All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
6
votes
0
answers
193
views
Bundles equivariant with respect to a transitive Lie algebra action
Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...
- 151
6
votes
0
answers
189
views
Infinite-dimensional BRST reduction
Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\...
- 1,391
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
6
votes
0
answers
304
views
Geometric interpretation of a formula for the induced character (fix point localization?)
Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
- 7,789
6
votes
0
answers
343
views
Are all stabilizer groups of the co-adjoint action smooth?
Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
- 1,702
6
votes
0
answers
268
views
duality between quiver variety and affine Grassmannian
Let $\frak{g}$ be a ADE type simple lie algebra. There are (at least) two geometric ways to get highest weight irreducible representations of $\frak{g}$. One is by considering constructible functions ...
- 849
6
votes
0
answers
291
views
Springer fibers and Weyl group
Let $\pi:\tilde{\mathfrak{g}}\rightarrow\mathfrak{g}$ the Grothendieck-Springer resolution of a semisimple Lie algebra $\mathfrak{g}$, over $\mathbb{C}$.
We know it's a small map, and that $\pi_{*}\...
- 3,472
6
votes
0
answers
354
views
Sporadic and Exceptional
I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...
- 151
6
votes
0
answers
223
views
$\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?
Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right $\...
user78172
6
votes
0
answers
327
views
Counting points on Hessenberg varieties over a finite field
Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...
- 1,856
6
votes
0
answers
161
views
LS paths construction
Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on $W/W_{...
- 61
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
6
votes
0
answers
319
views
Is it possible to describe the ideals of the Iwahori decomposition in a loop group using generalized minors?
Let $G$ be your favorite complex reductive algebraic group, and consider its (algebraic) loop group $G((t))$. A role very similar to the Borel $B\subset G$ is played in the loop group by the Iwahori
...
- 44.7k
6
votes
0
answers
460
views
Semistable reduction theorem over higher dimensional schemes
Let $k$ be a field, $S/k$ a smooth variety with function field $K$ and $U$ a nonempty open subscheme of $S$. For every finite separable extension $E/K$ we denote by $S^E$ (resp. $U^E$) the ...
- 1,673
6
votes
0
answers
304
views
How to decide if two surfaces occurring in Springer theory are isomorphic?
In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...
- 52.9k
6
votes
0
answers
402
views
What is known about line bundles on the tangent bundle of a flag variety?
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel ...
- 3,637
6
votes
0
answers
237
views
Moduli space of modules with fixed length
Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a $k$...
- 30.5k
6
votes
0
answers
264
views
When do equivariant sheaves on a formal neighborhood extend?
Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y \subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$...
- 1,038
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
5
votes
2
answers
775
views
Plucker relations in orthogonal Grassmannian
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known ...
- 121
5
votes
2
answers
1k
views
Describing the kernel of the exponential map as a homology group
I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I ...
- 5,637
5
votes
2
answers
694
views
Is a group scheme determined by its category of representations?
More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
- 299
5
votes
3
answers
945
views
algebraic groups and their Lie algebras
I have probably a stupid question about representations of algebraic groups:
Let $G$ be an algebraic group and $L$ be a Lie algebra of $G$. What is the connection between
categories of ...
- 143
5
votes
3
answers
1k
views
Tensor products of Weyl modules in positive characteristic
Let $G$ be a simple algebraic group over a field $k$, and let $U$ be the unipotent radical of a Borel subgroup $B$. Because $B$ normalises $U$, the group $H = B/U$ acts on the coordinate ring $\...
- 3,702
5
votes
1
answer
172
views
Extend a representation of a stabilizer group on a smooth DM stack to a locally free sheaf?
Consider a smooth tame Deligne-Mumford stack $[Y/G]$, a point $[p]$ on it with stabilizer group $H$. Is it true that every representation of $H$ can be extended to a locally free sheaf on $[Y/G]$?
...
- 157
5
votes
3
answers
963
views
$F_4$ flag variety
As flag variety or a homogeneous variety is a quotient $\Sigma=G/P$ of a reductive Lie group $G$ by one of its parabolic subgroups $P$. The subgroup $P$ fixes a flag of subspaces of standard ...
- 91
5
votes
1
answer
337
views
Invariants of cohomology of Springer sheaf
Let $G=Gl_n(\mathbb{C})$ and $\mathcal{N}$ be the nilpotent cone associated to it i.e nilpotent matrices inside $\mathfrak{g}=\mathfrak{gl}_n(\mathbb{C})$.
We have the variety $\tilde{\mathcal{N}}$ ...
- 1,061
5
votes
1
answer
458
views
When are finite maps quotients by finite groups?
Let $f: X \to Y$ be a finite map of projective varieties.
I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let ...
- 7,789
5
votes
2
answers
758
views
Equivariant Cohomology of a Complex Projective Variety
Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
- 4,920
5
votes
1
answer
466
views
Geometric properties of the adjoint action of a reductive group
$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...
- 605
5
votes
1
answer
166
views
Ideal of the boundary of $G/U \subset \overline{G/U}$
Let $G$ be a semi simple algebraic group, $B \subset G$ is a Borel subgroup and $U \subset B$ is the unipotent radical of $B$. We can consider the variety $G/U$. Let us also denote $\overline{G/U}:=\...
- 143
5
votes
2
answers
909
views
Representation theory over any field
I understand that representation theory of complex reductive groups is essentially combinatorial. By general principles, I imagine Galois theory then determines the theory over any field. For example, ...
- 51
5
votes
1
answer
733
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
- 639
5
votes
1
answer
408
views
Descent of vector bundle along branched cover of curve
Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ ...
- 1,761
5
votes
1
answer
541
views
Uniqueness of the wonderful compactification of a semi-simple group
Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)?
For instance, is the ...
user117617
5
votes
2
answers
812
views
representations of an algebraic group and extension of scalars
Let $G$ be an algebraic group over an algebraically closed field of characteristic zero $K$ and let $L$ be another algebraically closed field, together with an embedding $K \hookrightarrow L$.
Why ...
- 51
5
votes
1
answer
963
views
Singular/Smooth locus of Schubert variety of the affine grassmannian
Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat ...
- 627
5
votes
1
answer
736
views
When are orbits of semisimple group representations closed?
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Is there a nice description of those $v\in V$ for which the $...
- 4,920
5
votes
2
answers
1k
views
Is there a generalization of Borel-Weil-Bott theorem for not completely reducible vector bundles?
Let G be simple algebraic group, P its parabolic subgroup. Then the category of G-equivariant vector bundles on G/P is equivalent to the category of representations of P. Borel-Weil-Bott tells us how ...
- 420
5
votes
2
answers
549
views
Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space
What are the representations of $U(n)$ that induce (see link text) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known ...
- 3,399
5
votes
1
answer
464
views
Are degrees of polynomials in Weil's zeta function equal/bounded to/by dimensions of SOME cohomologies in non-smooth or non-projective case?
[Edit] Let me make question more focused. It is about details of Weil conjectures.
Rationality of zeta function does NOT require the manifold to be smooth & projective, so zeta function is a ...
- 24.9k
5
votes
1
answer
707
views
Cohomological interpretation of G-equivariant line bundles
In the theory of reductive algebraic groups, there is the following map (notation: $G$ reductive over an algebraically closed field, $T$ a maximal torus, $B$ a Borel, $X(T)$ the characters of $T$, $...
- 2,040
5
votes
1
answer
436
views
Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...
- 2,527
5
votes
1
answer
555
views
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and $S^-...
- 1,719
5
votes
1
answer
371
views
"Plucker" embedding of G/N, for reductive group G, affinization of quasiaffine varieties
I'll use "affinization" to describe the natural map of schemes $X \rightarrow \text{Spec}(\Gamma(X, \mathcal{O}_X))$. For quasi-affine varieties $X$ this is an open embedding.
Let $G$ be a reductive ...
- 1,423
5
votes
1
answer
469
views
If Spec(A) has a G-fixed point and a dense G-orbit, is Spec(A) a cone?
[Edited to include a dense orbit]
Let $X=Spec(A)$ be a normal affine scheme over an algebraically closed field $k$, with an action of a linearly reductive group $G$. Suppose $x\in X$ is a $G$-...
5
votes
1
answer
231
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
5
votes
3
answers
1k
views
Exceptional collections with many Exts
Background definitions:
Let $D$ be a triangulated category arising in nature (for instance as the cohomology category of a dg category). An object $E$ in $D$ is called exceptional if $RHom(E,E)$ is ...
- 1,859
5
votes
2
answers
223
views
Expressing properties of graded algebras in terms of the $\mathbb{G}_m$action
Let us fix a base ring $k$. The category of $\mathbb{Z}$-graded $k$-algebras is equivalent to the category of $\mathbb{G}_m$ equivariant affine $k$-schemes. The following 2 properties often come up ...
- 7,789
5
votes
1
answer
488
views
Isomorphism classes of sheaves which arise as extensions
Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in ...
- 7,377