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0
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0answers
22 views

generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$. Let $\mathfrak p$ be a parabolic subalgebra of ...
0
votes
1answer
44 views

Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? [on hold]

Let $G$ be a semisimple and simply connected linear algebraic group over $\mathbb{C}$. Let $H$ be a connected, Zariski closed and semisimple linear algebraic $\mathbb{Q}$-subgroup of $G$. Is $H$ a ...
0
votes
1answer
56 views

A bijection between Lusztig series induced by inflation

Context: Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F ...
6
votes
1answer
185 views

When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...
7
votes
0answers
203 views

Can an abelian variety/Q have no points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal ...
7
votes
1answer
201 views

Geometric interpretation of Cusps for general groups?

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$. Then for an automorphic form, ...
0
votes
0answers
73 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 ...
1
vote
1answer
231 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
8
votes
2answers
275 views

$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
8
votes
1answer
325 views

Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my ...
4
votes
0answers
138 views

Automorphisms of a quotient variety

Let $X$ be a variety, and $G\subset Aut(X)$ a subgroup of the automorphism group of $X$. Assume that the quotient $Y = X/G$ is a variety. Does there exist some simple relation between $Aut(X)$, $G$ ...
3
votes
2answers
195 views

A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$. I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...
2
votes
1answer
285 views

Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as ...
0
votes
0answers
86 views

About algebraic groups defined over Q [migrated]

I'm studying automorphic forms and there's something I don't understand, when we talk about a connected reductive algebraic group $G$ defined over $\mathbb{Q}$, connected means connected as an ...
9
votes
2answers
628 views

Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...
7
votes
1answer
228 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 ...
2
votes
1answer
163 views

Equivariant Derived Category

If $G$ is an connected unipotent group over $k$,and $X$ a scheme of finite type over $k$, (an algebraic closed field of positive characteristic) then we can define the bounded derived categorie of ...
2
votes
0answers
67 views

Dimension of affine Springer fiber and its functor of points as an ind-scheme

Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that ...
6
votes
0answers
215 views

What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific. As mentioned there, the ...
1
vote
0answers
130 views

rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$? ...
1
vote
0answers
71 views

Hyperspecial parahoric group schemes/Chevalley groups

Let $G$ be a simple group over $k=\mathbb{C}$, $A=k[[t]]$, $K=k((t))$, and consider the group $G(K)$. This group is a split reductive group over a local field, and therefore the results of Bruhat and ...
3
votes
0answers
84 views

Torsors under the group scheme over projective line

Consider the group scheme $\mathcal T$ over $\mathbf P^1$ given locally (variable $t$) by the equation $x^2 - f(t)y^2 = 1$ where $f(t)$ is a polynomial of degree $r$ with distinct roots (assume that ...
5
votes
0answers
114 views

LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...
4
votes
1answer
193 views

Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
5
votes
0answers
149 views

Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...
3
votes
0answers
164 views

Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...
5
votes
2answers
381 views

Left U_n-invariants of SL_n - an exercise in Kraft-Procesi

I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get ...
4
votes
3answers
314 views

What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise: Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...
3
votes
3answers
1k views

Can every parabolic subgroup be conjugated to its opposite by an element of the Weyl group?

Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic. Can we do the same thing if we choose a standard parabolic ...
3
votes
1answer
188 views

line bundle descends?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
8
votes
3answers
660 views

How to topologize X(R) when R is a topological ring?

Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
4
votes
1answer
333 views

Tannakian fundamental group of two explicit tensor categories

Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$): 1. The ...
2
votes
1answer
204 views

surjective homomorphism with compact kernel (Milne's note on Shimura varieties)

I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get. Let $G$ be a connected semisimple algebraic group $G$ over ...
3
votes
1answer
177 views

Special linear groups over function fields

Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements. What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
39
votes
4answers
4k views

Origin of terms “flag”, “flag manifold”, “flag variety”?

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
1
vote
0answers
126 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
2
votes
1answer
195 views

twists of algebraic groups

If $k$ is some field - for convenience, of characteristic 0 -, $\bar{k}$ is an alg. closure of $k$, and $G$ is some $k$-algebraic group, one can define a twist of $G$ to be some $k$-algebraic group ...
5
votes
1answer
189 views

Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel. Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...
2
votes
1answer
191 views

highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus. Let $\lambda,\mu,\nu$ be dominant characters of $T$. ...
1
vote
0answers
84 views

Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups

Let me start with the simplest version of the question since already there I don't know anything. For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb ...
0
votes
0answers
58 views

adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group. We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...
6
votes
1answer
172 views

Extensions of an abelian variety by a torus vs. extensions of their $\ell$-adic Tate modules

Let $K$ be a number field, let $A$ be an abelian variety over $K$, and let $H$ be a torus over $K$. For a prime $l$, we have the natural map $$\mathrm{Ext}^1(A, H) \otimes_{\mathbb{Z}} \mathbb{Z}_l ...
3
votes
0answers
314 views

A Step in the Proof of the Drinfeld-Simpson theorem

I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``$B$ ...
1
vote
0answers
72 views

Exponential map on a unipotent group

Let $G$ by a unipotent linear algebraic group defined over a field of characteristic $0$, with Lie algebra $\mathfrak{g}$. The exponential map $\mathfrak{g}\to G$ is bijective, and we can recover the ...
5
votes
1answer
181 views

Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent ...
6
votes
4answers
366 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 ...
2
votes
1answer
87 views

Rational Points of a Quotient of a Reductive Group by a Parabolic Subgroup

Let $G$ be a reductive group and let $P$ be a parabolic subgroup of $G$ all defined over $\mathbb{Z}$. Also, let $F$ be a number field, is it true (and if so, please provide a reference) that $$ ...
1
vote
1answer
121 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} ...
3
votes
1answer
159 views

How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
2
votes
0answers
152 views

affine schubert cells and bruhat order

Let $G$ be a simply 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 $\lambda\in X_{*}(T)^{+}$ be a ...