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2
votes
0answers
142 views

Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$. The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
3
votes
1answer
212 views

Existence of quotient variety for group implies existence of quotient for normal subgroups

Let $G=G_1\times G_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the ...
4
votes
2answers
191 views

Splitting field of a Torus

Let $T$ be a torus over a non-necessary perfect field $k$. Let $\bar k$ an algebraic closure of $k$. Is there a smallest extension $k'$ of $k$ in $\bar k$ such that $T \times_{{\rm spec}\, k} {\rm ...
3
votes
0answers
137 views

determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$. Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...
0
votes
0answers
98 views

a very elementary question on the conjugated matrices

Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero . We suppose that they have the same ...
3
votes
0answers
56 views

on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$. Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$. We consider $I_{\gamma}$ the group ...
3
votes
2answers
221 views

Why does the expression “the largest quotient of a linear algebraic group that is multiplicative type” make sense?

For the past few weeks I've been trying to get myself acquainted with the language and basic theory of linear algebraic group schemes. In an attempt to see whether I have learned enough to read a ...
2
votes
0answers
159 views

Is the Mumford-Tate group determined by its $\mathbb{Q}$-points?

Let $V$ be a $\mathbb{Q}$-Hodge structure and $MT \subset GL_V$ its Mumford-Tate group. Let $I \subset \mathcal{O}_{GL_V}$ be the ideal of functions vanishing on $MT(\mathbb{Q})$. Is ...
5
votes
1answer
135 views

A subgroup of the Weyl group

Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2. Let $Q=Q(D)$ denote the root lattice of $D$. Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
4
votes
1answer
139 views

One-parameter subgroups of symplectic group associated to roots

I'm having trouble sorting out some basic definitions concerning Chevalley groups. The groups I'm interested in are the simply connected groups of type $C_n$, so the groups $\text{Sp}_{2n}$. The ...
9
votes
2answers
358 views

Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...
3
votes
0answers
84 views

Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups? The ...
3
votes
0answers
164 views

K-theory of categories of group schemes and abelian varieties

Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group ...
35
votes
3answers
2k views

What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
3
votes
0answers
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 ...
17
votes
3answers
572 views

Spin group as an automorphism group

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with ...
2
votes
0answers
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
2answers
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 ...
1
vote
2answers
126 views

Rank of the character group of a maximal $K$-torus for semisimple and adjoint algebraic groups

I've been trying to understand some of the idiosyncrasies associated to algebraic groups over non-algebraically closed fields $K$ of characteristic $p > 0$. Let $G$ is a connected almost ...
2
votes
0answers
162 views

Certain central extensions of simply connected simple algebraic groups

An offbeat question involving Milnor's $K_2$ has come up recently. Start with an algebraically closed field $F$ (perhaps required to be of characteristic 0). Let $G$ be a connected, simply connected ...
0
votes
1answer
91 views

compact subset in linear algebraic group over local field

Let $F$ be a local field of characteristic zero, and $G$ a linear algebraic group with finite connected components over $F$. We will consider the $G(F)$, and give it $p$-adic topology. Let $C$ be some ...
3
votes
1answer
125 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 $F$ ?
1
vote
0answers
156 views

Compactifications of group schemes

Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
19
votes
2answers
711 views

Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...
6
votes
2answers
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
0answers
110 views

Product of Fixed points and kernel of Frobenius morphism

If $G$ is a reductive algebraic group over an algebraically closed field of positive characteristic $p$, and $G$ is defined over the prime field, we have the Frobenius morphism $F: G\to G$, which for ...
4
votes
1answer
168 views

Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero

I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic ...
5
votes
1answer
235 views

Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset ...
7
votes
1answer
358 views

Deformation theory of octonion algebras?

In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am ...
3
votes
1answer
153 views

counting the parabolic subgroups that contain a fixed semisimple element

Let $G$ be a finite group of Lie type , and let $P$ be a parabolic subgroup of $G$ containing a proper Borel subgroup $B$. Let $s$ be a semisimple element of $G$. I want to count the number of ...
3
votes
0answers
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 ...
4
votes
1answer
264 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 ...
3
votes
0answers
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 ...
6
votes
0answers
97 views

Rational points with small denominator in $U(n)$

Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$. Is there an ...
3
votes
1answer
300 views

Group scheme over a DVR whose special fibre is the image of points under reduction mod p

Let $R$ be a complete discrete valuation ring with maximal ideal $\mathfrak{p}$ and algebraically closed residue field $k$. Denote the field of fractions of $R$ by $F$. Let $G$ be an affine flat group ...
1
vote
0answers
238 views

For what fields is $GL_n(k)$ a rational variety?

I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational variety? Are there ...
3
votes
2answers
382 views

Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
1
vote
1answer
195 views

Smooth map to the stack of G-bundles

Let $G$ a semisimple group and $B$ a Borel subgroup. We denote by $Bun_{G}$ the stack of G-bundles. Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$? My question comes ...
0
votes
1answer
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 ...
2
votes
1answer
280 views

Quotient of algebraic groups in the étale topology

Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$. These statements ...
3
votes
1answer
202 views

differential of the characteristic polynomial

Let $\chi:GL_{n}(\mathbb{C})\rightarrow \mathbb{C}^{n}$ the map given by the coefficients of the characteristic polynomial. Let $A$ a regular semisimple matrix, do we have a formula for the ...
5
votes
1answer
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 ...
2
votes
1answer
181 views

On the $F$-rational points of the derived group of a connected reductive algebraic group

Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G_{der}$ denote the algebraic derived group of $G$; this is connected and ...
1
vote
0answers
48 views

open immersion, affine grassmanian and negative loop group

Let $G$ a semisimple group over $k=\bar{k}$. Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map: $LG^{-}\times ...
2
votes
2answers
325 views

Why are these parabolic subgroups opposed?

I am reading notes of Michel Brion on spherical varieties. Consider a reductive group $G$, a Borel $B$ in $G$, a finite dimensional $G$-module $M$ and a closed orbit $Y$ of $G$ in $\mathbb{P}(M)$. ...
7
votes
1answer
246 views

Counting conjugacy classes in simple groups of Lie type

Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split. ...
3
votes
0answers
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
0answers
76 views

do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
1
vote
0answers
149 views

Conjugacy classes of centralizers of semisimple elements in a finite group of Lie type

Let $G$ be a finite group of Lie type. By Deriziotis' and Carter's articles we know that conjugacy classes of connected centralizers of semisimple elements are parametrized by $(J,[w])$ where $J$ is a ...
2
votes
0answers
106 views

affine schubert cells and bruhat order

Let $G$ asimply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in G(k[[t]]), $T$ a maximal torus and $K=G(k[[t]])$. Let $\lambda\in X_{*}(T)^{+}$ a dominant ...