All Questions
Tagged with lie-groups ag.algebraic-geometry
225 questions
5
votes
1
answer
230
views
Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 6,622
2
votes
3
answers
181
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
- 6,622
1
vote
0
answers
71
views
Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
2
votes
2
answers
87
views
Computation of ideal of functions, given by explicit quadratic equations, vanishing on $G/P$ for the exceptional Lie group $G_2.$
In Section 10.6.6 of Procesi's "Lie Groups" he writes that a theorem due to Kostant tells us that for an algebraic group $G$ and a parabolic subgroup group $P,$ the ideal of functions ...
- 201
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
2
votes
0
answers
157
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
7
votes
0
answers
194
views
Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44.7k
3
votes
0
answers
163
views
Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
- 21.8k
12
votes
2
answers
634
views
Coordinate ring of universal centralizer (BFM space)
In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
- 303
7
votes
0
answers
1k
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
4
votes
0
answers
226
views
Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
- 7,300
11
votes
2
answers
1k
views
When does a group act effectively and holomorphically on some Riemann surface?
Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary ...
- 609
1
vote
0
answers
46
views
Weight of adjoint action on a lower central series extension
Let $\mathcal{U}$ be a unipotent Lie $\mathbb{Q}_p$-group scheme, whose associated gradeds from the lower central series filtration are $\mathcal{U}_0 = \mathcal{U}^{\text{ab}}$, $\mathcal{U}_1 = [\...
- 2,907
4
votes
2
answers
181
views
The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?
We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
- 309
3
votes
0
answers
502
views
The definition of a homogeneous vector bundle
For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
- 157
1
vote
0
answers
120
views
Geometric induction of modules for algebraic groups
Let $\Bbbk$ be an algebraically closed field (of any characteristic). Let $G$ be an algebraic group over $\Bbbk$, and $H$ a closed (hence algebraic) subgroup of $G$.
Let $V$ be a finite-dimensional $...
- 1,077
2
votes
1
answer
429
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 673
3
votes
0
answers
119
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,527
6
votes
0
answers
306
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,479
7
votes
1
answer
280
views
Non-homogeneous line bundles over a homogeneous space
Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...
- 257
12
votes
0
answers
247
views
Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
- 2,516
4
votes
1
answer
240
views
Bounded generation of group by unipotent radicals of opposite parabolic subgroups
Let $G$ be an almost $k$-simple group that is also simply connected (so that $G(k)^{+}=G(k)$). For opposite parabolic subgroups $P$ and $P^{-}$, it is known that $G(k)^{+}$ is generated by the ...
- 949
6
votes
0
answers
176
views
Functions of polynomial growth on linear algebraic groups
$\DeclareMathOperator\GL{GL}$Let $G$ be a complex linear algebraic group, i.e. a subgroup in $\GL_n({\mathbb C})$, defined by a system of polynomial equations
$$
p_i(x)=0
$$
(here $p_i$ are ...
- 7,426
5
votes
1
answer
234
views
A group in a neighbourhood of a Zariski dense subgroup
By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense.
Suppose we have a Zariski ...
- 9,237
2
votes
1
answer
160
views
Is the restriction of the Cartan 3-form on conjugacy classes exact?
Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...
- 443
8
votes
1
answer
221
views
Coordinates on $N_+ \backslash \overline{B_+ w B_+} / N_+$
Let $G = \text{GL}_n(\mathbb{C})$ and let $N_+$ be the subgroup of upper triangular matrices with $1$'s on the diagonal. Let $w$ be a permutation, let $B_+ w B_+$ be the Bruhat cell and let $\overline{...
- 156k
24
votes
2
answers
1k
views
Lie groups generated by finitely many Lie algebra elements
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
- 443
0
votes
0
answers
228
views
How do I detect whether a representation is (or is not) the adjoint representation?
Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
- 1,379
4
votes
1
answer
257
views
Question regarding semistability of a point of GIT quotient
$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
- 585
2
votes
0
answers
127
views
Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
2
votes
1
answer
241
views
An extension of algebraic torus
Let $T_1$ and $T_2$ be algebraic tori over a field of characteristic 0. Let $T$ be an extension of $T_1$ by $T_2$, namely
$$
1\longrightarrow T_1\longrightarrow T\longrightarrow T_2\longrightarrow 1.
$...
- 833
4
votes
1
answer
436
views
Universal covering groups of simple linear algebraic group schemes
Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
- 7,527
2
votes
0
answers
187
views
Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
- 1,061
6
votes
0
answers
201
views
Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
0
votes
0
answers
229
views
Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201
8
votes
0
answers
228
views
What can we say about the homogeneous spaces $E_8/E_7$ and $E_7/E_6$?
For the three exceptional compact Lie groups $E_8, E_7, E_6$ we have the inclusions
$$
E_6 \subseteq E_7 \subseteq E_8.
$$
What can we say about the the homogeneous spaces
$$
E_8/E_7, ~~~~ E_7/E_6?
$$
...
- 305
1
vote
0
answers
107
views
Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure
In this question, the following fact was used by the respondent
A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar
measure contains a coset of $G^0$, the connected component of
$G$ ...
- 2,118
3
votes
1
answer
454
views
Principal bundles from a fibration of homogeneous spaces
Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces
$$
G/H \twoheadrightarrow G/H'.
$$
Will it ...
7
votes
2
answers
774
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 159
2
votes
1
answer
213
views
When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?
This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ...
user341790
3
votes
1
answer
269
views
Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups
This is a question about the answer in this other post: Symplectic group over integers and finite fields.
In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
user341790
2
votes
0
answers
140
views
Integral functional on algebraic varieties
Suppose that $X$ is a smooth complex algebraic variety and $X_{\mathbb{R}}$ is a real form of $X$. If $X_{\mathbb{R}}$ is compact and oriented as a real manifold, then it will admit a natural ...
- 1,077
3
votes
0
answers
138
views
density of unipotent flows in algebraic groups
Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...
- 81
7
votes
0
answers
656
views
Geodesics on algebraic manifold
A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual embedded algebraic manifold) $M \subseteq \Bbb{R}^n$ that is also a nonsingular algebraic set (which means $...
- 1,543
13
votes
3
answers
1k
views
Extending group actions to vector bundles
Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$.
Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
- 139
2
votes
1
answer
237
views
Properties of stabilizers of adjoint action general linear group
Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...
- 1,061
4
votes
0
answers
289
views
Formal integration (?) in Chabauty’s method
In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”,
the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
- 1,364
3
votes
0
answers
190
views
Harmonic analysis on reductive groups over $\mathbb{R}$
A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
- 131