All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
4
votes
0
answers
162
views
Is the tangent bundle to the algebraic loop group of $GL_n$ ample?
I am trying to understand the tangent bundle to algebraic loop groups, particularly for $G=GL_n$, over arbitrary characteristic. Can anyone point me to existing literature related to this? In ...
user106243
4
votes
0
answers
240
views
On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type
A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.
Let $(G',X')$ be ...
- 133
4
votes
0
answers
173
views
Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?
Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
- 1,833
4
votes
0
answers
522
views
Equivariant sheaves over affine schemes
Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and
let $A$ be a commutative $k$-algebra which is acted on by $G$.
We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies ...
- 41
4
votes
0
answers
189
views
Fibers of torus equivariant moment maps
Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...
- 1,719
4
votes
0
answers
597
views
Euler characteristic, character of group representation and Riemann Roch theorem
I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in $Rep(G)$...
- 1,982
4
votes
0
answers
225
views
Linear independence of points under projection of Veronese re-embedding
Let $V$ be a complex vector space.
Let $x_1,...,x_k\in PV$. Let $v_d: PV\rightarrow PS^dV$
be the Veronese. Then $v_d(x_1),...,v_d(x_k)$ are in general linear position
as long as $k\leq d-1$.
Now let $...
- 835
4
votes
0
answers
351
views
Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ ...
- 41
4
votes
0
answers
140
views
Scaling-Invariant Orbits of Semisimple Group Representations
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...
- 4,920
4
votes
0
answers
186
views
Is there an arithmetic analogue of Drinfeld's count of a number of 2d irreps of fundamental group of a curve ?
There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field".
It gives a formula for this ...
- 24.9k
4
votes
0
answers
386
views
Reference request for character formula between tensor products of Weyl modules.
So it is well known that when you tensor together two induced modules for an algebraic group
$\nabla(\lambda) \otimes \nabla(\mu)$ that the result has a filtration by other induced modules, (I.e. it ...
- 41
4
votes
0
answers
179
views
Equivariant version of a spectral sequence in Beilinson-Ginzburg-Soergel
In Beilinson, Ginzburg, and Soergel, "Koszul Duality Patterns in Representation Theory" (comment 3.4), the authors outline a spectral sequence as follows:
Given a filtered complex algebraic variety $...
- 103
4
votes
0
answers
360
views
On a resolution of sections of line bundles on the cotangent bundle of a flag variety
Background
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. Let $B \subseteq G$ be a Borel subgroup and let $U \subseteq B$ be its unipotent radical....
- 3,637
4
votes
0
answers
266
views
ind schemes and formally smoothness
In Beilinson-Drinfeld (Hitchin System, lemma (362)) they show that if f:X->Y is a morphism between formally smooth ind-schemes of ind-finite type such that the differential is surjective then f is ...
- 41
4
votes
0
answers
178
views
İs the deligne lusztig variety corresponding to group of type G(2) computed?
I know that the deligne kusztig varieties corresponding to Suzuki group, Ree group and PGU-_2'(q) are explictly computed. Are there any result for the group G(2). Here 'result' means equation of ...
- 225
4
votes
0
answers
463
views
Covers of nilpotent orbits as degenerations of semisimple orbits
Let $G$ be a complex semisimple group, $P$ a parabolic subgroup, $\mathfrak{p}$ its Lie algebra, $\mathfrak{l}$, $\mathfrak{n}$ its Levi part and the nil radical, respectively. Let $\mathfrak{k}$ the ...
- 6,123
4
votes
0
answers
331
views
What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
4
votes
0
answers
118
views
Representations with small dual
I want to construct an irreducible representation $V$ of any group $G$ such that there exist a mapping of form $\sum_{i=1}^{100} a_i g_i, a_i \in C, g_i \in G$ with kernel of dimension
$dim V-1$. In ...
- 41
4
votes
0
answers
420
views
Etale cohomology analogue for the semistable reduction theorem
Let $K$ be a field, $X/K$ a smooth projective variety, $l\neq char(K)$ a prime number and $q\ge 0$. Then we define $\overline{X}:=X_{K_{sep}}$ and denote by
$\rho_{X, l}^{(q)}$ the representation of $...
- 1,673
3
votes
2
answers
1k
views
Hitchin fibration outside of type A
I've been learning a bit about the Hitchin fibration, and I wanted to ask about how it works outside of type A.
Background: In type A, the Hitchin fibration is reviewed on pg 14 of this paper of ...
- 2,216
3
votes
2
answers
1k
views
Indexing the Line Bundles of a Flag Manifold
Following on from this question link text, how are the line bundles of a complex flag variety indexed? Are they still labeled by the integers? If so, why? A representation theory explanition in terms ...
- 3,399
3
votes
1
answer
1k
views
Quotients of Grassmannians
Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus. Then the Grassmannians $Gr(r,n)$ and $G(n-r,n)$ are isomorphic. Now for the left action of the torus on each of them can we say that the GIT ...
- 121
3
votes
2
answers
1k
views
Rational Characters of a reductive group have the same rank as split component
Let $G$ be a connected reductive group defined over a perfect field $F$. The split component $A$ of $G$ is the unique maximal $F$-split subtorus of the radical of $G$. For an algebraic group $H$ ...
- 6,180
3
votes
1
answer
547
views
Is D-module on flag variety of Lie algebra a scheme?
This question was motivated by the answers in D-module as quasi coherent sheaves on deRham stack. What I am interested in is the case of D-module on flag variety of Lie algebra. So,in this case, if we ...
- 5,515
3
votes
1
answer
249
views
Completely reducible subgroups over local field in terms of closed orbits
$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...
- 667
3
votes
1
answer
645
views
Cohomology of Springer resolution
This question is elementary. Let $G$ be a simple algebraic group over $\mathbb{C}$, and let $B$ be a choice of Borel subgroup, with unipotent radical $U$ with Lie algebra $\mathfrak{n}$. Then the ...
- 1,209
3
votes
2
answers
707
views
$k$ structures on $K$ vector spaces
The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...
- 1,553
3
votes
1
answer
322
views
Pairs of quadratic forms and $\mathbf{A}^8/\mathrm{SL}_2^{\times 3}$
$\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$Fix the base field to be the complex numbers $\mathbf{C}$. Let $\std = \mathbf{A}^2$ denote the standard ...
- 5,760
3
votes
1
answer
137
views
Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$
It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$.
For ...
- 1,343
3
votes
1
answer
335
views
Maximal Coset representative for the Weyl group of a Parabolic
Let $G=SL_n$ and let $P_i$ be a maximal parabolic corresponding to a simple root say $\alpha_i$. Let $W_{P_i}$ be the Weyl group of $P_i$. Is there an efficient way to compute the longest coset ...
- 43
3
votes
1
answer
310
views
Description of the algebra of $G$-invariant polynomials by generators and relations
Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = \text{...
- 45
3
votes
2
answers
616
views
counting points on nilpotent Springer fiber
Computing $p$-adic orbital integral I come to the following question. My ground field $k$ is the residue field of a non-arch local field, i.e. a finite field. I am happy to put any assumption on $\...
- 1,856
3
votes
2
answers
395
views
Cohen-Macaulay Representations
I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research.
If yes, then
what are some of ...
- 839
3
votes
1
answer
1k
views
Is base affine space a trivial fibration?
I am very confused about the base affine space. Let $G$ be a complex reductive algebraic group and $H=B/U$ the abstract torus.
If I understand it correctly (Edit: which turns out not to be the case! ...
- 13.2k
3
votes
2
answers
762
views
Vanishing cohomology of line bundles on the Springer resolution
My question is regarding Broer's paper "Line bundles on the cotangent bundle of the flag variety" (see http://www.springerlink.com/content/t41418q436524515/). Given the Springer resolution, and its ...
- 2,216
3
votes
1
answer
337
views
Homogeneous vector bundles with zero chern classes
We know that a line bundle $L$ on the complex flag variety $G/P$ is trivial iff $c_1(E) = 0$. But if we have a homogeneous vector bundle $E$ of higher rank, then is it true that $c_i(E) = 0$ $ \...
user69183
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 31
3
votes
1
answer
195
views
How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?
Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
p:S^2S^3\mathbb{C}...
- 2,527
3
votes
2
answers
704
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,719
3
votes
1
answer
884
views
Why is the semisimple quotient of a reductive group with semisimple rank 1 equal to PGL2?
Lets work over $\mathbb{C}$. Let $G$ be a reductive group with semisimple rank 1 (this means that a maximal torus in $G / R(G)$ has dimension 1). In section 25 of Humphreys book "linear algebraic ...
- 3,830
3
votes
1
answer
385
views
Is Mazur's deformation ring R integral?
Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon G_{\...
- 111
3
votes
1
answer
464
views
Action of Non-Split Torus in Deligne-Lustzig induction
Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}_p).$
In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ ...
- 565
3
votes
1
answer
251
views
About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
- 133
3
votes
1
answer
198
views
What is the minimum possible k-rank of a quasi-split reductive group over a field?
It is not possible for a quasi-split reductive group $G$ over a field $k$ to be anisotropic (unless it is solvable, hence its connected component is a torus). Indeed, there exists a proper $k$-...
- 605
3
votes
1
answer
397
views
Algebraic representations and vector bundles
This might seem like a silly question considering my relatively elementary knowledge of representation theory.
The question is regarding Eugen Hellman 's paper titled "On the derived category of ...
- 821
3
votes
2
answers
578
views
Moduli stack of quiver representations
Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT ...
- 1,061
3
votes
1
answer
223
views
What is the geometric quotient of the abelian threefold?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$.
Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant ...
- 1,833
3
votes
1
answer
354
views
Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?
Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
- 7,789
3
votes
1
answer
359
views
Tensor and symmetric invariants of Symmetric group
For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \...
- 125
3
votes
1
answer
201
views
A fact about $t/W$ and the centralizer bundle on $\mathfrak{g}^{\text{reg}}$
Let $\mathfrak{g}$ be a simple Lie algebra; let $R = \mathfrak{g}^{*, reg}$ denote the regular locus in the dual Lie algebra. Consider the vector bundle $\mathfrak{z}$ over $R$, whose fiber over a ...
- 2,216