All Questions
2,543 questions
2
votes
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350
views
What is the explicit version of the Peter Weyl Theorem?
While the name "Peter-Weyl" is reserved for the compact group case, I prefer to talk in greater generality. Let $G$ be a unimodular type I topological group with a fixed Haar measure. The ...
- 2,081
2
votes
0
answers
190
views
Conjugacy classes in centralizers
Let $G$ be a complex reductive group, let $g$ be an element, and let $C$ be the connected component of its centralizer. I'm curious about what is known about the intersection of conjugacy classes in $...
- 491
2
votes
0
answers
152
views
What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?
Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...
- 1,066
2
votes
0
answers
130
views
Dynkin automorphism of $\mathrm{SO}_{2n}$
Let $\sigma$ be a Dynkin automorphism of $G=\mathrm{SO}_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to ...
- 21
2
votes
0
answers
209
views
Are Milnor K-groups algebraic groups?
Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is,
$$
K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I,
$...
- 479
2
votes
0
answers
102
views
lie algebra bundle and underlying vector bundle
Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$.
As a vector bundle it is trivial, ...
- 3,472
2
votes
0
answers
291
views
Automorphisms group of complex and real simple Lie algebras
$\DeclareMathOperator{\Inn}{\operatorname{Inn}}\DeclareMathOperator{\Aut}{\operatorname{Aut}}\DeclareMathOperator{\Out}{\operatorname{Out}}\DeclareMathOperator{\g}{\mathfrak{g}}$According to Wikipedia,...
- 2,147
2
votes
0
answers
80
views
Reference request: additive basis of $\mathbb{C}[N]$
Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$:
$$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and ...
- 6,211
2
votes
0
answers
192
views
Maximal connected subgroup of orthogonal group
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$ with $\dim(V) \geq 3$
Define
$$ SO_Q:= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\...
- 63
2
votes
0
answers
139
views
Bruhat-Tits theory: how does the normalizer act on an apartment?
Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...
- 6,180
2
votes
0
answers
175
views
Jordan normal form in a reductive group
Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
- 491
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
2
votes
0
answers
48
views
Carter Payne homomorphisms and reduced expressions
Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...
- 1,413
2
votes
0
answers
95
views
Metaplectic group $Mp(2n)(\mathbb{A}_F)$ splits over $Sp(2n)(F)$?
My question is the title.
In some literature, authors seem to use this without assumption.
Is it ture in general?
- 1,759
2
votes
0
answers
206
views
Diagonal action on external product of trivial principal bundles
(Title, tags and phrasing of this problem might not very good, so please feel free to edit it.)
In the course of writing a long and technical proof, I recently came across the following problem:
Let ...
- 21
2
votes
0
answers
77
views
roots and embeddings
Let $G$ be a connected reductive group over an algebraically closed field, can we always find an embedding let $\rho:G\rightarrow GL_n$, that sends a Borel pair $(B_G,T_G)$ to $(B,T)$ and center to ...
- 3,472
2
votes
0
answers
235
views
Full automorphism group of a Bruhat-Tits building
If we start with a semisimple algebraic group $G$ defined over a non-archimedean local field and want to understand the relationship of this group with the full type-preserving automorphism group of ...
- 2,125
2
votes
0
answers
201
views
Restriction to the maximal torus
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}$Let me say that I am kind of sure that all the things I ...
- 2,152
2
votes
0
answers
119
views
relative rank two group: structure of parabolic subgroup-- high-level Jacobson--Morozov sl_2 triple
Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ ...
- 301
2
votes
0
answers
306
views
Fiber product arising from reductive group action on varieties
Let $G$ be a reductive group acting on a smooth projective $X$. Let $P$ be a parabolic subgroup of $G$, and $Y$ a locally closed subvariety invariant under $P$. Assume in addition $Y$ is smooth. ( ...
- 815
2
votes
0
answers
73
views
Does $\mathfrak{g}^*$ split off from the augmentation ideal
(Note: I had this posted on MSE for a while but didn't get much of a response... so I'm posting it here now.)
Let $G$ be an affine algebraic group over an algebraically closed field $k$ of ...
- 1,077
2
votes
0
answers
98
views
On a rationality question about morphic actions of a unipotent algebraic group defined over a non-archimedean local field
Fix an algebraically closed field $K$, and let $G$ be a unipotent linear algebraic group over $K$ acting morphically on an affine variety $X$. According to [1, Prop. 2.5] we have the following result:
...
- 353
2
votes
0
answers
150
views
Projection of conormal bundle of Schubert variety under Springer resolution
Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ ,
$\mu:T^*(G/B)\to \mathcal{N}$ ...
- 849
2
votes
0
answers
98
views
Query about Bruhat-Tits buildings over completions of fields with respect to a valuation, but the residue class field is not necessarily finite
I'm reading Soulé's article "Chevalley groups over polynomial rings", and he has a situation where $k$ is an arbitrary field, not necessarily finite, then you take a simple transcendental extension of ...
- 2,125
2
votes
0
answers
275
views
Étale group scheme exact sequence
Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$:
$$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\...
user141691
2
votes
0
answers
111
views
A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$
Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.
From "A note on generators for arithmetic subgroups of algebraic groups" by ...
- 332
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
2
votes
0
answers
62
views
What are the compact Aut(A) of an algebra A(G), G finite, that contains the identity?
If we have an algebra A over a finite group G, then if G is non-abelian we can have a non-trivial set of compact automorphisms of A that map the elements of G onto a set isomorphic to G. It may be ...
2
votes
0
answers
242
views
Intersection of Levi subgroups via intersection of their Weyl groups
Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
- 163
2
votes
0
answers
173
views
Characterize an element of $\operatorname{SL}_n(\mathbb Z)$
I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$.
In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where
$$e_{1,n}(m)=
...
- 332
2
votes
0
answers
261
views
Vector extension for p-divisible group
Background:
I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention.
Reference:
Messing, The crystals associated to Barsotti-Tate ...
- 397
2
votes
0
answers
118
views
Explicit example for Display Theory for p-divisible group
Recently I am studying the display theory of formal p-div groups ([1] )by Zink.
I would like to study by working on an example. As far as I understood, the display theory is a generalization of ...
- 397
2
votes
1
answer
221
views
When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]
Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure
$$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
- 311
2
votes
0
answers
196
views
Trying to understand why Eisenstein series is well defined
I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...
- 3,625
2
votes
0
answers
103
views
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 ...
- 953
2
votes
0
answers
162
views
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 ...
- 463
2
votes
0
answers
139
views
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 ...
2
votes
0
answers
143
views
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.
...
- 403
2
votes
0
answers
60
views
A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$
THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...
- 5,422
2
votes
0
answers
104
views
Alternatives to the ring of invariants depicting the orbit closures?
Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
- 3,031
2
votes
0
answers
291
views
tangent space to a (not necessarily algebraic/Lie/..) group
Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``...
- 2,232
2
votes
0
answers
100
views
Classical reductive group schemes vs. unitary groups of separable algebras with involution --- reference request
Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...
- 2,928
2
votes
0
answers
104
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 ...
- 121
2
votes
0
answers
141
views
Are Generalized Involution Varieties hyperlane sections of Generalized Severi Brauer Varieties?
Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.
Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.
...
- 1,479
2
votes
0
answers
304
views
Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 203
2
votes
0
answers
121
views
Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
2
votes
0
answers
170
views
Orbits under an algebraic group inside a Shimura variety
Let $(G,X)$ be a Shimura datum, $K$ a compact open subgroup of $G(\mathbb A_f)$. Let $H\subset G$ be a $\mathbb Q$-subgroup. Choose a point $x\in X$. What can be said about the orbit $H(\mathbb R)\...
- 796
2
votes
0
answers
382
views
Integral smooth model of unramified reductive groups
My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$...
- 193
2
votes
0
answers
264
views
etale cohomology of tori
Let $k$ be an algebraically closed field.
Let $A$ be a strictly henselian local ring which is a $k$-algebra.
Let $T$ a torus over $k((t))$.
Can we compute $H^{1}(A((t)),T)$?
- 3,472
2
votes
0
answers
440
views
Finite Flat Group Schemes are Syntomic
Let $G$ be a group scheme which is finite and flat over a base $S$. Then is $G$ always syntomic over $S$, i.e. is $G$ a local complete intersection over $S$?
This reduces immediately to the case ...
- 143