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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

7
votes
0answers
134 views

Decomposition of linear groups into free products

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
4
votes
0answers
124 views

Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn! Could anyone provide a reference request about extensions of finite group schemes / Ext groups. As far as I know the category ...
4
votes
1answer
62 views

(Euclidean) open orbit in an irreducible real algebraic set

Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
6
votes
0answers
133 views

representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$

Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...
-1
votes
1answer
209 views

$A[x]$ points of an algebraic group

Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$. Is there any reference for such kind of results?
-1
votes
0answers
105 views

Finite groups with trivial center [closed]

I know the Symmetric groups, Alternating groups and Frobenius groups of order $pq$, where $p$ and $q$ are distinct prime number, have trivial center. I want to know a classification of finite groups ...
3
votes
2answers
106 views

Generating Irreducible representations of a simple lie algebra with Schur functors

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
9
votes
0answers
89 views

Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?

Let $R$ be a connected (commutative) ring with $2\in R^\times$. Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...
2
votes
0answers
51 views

Valuations of root group elements appearing in the intersection of Iwasawa and Cartan double cosets

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\val}{\mathit{val}}$Let $F$ be a local non-Archimedean field with valuation $\val$ and $G$ be (the $F$-points ...
4
votes
4answers
299 views

Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group

Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...
4
votes
1answer
62 views

Existence of Regular Semisimple elements of reductive groups in characteristic 0

Suppose $G$ is a connected reductive group defined over a field $F$ of characteristic $0$. Does every maximal torus contain a regular semisimple element defined over $F$? I know that over an ...
1
vote
3answers
206 views

References request: representations of classical groups

Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$? I would like to know explicit formulas of the ...
1
vote
1answer
92 views

Lifting Lang-Steinberg to DVR's in Characteristic 0

Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
2
votes
1answer
86 views

Do the absolute roots restricting to a given root form a Galois orbit?

Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and ...
5
votes
0answers
76 views

Restricting projective representations of Lie groups to lattices

Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
3
votes
0answers
79 views

Special unitary group simply connected?

Let $C'\rightarrow C$ be a double cover of smooth projective curves over $\mathbb C$. Let $K'/K$ the corresponding function fields extension. Denote by $\tau$ the Galois involution on $K'$. Let $SU(K')...
2
votes
1answer
94 views

Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?

Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal Levi subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2....
4
votes
0answers
64 views

Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
1
vote
0answers
81 views

Group schemes and Hyperspecial maximal compact subgroups

Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
4
votes
0answers
125 views

Integral structures via lattices

I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
0
votes
0answers
23 views

What are the corner minors in $Sp(4)$?

This question relates to the question and the question. Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 &...
0
votes
0answers
37 views

What is the natural projection of Parabolic subgroups in $Sp(4)$?

Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{2,1} & x_{2,2} & 0 & 0\\ x_{...
0
votes
2answers
76 views

What is the longest word in type $C_2$ Weyl group written in terms of a matrix?

The longest word in type $A_3$ Weyl group written as a matrix is \begin{align} w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & ...
1
vote
0answers
108 views

Borel subgroup of $Sp(4,\mathbb{C})$

I am trying to understand Borel subgroups of $Sp(4,\mathbb{C})$. I think that the following is a Borel subgroup of $Sp(4, \mathbb{C})$: the subset of $Sp(4, \mathbb{C})$ of all lower triangular ...
1
vote
1answer
82 views

A non trivial example of an anti -affine algebraic group

An anti- affine group $G$ is defined to be an algebraic group with no global sections. Examples include abelian varieties and non trivial extensions of abelian varieties by torus (in characteristic $\...
2
votes
1answer
76 views

Minuscule cocharacter for reductive groups

I have a question about minuscule cocharacters, which might sound trivial to the experts: Let $G$ be a smooth affine group scheme over $\mathbb{Z}_{p}$. Furthermore, consider a cocharacter $$\mu\...
3
votes
1answer
200 views

Possible groups appearing in a Shimura datum

Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
5
votes
1answer
122 views

Presentation of special linear group over localizations of the integers

I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\...
3
votes
1answer
143 views

Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper

In Kottwitz's 1985 Compositio paper, Isocrystals with additional structure, first page, paragraph 4: Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
5
votes
0answers
65 views

Hilbert series for invariant ring

I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with). To put it in some broader context, let $G$ ...
9
votes
0answers
212 views

Naive question about classification of unipotent character sheaves

Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in ...
4
votes
0answers
58 views

$G$-Invariant Differential Operators

Let $G$ be a complex algebraic group, $K$ a closed subgroup so that $X=G/K$ is a homogeneous space. Let $\mathcal{D}(X)$ denote the algebra of differential operators on $X$. The group $G$ acts on $\...
4
votes
0answers
154 views

Etale cohomology of variety of matrices with given characterestic polynomial

A continuation of Number of points of the nilpotent cone over a finite field and its cohomology. Let $k=\Bbb F_q$ be a finite field, $p=\text{char}(k),$ $P \in k[\lambda]$ be a monic polynomial of ...
7
votes
2answers
315 views

Number of points of the nilpotent cone over a finite field and its cohomology

Let $k=\Bbb F_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$. Motivation ...
0
votes
0answers
68 views

Is there a projection from $G$ to the Levi subgroup of a Parabolic subgroup?

Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel ...
2
votes
0answers
53 views

Under what conditions are superspecial abelian surfaces isomorphic over a finite field?

Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
1
vote
0answers
83 views

Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
17
votes
1answer
459 views

Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$?

Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$? Generally, let $R$ be a discrete valuation ring and $K$ its fraction field. Let $\widehat{R}$ be the completion and $\widehat{K}$ the ...
4
votes
1answer
116 views

Legendre's symbol in Schrödinger model for the Weil representation

I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$. The way to present the action of the Weil representation $\omega$ of $Sp(2n,\...
0
votes
0answers
77 views

Reference request for a property of Bruhat decomposition

Let $G$ be a semi-simple algebraic group, $B$ a Borel subgroup of $G$, $T$ a maximal torus of $G$, and $U$ the unipotent radical of $B$. Let $I$ be the set of the vertices of the Dynkin diagram of $...
3
votes
1answer
130 views

Real automorphisms of the “quaternionic” real group ${\rm SO}^*(4m)$

Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$. Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...
4
votes
0answers
157 views

Why the hyperoctahedral group is a ``reductive'' group?

Sorry for the misleading title, I actually mean the following: The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
3
votes
1answer
256 views

What is $\rho^{\vee}(-1)$?

I am trying to understand the notation $\rho^{\vee}(-1)$. Let $T$ be a maximal torus of a semi-simple algebraic group $G$ and $\mathbb{G}_m$ the multiplicative group. I think that $\rho^{\vee}$ is a ...
7
votes
1answer
315 views

Example of a connected finite group scheme which is not solvable

What would be an example of a connected finite group scheme over a field $k$ that is not solvable? Here $k$ is algebraically closed. Let $\operatorname{GL}_n$ be the general linear group scheme over ...
1
vote
2answers
127 views

How to decompose an map $\phi: \mathbb{G}_m \to T$ as the product of a cocharacter $\phi'$ and a map $\phi'':\mathbb{G}_m \to T$?

Let $\mathbb{G}_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^*(T)=\{ \phi: T \to \mathbb{G}_m \}$ be the set of characters and $X_*(T)=\{ \phi^{\vee}: \mathbb{G}...
3
votes
1answer
283 views

Tits building of a linear algebraic group

I have a basic (probably naive) question about Tits buildings. Let $G$ be a (connected) linear algebraic group over a field $k$ (I am interested in the case where $k$ is algebraically closed but I ...
12
votes
0answers
483 views

Representation theory of finite groups with additional structures

Let $H$ be a finite group, representation theory of $H$ over $\Bbb C$ essentially determines $\operatorname{Hom}(H,GL_n(\Bbb C))$ up to conjugation action of $GL_n(\Bbb C)$ for each $n$. If we replace ...
1
vote
1answer
101 views

Functorial description of a certain subgroup scheme

We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for ...
1
vote
1answer
210 views

Complexification of compact Lie Groups and complex algebraic linear reductive groups

I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker". I found on internet that there is a bijection between complexifications of compact Lie ...
3
votes
0answers
109 views

$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property

Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...