The reductive-groups tag has no usage guidance.

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486 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...

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59 views

### Product of standard parabolic subgroups

Consider a quasi-split reductive group $G$ over a field $k$. Let $B$ be a Borel subgroup of $G$, and let $P, Q$ be two parabolic subgroups of $G$ that contain $B$. Is the product set $PQ = \{xy| x \in ...

**4**

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**1**answer

62 views

### Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$

I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of ...

**3**

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**2**answers

210 views

### degeneration of reductive group

If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...

**1**

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**1**answer

88 views

### Reference requests for complex duals of connected reductive groups

In Kottwitz's paper "Stable trace formula: cuspidal tempered terms", it is said that if $$1 \rightarrow G_1 \rightarrow G_2 \rightarrow G_3 \rightarrow 1$$ is an exact sequence of connected reductive ...

**4**

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**1**answer

159 views

### Braid relations $n_\alpha n_\beta n_\alpha \ldots = n_\beta n_\alpha n_\beta \ldots $ in arbitrary reductive groups

I'm currently trying to prove or disprove the following claim. First let me set up some notation.
Let $G$ be a connected reductive group over a field $K$, let $S \leq Z \leq N \leq G$ be respectively ...

**3**

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**2**answers

288 views

### Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...

**10**

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**1**answer

408 views

### Primer on Eisenstein series

My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.
In my continuing journey of ...

**1**

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**1**answer

151 views

### Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...

**0**

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**1**answer

125 views

### Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...

**7**

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319 views

### Is every connected reductive group over a local field already defined over a global field?

Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...

**2**

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**1**answer

145 views

### Unitary representation with fixed Casimir

Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ...

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391 views

### Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...

**4**

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**1**answer

257 views

### Torsors in the analytic topology versus torsors in the etale topology

Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over ...

**12**

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**2**answers

860 views

### Is there a scheme parametrizing the closed subgroups of an algebraic group?

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...

**7**

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**1**answer

605 views

### Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...

**2**

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**0**answers

85 views

### Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...

**2**

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**2**answers

204 views

### Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$

Let $K$ be a global field and set $O := \prod_{v\nmid \infty} O_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}_n(O) = \prod_v \mathrm{GL}_n(O_v)$ with the product of the $v$-adic ...

**2**

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108 views

### $X$-points of reductive group schemes, if $X$ is a proper smooth curve over a finite field

Let $X$ be a connected proper smooth curve over a finite field (so the generic point of $X$ is the spectrum of a global field $K$), and let $G \rightarrow X$ be an affine $X$-group scheme of finite ...

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**1**answer

177 views

### Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?

**0**

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**1**answer

64 views

### Decomposition of Lie subspaces

If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$
where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and ...

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**1**answer

251 views

### Decomposition of $S^7=Spin(7)/G_2$

The seven sphere can be written as the reductive space $S^7=Spin(7)/G_2$. Has the decomposition $Spin(7)=G_2\times S^7$ been calculated somewhere; maybe in terms of Cayley numbers?

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188 views

### Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...

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**1**answer

219 views

### $\Gamma$-action on maximal tori in Borel-Tits

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...

**3**

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**1**answer

181 views

### Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center.
Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$.
Let $\chi : T(\mathbf{Q}_p) \to ...

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90 views

### Splitting for Subsequence of Automorphism Sequence for Algebraic Groups

Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups
$$
1\to \mathrm{Inn}(G)\to ...

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143 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

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**1**answer

77 views

### Strictly contracting elements in the center of a Levi subgroup

Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...

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186 views

### Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...

**3**

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**1**answer

166 views

### Supercuspidal with Iwahori fixed vector

Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, ...

**0**

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**1**answer

109 views

### Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...

**3**

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**1**answer

285 views

### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

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**1**answer

206 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**0**

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**1**answer

170 views

### number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...

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171 views

### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

**8**

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**1**answer

509 views

### Why people usually consider reductive groups in GIT?

Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT?
I found that the property of completely reducible of ...

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**2**answers

296 views

### Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...

**7**

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**1**answer

239 views

### Open cell decomposition after applying a Weyl group element

Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For ...

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93 views

### Invariant vectors in supercuspidal representations of GL_2(Zp)

Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the ...

**3**

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**1**answer

159 views

### Semisimple group not split by an unramified extension?

Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?

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585 views

### Reference request: expository text on the structure of reductive groups over non-archimedean local fields

I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner ...

**3**

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170 views

### How to think about non-connected reductive groups

Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...

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**1**answer

315 views

### How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the ...

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225 views

### Principal series of finite group of Lie type

I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field ...

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140 views

### Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
...

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**1**answer

288 views

### Weyl group of the restriction of scalars of split reductive group

Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.
Set $G' = ...

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268 views

### Steinberg reps of reductive groups over local fields vs finite fields

Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$.
Edit: The statements only make sense modulo tensoring by one-dimensional representations.
Are the unitary, ...

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227 views

### a conjugacy question in quasi-split reductive groups

I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over ...

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**1**answer

1k views

### Reconciling Lusztig's results with the Langlands philosophy

Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = ...

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1k views

### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...