# Tagged Questions

**6**

votes

**1**answer

94 views

### Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...

**2**

votes

**1**answer

87 views

### dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...

**16**

votes

**3**answers

375 views

### Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...

**1**

vote

**1**answer

122 views

### Tables of data associated to reductive algebraic groups?

I am looking for a reference that contains lots of calculations for specific examples of various objects one can associate to a reductive algebraic group. For example, given a (specific) linear ...

**3**

votes

**1**answer

203 views

### Quasi-coherent sheaves on classifying stacks

Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq ...

**6**

votes

**3**answers

397 views

### $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) ...

**1**

vote

**4**answers

208 views

### Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...

**6**

votes

**1**answer

229 views

### Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...

**4**

votes

**1**answer

98 views

### Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...

**1**

vote

**0**answers

47 views

### Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...

**0**

votes

**1**answer

196 views

### Quotient of an algebraic group by a closed algebraic subgroup

Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" ...

**4**

votes

**1**answer

231 views

### Bruhat decomposition for reductive groups in characteristic zero?

Let $G$ be a reductive, linear algebraic group (variety) over an algebraically closed field $\Bbbk$ of characteristic zero. If $G$ is connected, I know from Humphrey's book that for any Borel subgroup ...

**7**

votes

**2**answers

374 views

### Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...

**3**

votes

**1**answer

74 views

### Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$;
...

**3**

votes

**3**answers

292 views

### Topological properties of $K$ orbits in $G/B$

I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...

**8**

votes

**1**answer

341 views

### Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...

**3**

votes

**0**answers

121 views

### Jordan decomposition for non algebraically closed fields

Let $G$ be a (linear?) algebraic group defined over some field $k$ (not necessarily algebraically closed). For $g\in G$ we have the Jordan decomposition $g=su$ in the semisimple part $s$ and the ...

**2**

votes

**0**answers

257 views

### Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two.
I am looking for a reference that explains how to ...

**3**

votes

**2**answers

248 views

### Simple representations of products of algebraic groups

I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.
Let $G_1$ and $G_2$ be affine algebraic group schemes ...

**6**

votes

**2**answers

395 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**

votes

**0**answers

139 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 ...

**3**

votes

**0**answers

254 views

### Are principal bundles isotrivial?

Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such ...

**0**

votes

**1**answer

140 views

### Reference on elements of finite order in principal congruence subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...

**5**

votes

**1**answer

222 views

### Levi decomposition in disconnected linear algebraic group (characteristic 0)?

For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I postpone further my ...

**3**

votes

**0**answers

130 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 :
...

**2**

votes

**2**answers

196 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 ...

**2**

votes

**1**answer

123 views

### A question about $R$-points of an complex reductive group.

I hope somebody can give me a good reference for the following:
Let $G$ be a complex reductive group $H$ be a closed subgroup. Let further $R$ be any $\mathbb{C}$-algebra. Then the canonical map
...

**12**

votes

**0**answers

228 views

### Which p-adic algebraic groups are type I?

It was proved by Jacques Dixmier (Sur les reprÃ©sentations unitaires des groupes de Lie algÃ©briques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820 , Zbl ...

**2**

votes

**2**answers

460 views

### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

**5**

votes

**2**answers

370 views

### Normal subgroups of $SL_2$ of a polynomial ring

What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction ...

**3**

votes

**1**answer

243 views

### Rational automorphisms of semisimple algebraic groups

Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...

**17**

votes

**1**answer

992 views

### Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...

**3**

votes

**0**answers

203 views

### When is a subgroup the Weil restriction of another subgroup?

I asked this question on Math.StackExchange, to no avail. I try my chance on this one.
Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ ...

**3**

votes

**1**answer

189 views

### Structure of abelian connected complex linear algebraic groups?

Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated ...

**1**

vote

**1**answer

478 views

### Conjugacy classes in Aut(G)

Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.
Now, I'd like to know the ...

**2**

votes

**4**answers

319 views

### A polynomial homomorphism from Gl to the group of units is a power of the determinant

I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism ...

**2**

votes

**1**answer

134 views

### Reference request for Cartier Duality of algebraic tori

Hi,
I need a reference for the following result:
Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F_X :S'\mapsto Hom_{S'}(X\times S',\mathbb{G}_M\times S')$ is ...

**1**

vote

**0**answers

273 views

### Has anyone used this theorem of P. Cartier?

In "Groupes Algebriques et Groupes Formels", Conf. au coll. sur la theorie des groupes algebriques, Bruxelles 1962, P. Cartier proves the following in Section 9, Theoreme 1:
(What follows is my ...

**3**

votes

**0**answers

130 views

### Does the following characterization of subgroups of $GL_2(\mathbb{F}_p)$ generalise?

Let $p$ be a prime number. By a Cartan subgroup of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.
When $n=2$, it is well-known* that, for $G \subset ...

**13**

votes

**3**answers

1k views

### Questions about the Bernstein center of a $p$-adic reductive group

Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of ...

**2**

votes

**1**answer

223 views

### Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following:
By Weilâ€™s theorem $[17]$, any local algebraic group is birationally
equivalent to an algebraic group.
Where
$[17]$ A.Weil. On ...

**5**

votes

**2**answers

428 views

### abelian centralizers in almost simple groups

Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...

**2**

votes

**2**answers

343 views

### Reference request: The geometry of $GL_2(\mathbb{R})$ and related questions

Can anyone please recommend some good reading on the geometry of linear groups and their actions?
An example of the kind of question I am interested in: Explicitly describe a fundamental domain for ...

**15**

votes

**3**answers

2k views

### A reference for geometric class field theory?

The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...

**0**

votes

**0**answers

525 views

### Book on linear algebraic groups in scheme language

Is there a book on linear algebraic groups using the scheme language (i.e. not Springer or Borel, but like Waterhouse, but more in-depth)?
The book should discuss topics like Borel subgroups etc.
...

**9**

votes

**2**answers

1k views

### Literature on the Springer resolution

Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper?
Unfortunately googling for "Springer" and ...

**7**

votes

**1**answer

434 views

### Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such
as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...

**5**

votes

**2**answers

490 views

### Reference request: representations of unipotent groups have a fixed point.

I'm looking for a reference for the following standard result:
Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic ...

**48**

votes

**9**answers

4k views

### What are “classical groups”?

Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...

**11**

votes

**3**answers

2k views

### The algebraic fundamental group of a reductive algebraic group

For a connected reductive algebraic group $G$ over a field $k$, other than the \'etale fundamental group of $G$ (regarded just as a scheme), there seems to be another notion, usually called the ...