The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
166 views

Is there any way to determine the “effeciency” of Jantzen's sum formula?

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a reductive algebraic group over $k$. In order to determine the structure of the Weyl modules $V(\lambda)$, ...
3
votes
2answers
274 views

Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups. The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...
2
votes
1answer
172 views

Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
0
votes
1answer
92 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
votes
1answer
260 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 ...
1
vote
1answer
165 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 ...
8
votes
0answers
214 views

Earliest use of the term “linearly reductive”?

Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...
12
votes
1answer
559 views

automorphisms of fat points

Let $k$ be an algebraically closed field. I am looking for an easily quotable description of automorphism groups of $\mathrm{Spec} k[x]/(x^n)$. I could compute explicit matrix representations for ...
1
vote
0answers
161 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 ...
3
votes
1answer
230 views

Weyl group invariants in a maximal torus

Suppose G is a semi-simple adjoint group over complex numbers. Suppose T is a maximal torus in G. Does one know what are the W invariant (non-trivial) elements in T? Perhaps I might give a few ...
6
votes
2answers
280 views

Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...
1
vote
0answers
131 views

Restriction of the Steinberg representation

Let $G$ the group $GL(n,F)$ where F is a locally compact non Archimedean field, and $G^{0}$ the subgroup of $G$ consists of elements $g$ in $G$ such that $\det(g)$ in $\mathcal{O}_{F}^{\times}$, where ...
5
votes
2answers
282 views

conjugacy classes in anisotropic semisimple groups

You have an anistropic semisimple algebraic group $G$ defined over a non-archimedean local field $k$. When can you say that the $k$-rational part of the conjugacy class of a $k$-rational point is ...
2
votes
0answers
85 views

radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers. Let $B$ a Borel subgroup and $I$ the corresponding Iwahori. Let ...
6
votes
1answer
473 views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
1
vote
1answer
124 views

Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$, \begin{align} ...
0
votes
0answers
106 views

Enveloping algebras of map algebras as hyperalgebras of algebraic groups

This is a continuation of various questions about Chevalley groups over rings, cf these two questions (and a rather bad question of mine here). Consider a semisimple Lie algebra $\mathfrak g$ over ...
0
votes
0answers
95 views

The centralizer $Z_G(X)$ of a nilpotent element in a real simple Lie group

I am looking for the description of the centralizer $Z_G(X)$ , where $G$ is a real simple Lie Group and $X\in \ Lie (G) $ such that $X^d=0,\ X^{d-1}\neq 0 $. It is is helpful to me any references or ...
3
votes
1answer
120 views

Non-abelian group from affine hermitian curve

I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve: $(a,b) * (c,d) = ...
1
vote
1answer
217 views

Form of a block upper triangular matrix of finite order

If I take a block upper triangular matrix and the matrix is known to be diagonalizable and the leading blocks in the diagonal are each of finite order, is it true that away from the leading block ...
7
votes
1answer
231 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 ...
2
votes
1answer
173 views

Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...
3
votes
2answers
342 views

Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$, $$G\times(X\times...\times ...
0
votes
0answers
67 views

rational conjugacy classes

Say that $G$ is a reductive group over a local field $F$ and $g\in G(F)$. Can we show that the conjugacy class of $g$ in $G(F)$ has finite index in the $F$-rational part of the conjugacy class of $g$ ...
2
votes
0answers
89 views

Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...
2
votes
1answer
129 views

centralisers of maximal split tori

Suppose that $G$ is a reductive group defined over a field $k$ which is not quasisplit. Suppose that $S$ is a maximal $k$-split torus. Let $\mathcal{L}(S)$ be the centraliser of $S$ and ...
3
votes
2answers
272 views

isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$. By ...
3
votes
1answer
457 views

Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...
1
vote
2answers
177 views

Preimage of a maximal compact open subgroup in the simply connected cover

Let $G$ be a semi-simple algebraic group over $Q_p$ and $K$ in $G(Q_p)$ a maximal compact open subgroup. Let $\tilde{\pi}\colon \tilde{G}\rightarrow G$ be the simply connected cover. Then ...
5
votes
1answer
206 views

When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$. Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...
4
votes
2answers
280 views

Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
1
vote
1answer
85 views

Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and $I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$ the scheme of centralizer. Is $I$ a Cohen-Macaulay scheme ...
2
votes
0answers
183 views

Chevalley groups over $k[t]/t^n$

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and ...
6
votes
1answer
246 views

Mumford-Tate groups of products of Hodge structures

Let $V_1$, $V_2$ be two polarised simple $Q$-Hodge structures which are non-isomorphic. I am assuming that the Mumford-Tate groups of $V_1$ and $V_2$ are semi-simple adjoint. Is it true in this case ...
3
votes
1answer
275 views

A question on algebraic loop groops

Setup: Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...
4
votes
1answer
282 views

Zariski dense subgroups of linear algebraic groups

The theorem of Matthews, Vaserstein and Weisfeiler asserts that if $G$ is a simply connected absolutely almost simple groups over $Q$ and $\Gamma$ a finitely generated subgroup of $G(Q)$ which is ...
5
votes
1answer
319 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 ...
2
votes
1answer
117 views

Subgroup containing a given torus as a maximal torus

Let T be a non-trivial torus in a semisimple group G of adjoint type defined over Q. I was wondering whether there exists a reductive subgroup H of G (defined over Q !) such that T is a maximal torus ...
2
votes
2answers
254 views

Centralizer of a subtorus in a reductive group is Levi?

Questions a bit similar to this one have already appeared I think on the forum but I couldn't find the answer to my question using those answers. I must say from the beginning that my knowledge of ...
7
votes
0answers
241 views

Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
6
votes
1answer
166 views

regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$). What can ...
3
votes
0answers
43 views

points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...
4
votes
1answer
285 views

When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...
0
votes
0answers
99 views

semisimple conjugacy classes over general bases

Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism. We know that if ...
4
votes
1answer
172 views

Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices

I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...
7
votes
2answers
403 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 ...
5
votes
1answer
232 views

Smooth affine algebraic subgroups as complete intersections

Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular ...
0
votes
0answers
167 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
3
votes
1answer
81 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 $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$; ...
1
vote
1answer
162 views

Equivariant fibre product

Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product ...