subgroup of $G$ and $\sigma$ is an ...
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4 votes
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Are the stabilizers of facets in a Bruhat-Tits building pairwise distincts?

Let $G$ be a reductive $p$-adic group and let $\mathcal B$ denote its Bruhat-Tits building. For $x \in \mathcal B$, denote by $\overline x$ the (closure of the) minimal facet containing $x$. We ...
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2 votes
1 answer
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Structure of the Weil restriction

I am trying to understand how Weil restriction affects the structure theory of a reductive group over the two fields involved. As a toy example, I looked at the following: Consider $SL_2$ as an ...
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Finiteness of the volume $G(k) \backslash G(\mathbb A)^1$ from that of $G(k) Z(\mathbb A) \backslash G(\mathbb A)$

It follows from the construction of a Siegel domain that $G(k) Z(\mathbb A) \backslash G(\mathbb A)$ has finite volume, where $G$ is a connected, reductive group over a number field $k$, $Z$ is the ...
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6 votes
2 answers
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How parabolic subgroups are determined by roots?

I am interested in understanding the general construction of important subgroups in reductive groups, and how they are parametrized (Borel, Levi, parabolic, etc.). But for simplicity I take the ...
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Conjugating a root subgroup by a Weyl group element

Fix a field $k$. This is perhaps unnecessary, but assume $\operatorname{char} k = 0$. Let $G$ be a reductive isotropic quasi-split algebraic $k$-group. Let $S \subset G$ be a maximal split torus (of ...
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5 votes
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Bernstein center

I am trying to read the lecture notes of Joseph Bernstein on the representations of $p$-adic groups and struggling to understand a certain claim regarding the Bernstein center. Suppose $G$ is a ...
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2 votes
1 answer
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What is the canonical isogeny?

Given a (split connected reductive) root datum $\Psi$, there is another associated root datum $\Psi'$ and a morphism of root data $\Psi \to \Psi'$ called the canonical isogeny. My question is, is the ...
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Reductive quotient of parahoric subgroups of a simply connected simple group.

Let $G$ be a simply-connected simlpe group over a local field $K$ and $P_x$ a maximal parahoric subgroup of it. Let $P_x^+$ be the pro-unipotent radical of $P_x$. Then $P_x/P_x^+$ is a connected ...
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7 votes
0 answers
184 views

Are the Deligne-Lusztig varieties for $\mathrm{GL}$ and $\mathrm{SL}$ the same?

Let me first give some definitions for reference. Let $\mathbf G$ be a connected reductive group over an algebraic closure $\overline{\mathbb F}$ of a finite field $\mathbb F_p$, where $p$ is a prime ...
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My professor is asking us to find the semigroup reduct of the dihedral group D3, what is that?

Im not exactly sure what a semigroup reduct is, I know what a semigroup is but thats it, and im not exactly sure how to apply that to the dihedral group D3 anyway. Can someone help me here?
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2 votes
1 answer
150 views

Parabolic subalgebra conjugate under Weyl group

Let $G$ be a semisimple (or reductive (using the definition of Knapp)) Lie group. Then any two minimal parabolic subalgebras resp. subgroups are conjugated under $\operatorname{Ad}(K)$ resp. $K$ ($K$ ...
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3 votes
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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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2 votes
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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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2 votes
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42 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 ...
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1 vote
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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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2 votes
0 answers
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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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