Questions tagged [algebraic-groups]
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.
2,105
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Homeomorphisms of Springer fibers
Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...
2
votes
1
answer
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Lie algebra of an algebraic group generated by connected subgroups
Let V be a vector space over an algebraically closed field.
Let $\{H_i\}_{i \in I}$ be a collection of closed connected subgroups of $\operatorname{GL}(V)$ (wrt. Zariski topology). It is a basic ...
1
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0
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195
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Trying to understand an argument to put a topology on $GL_n(R)$ when $R$ is a topological ring
I'm reading this set of notes and I'm trying to understand this passage where they explain how to put a topology on $GL_n(R)$ when $R$ is a topological ring, which I am not completely following. The ...
9
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1
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527
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Geometric interpretations of nil-Hecke ring and affine Hecke algebra
I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^*$-equivariant $K$-theory of flag varieties.
Let $G$ be a semisimple, simply connected ...
5
votes
1
answer
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What does Rosenlicht mean by a "point"? By $k(v_1,v_2)$?
This is cross-posted from Math.SE at the recommendation of a commenter.
I'm reading M. Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" [link], and having trouble with some of the ...
5
votes
1
answer
518
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Unipotent completion of free group
Whilst I am reading articles on unipotent completion to understand its basic construction, I found something confusing. Let $F$ be a free group of rank 2 whose generating letters are $x$ and $y$ and ...
15
votes
3
answers
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Finitely generated matrix groups whose eigenvalues are all algebraic
Let $G$ be a finitely generated subgroup of $GL(n,\mathbb{C})$. Assume that there exists a number field $k$ (i.e. a finite extension of $\mathbb{Q}$) such that for all $g \in G$, the eigenvalues of $...
13
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4
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exponential/logarithm for unipotent algebraic groups
Let $k$ be a field (of possibly positive characteristic), let $U_n$ denote the space of all $n \times n$ unipotent upper triangular matrices over $k$, and let $G$ be an algebraic subgroup of $U_n$ (...
2
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1
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164
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If $H \subset \operatorname{GL}(n)$, can we realize $\operatorname{Res}_{K/k} H$ inside $\operatorname{GL}([K : k]n)$?
Let $K/k$ be a finite separable extension. If necessary, we can assume $[K : k] = 2$. Let $H$ be a $K$-closed subgroup of $\operatorname{GL}_n$, and let $\tilde{H} = \operatorname{Res}_{K/k}H$. ...
3
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1
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Is this unipotent group, over characteristic 2, connected?
Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
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Does every character from group factor through largest central subgroup?
Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.
Let $X(G)_{\mathbb{Q}}$ be the addtive group of ...
4
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3
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633
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Real points of reductive groups and connected components
Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...
3
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174
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Relative position on flag variety
Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...
8
votes
1
answer
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Orbits of action of the split group of type $F_4$
Let the split group of type $F_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F_4\times \mathbb{G}_m$ on $A$, given by letting $\mathbb{G}_m$ act by scalar ...
4
votes
2
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512
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Variety of conjugacy classes
Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
1
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1
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257
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Element in finite number of Borel subgroups
Let G is a linear algebraic group over algebraic closed field, B is an
Borel subgroup of G. Does there exist g$\in$G which is only in a finite
numbers of conjugates of B (they are also Borel ...
2
votes
1
answer
292
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Levi subgroup of Siegel parabolic of GSpin
I consider the group $G=\mathrm{GSpin(V)}$ as in this question.
We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using ...
3
votes
2
answers
700
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Criterion for vector bundle to descend to GIT quotient in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic. Let $G$ be an affine reductive group acting on a smooth projective variety $X$. Let $E$ be a vector bundle on $X$ such that the ...
8
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Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?
Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
2
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0
answers
100
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Component Groups of Reductive Groups
Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can ...
9
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Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
5
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0
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What is an example of a cokernel $B/\phi(A)$ in group schemes which does not have $A=\mu_d$ and requires the fppf topology to be a sheaf?
Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $...
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115
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Example of a spherical homogeneous space $G/H$ with a pairs of colors and with the center of $G$ not contained in $H$?
Let $G$ be a simply connected simple algebraic group over $\mathbb C$,
$B\subset G$ a Borel subgroup, and $T\subset B$ a maximal torus.
Let $\mathcal{S}=\mathcal{S}(G,T,B)$ denote the set of simple ...
3
votes
1
answer
134
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Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$
I have the following question: Consider the following two finite groups: $GL_{n}(q)$ and $SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of $SL_{n}(q)$. Now, ...
0
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0
answers
205
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How is this pairing $\langle\,,\rangle$ defined of cocharacter and character of an algebraic group?
Let $G$ be a semisimple linear algebraic group. Let $X^*$ be the group of characters and $X_*$ be the group of cocharacters. Then I know that there exists a pairing $\langle\,,\rangle : X^*(G) \times ...
2
votes
0
answers
152
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Most important results for Shalika germs
This is more of a general question, but what do you think are the most important results for Shalika germs if you were giving a presentation? You can assume the target audience to be 2nd-3rd year ...
0
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145
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Groups implementable by finite field
I'm interested in finding all groups for which the group operation (and inverse map) may be implemented using finite field arithmetic.
I've done some searching and have come across "algebraic groups",...
10
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1
answer
542
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How is the sheaf defined for $G/H$ where $G$ is an algebraic group and $H$ is a normal closed subgroup?
I have learned that if $G$ is an algebraic group and $H$ is a normal closed subgroup then $G/H$ is also an algebraic group satisfying:
for any morphisms $\phi : G \rightarrow X$ constant on the ...
4
votes
0
answers
420
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Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$
Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
2
votes
0
answers
132
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Maximal split tori in quasisplit groups
Let $k$ be a number field. Let $G$ be a quasisplit (but not split) semisimple group over $k$. Let $S$ be a maximal $k$-split torus in $G$. Let $T$ be the centralizer $Z_G(S)$ of $S$; it is a maximal ...
3
votes
1
answer
422
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On connectedness of intersection of subgroups
I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...
1
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0
answers
88
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How does this $\chi \in X^*(P)$ define a line bundle on $P \backslash G$, where $G$ is a semisimple linear algebraic group
Let $F$ be a number field, and $G$ be a semisimple linear algebraic group over $F$. We let $P_0 \subseteq G$ be a minimal $F$-rational parabolic subgroup.
Let $P$ be a standard (i.e. containing $P_0$...
29
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3
answers
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What is the defining property of reductive groups and why are they important?
Having read (skimmed more like) many surveys of the Langlands Program and similar, it seems the related ideas apply exclusively to groups that are "reductive".
But nowhere, either in these surveys or ...
6
votes
1
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502
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How to compute Galois representations from etale cohomology groups of a generalized flag variety?
Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
3
votes
0
answers
90
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Isogeny to a semi-simple group
Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ?
I have in mind the case of $\text{GL}...
9
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0
answers
159
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Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...
8
votes
1
answer
396
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L-packets in the local Langlands correspondence: why finite sets?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
8
votes
1
answer
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How should the local Langlands correspondence for general reductive groups take into account different inner forms?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
3
votes
1
answer
157
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Is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ a unitary group?
Let $F$ be an p-adic field, and $E$ be a quadratic extension of $F$, then is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ isomorphic to some unitary group $U_{E/F}(2)$? ...
2
votes
1
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601
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The centralizer of a semisimple element which is not contained in any proper parabolic subgroups
Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the ...
5
votes
1
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182
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Reference for showing that $\mathcal{O}(G) \cong U(\mathfrak{g})^{\circ}$ when $G$ is connected and simply connected
Let $G$ be an algebraic group. We can try to reconstruct $G$ from its lie algebra $\mathfrak{g}$, but the best we get in general is a formal group scheme $\operatorname{Spf}(U(\mathfrak{g})^*)$, where ...
19
votes
1
answer
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What are the equations for $SL_3/SL_2$?
Consider $SL_2$ embedded into $SL_3$ as upper left block matrices. The quotient $SL_3/SL_2$ is an affine variety, as is any quotient of reductive groups. How does one describe $SL_3/SL_2$? What are ...
8
votes
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368
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Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
9
votes
1
answer
511
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Polynomial invariants for simple algebraic groups
Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...
4
votes
0
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177
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Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?
Prove or disprove:
Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$?
Note that then $G_K \...
2
votes
1
answer
219
views
Relative weight lattice
Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, ...
1
vote
0
answers
139
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Do we have $K \cap P = (K \cap M)(K \cap N)$?
Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
2
votes
0
answers
135
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Homology of SL(2,R) with finite coefficients
Consider the third homology group of a real special linear group
$H_3 (SL(2,\mathbb{R}),\mathbb{F}_p)$. It is known$[1]$ that for $p=2$ the third homology group of $SL(2,\mathbb{R})$ vanishes.
...
1
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0
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68
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When are "square spans" not transversal?
Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
5
votes
0
answers
200
views
Does $G$ act 2-transitively on its Bruhat-Tits building?
Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$.
Question: If $x,y,x',y'$ are vertices, ...