All Questions
2,543 questions
1
vote
0
answers
98
views
Cohomology with coefficient in sheaf of morphisms of an algebraic group
Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
- 1,177
1
vote
0
answers
152
views
Properness of Hom-schemes for finite group schemes
In SGA 3 XI proposition 3.12 (b) (https://webusers.imj-prg.fr/~patrick.polo/SGA3/Expo11.pdf)
It is shown that:
If $G$ is an affine group scheme over a base $S$ and $H$ is a finite group scheme over $S$...
- 595
1
vote
0
answers
102
views
How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?
Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$.
Let $\tau$ be an irreducible cuspidal automorphic ...
- 1,019
1
vote
1
answer
346
views
Is the manifold of complex points of a quotient of compact groups just the tangent bundle?
In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
- 4,081
1
vote
0
answers
143
views
Degree of a regular function on an algebraic group
Let $k$ be a field and $G$ be an affine algebraic group. We
assume that $G$ is a $k$-subvariety of
$\operatorname{GL}(V) \subset \operatorname{End}(V)$ for appropriate vector space $V$
of dimension $d=...
- 6,006
1
vote
0
answers
86
views
If algebraic group $G$ acts faithfully on a $G$-qp variety $X$, then $G$ has a Faithful projective representation
In Michel Brion's survey on Linearization of algebraic group actions
is stated in Examples 3.2.2.(iv) following claim p 17
without proof:
We fix an algebraic group $G$ over field $k$ (of arbitrary ...
- 6,006
1
vote
0
answers
95
views
Why the trilinear GL_2 model is spherical?
Consider the homogeneous space $X:=GL_2\times GL_2\times GL_2/ H$ where $H=GL_2$ is diagonally embedded into $GL_2\times GL_2\times GL_2$. My question is why $X$ is spherical (i.e., there is a Borel ...
- 1,121
1
vote
0
answers
362
views
Invariant ring of linear algebraic groups
Let $G$ be a connected linear algebraic group. This question concerns Hilbert's 14th Problem for the adjoint action of $G$ on itself. Let $k[G]^G$ denote the algebra of regular functions on $G$ ...
- 2,751
1
vote
0
answers
91
views
Is this equivariant function constant?
Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
- 949
1
vote
0
answers
134
views
What is the analogue of Leibniz's rule for universal enveloping algebra?
Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra.
Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...
- 1,759
1
vote
0
answers
59
views
Distributivity property for smooth parabolic induction
Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...
- 473
1
vote
0
answers
88
views
$H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$
I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...
- 6,180
1
vote
0
answers
90
views
The splitting pattern of the Killing form of an algebraic group and the Tits index
Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...
- 1,479
1
vote
0
answers
216
views
Blow up of a flag variety at a point
What is the description of the blow-up space of $G/B$ at $B.$ Specifically:
(1) Can we describe it explicitly using an embedding of $G/B$ into some projective space associated to a dominant character ...
- 39
1
vote
0
answers
142
views
Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
- 11
1
vote
0
answers
118
views
Kazhdan-Lusztig Conjecture over non-algebraically closed field
Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or ...
- 473
1
vote
1
answer
239
views
Two different formulations of the Bott–Samelson resolution
There seem to be two formulations of the Bott–Samelson resolution flowing around. For concreteness, let $ G = \mathrm{GL}_{n} ( \mathbb{C} ) $ with the Borel subgroup $ B \subset G $ of upper ...
- 295
1
vote
0
answers
171
views
How to interpret divided powers in Kostant Z Form when passing to a field of characteristic p > 0?
Let $G = GL_n(\mathbb{K})$, where $\mathbb{K}$ is a field of characteristic $p > 0$. Let $\mathfrak{g} = \mathfrak{gl}_n(\mathbb{C})$. Let $e_{ij}$ denote the elementary matrices which are a basis ...
- 239
1
vote
0
answers
172
views
Is every connected solvable group Borel?
Is every connected solvable algebraic group a Borel subgroup of a reductive group? If a counterexample exists, I would ideally like it to be over $\Bbb C$.
- 3,079
1
vote
0
answers
51
views
Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
1
vote
0
answers
121
views
Embedding of wreath product
Consider the wreath product $G=\mathbb{Z}_2\wr O_n(\mathbb{R}),$ where $O_n(\mathbb{R})$ is the set of orthogonal groups over reals. Can we show $G$ embeds in a nice enough group (for example, some ...
- 521
1
vote
0
answers
53
views
Stabilizer group uniquely determines subspace
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$.
Let
$$ SO_Q(k):= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\sigma) = 1 \}$$
Let $L \...
- 63
1
vote
0
answers
257
views
Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?
I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices.
Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
- 885
1
vote
0
answers
91
views
Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?
Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...
- 479
1
vote
0
answers
335
views
Any finite flat commutative group scheme of $p$-power order is etale if $p$ is invertible on the base
This question is immediately related to Discriminant ideal in a member of Barsotti-Tate Group
dealing with Barsotti–Tate groups and here I
would like to clarify a proof presented by
Anonymous in the ...
- 6,006
1
vote
0
answers
165
views
Discriminant ideal in a member of Barsotti-Tate Group
Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$
is an inductive system
...
- 6,006
1
vote
0
answers
94
views
Is there an infinite order $\mathbb{F}_{p}$-section for a certain elliptic surface $\mathcal{E}_n$?
Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
- 1,833
1
vote
0
answers
62
views
Reference request for finite simple exceptional group of lie type $E_7(q)$ and its Schur covering group $2.E_7(q)$?
Does anyone have the paper named 'Génerateurs, relations et revêtements de groupes algébriques' written by Robert Steinberg in 1962, or any other reference for simple groups of Lie type $E_7(q)$ and ...
- 271
1
vote
0
answers
77
views
simple Lie groups over C [closed]
For an affine algebraic group over $\mathbb{C}$ which is simple, in the sense of no normal subgroups closed in the Zariski topology except for finite central subgroups and the whole thing, I'm trying ...
- 2,125
1
vote
0
answers
153
views
Descent of projective bundles
A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients.
It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
- 297
1
vote
0
answers
126
views
Which quotients of surface groups are linear?
Let $S$ be a compact connected Riemann surface, and let $\pi = \pi_1(S)$ be its fundamental group. Let $\pi \to G$ be a surjective homomorphism.
Is $G$ linear? (That is, does $G$ admit a ...
- 513
1
vote
0
answers
110
views
Reference request: Commutator relations for the exceptional group F4
Is there any standard reference for the commutator relations for the exceptional group of type $F_4$?
If this question is not appropriate here, please let me know and I will delete it.
Thanks in ...
- 960
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
vote
0
answers
150
views
Formal group as a limit of its finite subgroups
I'm reading Manin's article on formal groups and I have a problem with Lemma 1.1.
Consider $k$ a prefect ring of characteristic $p$ and $(A,m,k)$ a noetherian complete local ring of the same ...
- 1,093
1
vote
0
answers
146
views
Factoriality of schubert cells in affine flag variety
Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one.
For each $...
- 3,472
1
vote
0
answers
64
views
Conjugacy classes and normal form of $O_n$ and $U_n$
I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups.
...
- 11
1
vote
0
answers
85
views
Galois orbit of a $k_{s}$ - torus
I have some trouble while reading a proof of a lemma in the book
Conrad, Brian; Gabber, Ofer; Prasad, Gopal, Pseudo-reductive groups., New Mathematical Monographs 26. Cambridge: Cambridge University ...
- 11
1
vote
0
answers
95
views
Units in the coordinate ring on a reductive group
Let $K$ be a field and $G$ a connected reductive group over $K$.
Can we describe $K[G]^{*}$?
- 3,472
1
vote
0
answers
68
views
Partitions corresponding to unipotent elements in simple classical algebraic groups
Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.
For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid ...
- 121
1
vote
0
answers
258
views
Bruhat cell of a Coxeter element
If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a ...
- 332
1
vote
0
answers
112
views
Nontrivial relations of the irreducible root systems
For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...
- 332
1
vote
0
answers
371
views
Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
- 332
1
vote
0
answers
153
views
Is the Bruhat cell Zariski open in a connected algebraic group $G$? [closed]
Is the Bruhat cell Zariski-open in a connected algebraic group $G$?
Specifically, is the big Bruhat cell Zariski-open (and maybe Zariski-dense)?
Is it true for all the Bruhat cells?
- 332
1
vote
0
answers
140
views
Describing compact Lie groups in purely topological terms
Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
- 993
1
vote
0
answers
63
views
Question on the proof that the Jacquet module preserves admissibility
Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
- 6,180
1
vote
0
answers
195
views
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 ...
- 3,625
1
vote
0
answers
116
views
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 ...
- 14.2k
1
vote
0
answers
88
views
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 $P_0$...
- 3,625
1
vote
0
answers
141
views
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 ...
- 6,180
1
vote
0
answers
68
views
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\...
- 20.2k