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.

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54 views

étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$. Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$. For ...
11
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2answers
377 views

1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
5
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0answers
69 views

Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
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0answers
138 views

exactness of induction functor [on hold]

I am studying induction functor of algebraic groups. In particular, let $G$ be a reductive group scheme defined over an algebraically closed field $k$ of prime characteristic and $T$ a maximal torus ...
3
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0answers
57 views

Correspondence between dual center and linear characters of finite reductive group

Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq ...
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0answers
433 views

Quotient of an algebraic group by the connected component containing identity

Suppose $G$ is a finite flat group over scheme $S$, let $G^0$ be the connected component containing identity. Is it true that the quotient sheaf $G/G^0$ is always representable by a group scheme over ...
2
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2answers
246 views

Epimorphisms between affine group schemes

Does there exist a simple characterization of epimorphisms between affine group schemes over a field ? Are they faithfully flat morphisms ?
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2answers
165 views

Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations?

The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general ...
3
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1answer
112 views

A more precise description of conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups ...
8
votes
1answer
192 views

Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number ...
7
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1answer
190 views

Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$. However, it is quite possible ...
4
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1answer
239 views

About the conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups ...
4
votes
2answers
226 views

Adjoint semi-simple algebraic groups over non-algebraically closed fields

Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$. On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes algébriques ...
4
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2answers
213 views

A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any ...
4
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1answer
172 views

inductive construction of unipotent radicals

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are ...
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1answer
113 views

cubic forms and finiteness of $k^*/(k^*)^3$

In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms. If $k^*/(k^*)^2$ is finite then there are only finitely many ...
4
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1answer
58 views

reduced norm from degree 3 division algebra

Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3. Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^* ...
7
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2answers
305 views

Examples to keep in mind while reading the book 'The Admissible Dual…' by Bushnell and Kutzko and the importance of Interwining of representations

I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko. Let me first describe the book a ...
4
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1answer
111 views

How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat ...
4
votes
1answer
275 views

How to compute the tangent space of a quotient by a finite group

Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
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2answers
328 views

Perverse sheaves and tensor product

If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so ...
9
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198 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
4
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1answer
105 views

Cartan subspaces for general algebraic representations

So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have? To be precise, let $G\curvearrowright V$ be an algebraic ...
3
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1answer
89 views

Extension property for unipotent linear groups over rings

This is my first question, so my apologies if it is too simple/poorly motivated. During the course of some recent research I came across a particular variant of the following problem. Let $G$ ...
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0answers
209 views

How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much. For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...
1
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1answer
429 views

Rational points

Let $G$ be an affine algebraic group defined over a field of characteristic zero $K$. Suppose $G$ has only one single $K$-point, can we conclude that $G$ does not have more points?
30
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1answer
857 views

Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own. I'm becoming increasingly fascinated by stuff ...
11
votes
3answers
619 views

Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...
1
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1answer
69 views

Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)

Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GSp}_{2g}$, reductive and defines a Shimura subdatum of $(\mathrm{GSp}_{2g},\mathfrak{H}_g)$. Let $V$ be the natural representation of ...
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70 views

Notation clash between a representation and spectral radius

I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by ...
5
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3answers
239 views

structure of maximal tori in semisimple algebraic groups

I feel experts might be able to answer this question immediately. Let $G$ be a connected $\mathbb Q$-simple and $\mathbb Q$-isotropic algebraic group. Let $S$ be a maximal $\mathbb Q$-split torus ...
2
votes
1answer
283 views

When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?

I am trying to learn character sheaf theory, and encounter the following question: (*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}_q$, ...
3
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1answer
83 views

Is Koszulity equivalent to the Lusztig character formula holding?

Let $\pi$ denote a saturated set of weights. Let $S_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true: Claim: The algebra $S_q(\pi)$ is ...
18
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3answers
1k views

Small-index subgroups of SL(3,Z)

I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$. The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$, along the bottom row. I could not figure out ...
6
votes
2answers
140 views

Can we count the number of simple modules for a reduced enveloping algebra?

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple ...
3
votes
1answer
188 views

system of complex equations

I am working on a system of complex equations The question is the following: Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that $$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} ...
10
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1answer
183 views

commutators in upper triangular matrices

Consider the group $T_p(n)$ of all non-singular upper triangular matrices with entries in $\mathbb{F}_p.$ Its commutator subgroup is $U_p(n)$ (all elements in $T_p(n)$ with $1$s on the main diagonal). ...
4
votes
0answers
140 views

Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...
3
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2answers
199 views

open subgroup scheme closed

Let $G/S$ be a group scheme and $H \leq G$ an open subgroup scheme. Is $H \subseteq G$ closed? I want to apply this to $G^0 \leq G$ (see SGA 3, VI_B, Théorème 3.10) for $G$ commutative. (*) If $S = ...
4
votes
1answer
171 views

Is the Luna slice theorem valid for any orbit with a reductive stabilizer?

The Luna slice theorem states that if a reductive group $G$ acts on an affine space $X$ and $O$ is a closed orbit, then (in the etale topology) there exists a $G$-invariant negihborhood of $O$ with a ...
8
votes
1answer
415 views

Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein here. For other root systems the original question should be modified, leading to a precise ...
18
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1answer
219 views

How many ways can I factor a matrix (over $\mathbb{Z}$)?

Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$. Question 1 How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$? The answer to this question is pretty clearly ...
4
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1answer
169 views

Smooth algebraic stacks with precisely two $\mathbb C$-objects

In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one ...
4
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0answers
155 views

Descent of line bundles to the quotient

If a finite group acts $G$ on a variety $X$, consider the quotient $X/G$. I would like to understand which line bundle on $X$ descends to $X/G$. The action is not free. Can anyone direct me to some ...
2
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0answers
104 views

Quotients of quasi affine varieties and extension of scalars

I have some questions about GIT quotients and extensions of scalars of categorical quotients: 1) Let $X$ be a complex algebraic quasi-affine variety, $G$ an algebraic reductive group over ...
0
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1answer
77 views

presentation for a nilpotent group associated to the square of a coxeter element

This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups. Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...
0
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0answers
97 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...
3
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0answers
119 views

reduction mod $p$ of Weyl modules

Let $G$ be a reductive algebraic group defined over a non-Archimedean field $F$. Let $k_F$ be its residue field, of characteristic $p$. Assume $G$ is unramified over $F$, then it admits a hyperspecial ...
11
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1answer
224 views

Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group?

This question is a followup to two of my previous questions, see here and here. Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using ...
1
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1answer
107 views

Concept of Facets in the structure of reductive algebraic groups

Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...