All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
3
votes
1
answer
260
views
Invariants of general linear groups under torus action
Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...
- 1,061
3
votes
1
answer
564
views
Computing affine Springer fibers
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\Gr{Gr}\DeclareMathOperator\SL{SL}$I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $...
- 3,019
3
votes
1
answer
180
views
Kernel of restriction for ring of functions on reductive groups
Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then ...
- 689
3
votes
1
answer
161
views
How to show that a map which relates to Donaldson–Thomas invariants is an automorphism?
I am reading the lecture notes INTRODUCTION TO DONALDSON–THOMAS INVARIANTS. I have a question in the end of page 1 about the proof of a map is an automorphism.
Let $m>0$ be an integer. Let $\...
- 6,201
3
votes
1
answer
203
views
Symplectic representation of modular group
The modular group $\Gamma_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a ...
- 33
3
votes
1
answer
431
views
variations of finite stabilizer in the action of an algebraic group on an affine variety
Assume that $G$ is an affine reductive algebraic group (I am mostly interested in the case $GL_n$) over an algebraically closed field $K$ of characteristic zero. Assume also that $G$ acts on an affine ...
- 5,039
3
votes
2
answers
526
views
Algebraic Groups, Modules, and Comodules
Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...
- 401
3
votes
1
answer
172
views
On the linearizability of the action of a finite group on a formal polydisc
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
- 541
3
votes
1
answer
187
views
Regular embeddings of a reductive groups with induced center
Let $G$ be a reductive group over the finite field $\mathbb{F}_q$. Then a regular embedding of $G$ is an $\mathbb{F}_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group ...
- 755
3
votes
1
answer
152
views
Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?
If $ G $ is a finite cyclic group, $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ is a linear $ n $ dimensional representation of $ G $, and $ \{x_{1},\dots,x_{n}\} $ is a basis of $ \mathbf{V}^{\ast} ...
- 782
3
votes
1
answer
304
views
A question on algebraic loop groops
Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in $\...
- 2,554
3
votes
1
answer
501
views
Equivariant cohomology of nilpotent orbits
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, and let $N$ be a nilpotent orbit of $\mathfrak{g}$. What is the equivariant cohomology of its closure, $H^*_G(\overline{N})$, with ...
- 6,123
3
votes
1
answer
252
views
Large modules with non-trivial cohomology
Let $p$ be a prime and $F$ algebraic closer of $F_p$.
I want to know if it is possible to construct family of groups $\{G_i\}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of ...
- 81
3
votes
1
answer
473
views
Borel–Weil–Bott for partial flag varieties
Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
- 3,079
3
votes
1
answer
242
views
Notions of integrability for affine Lie algebras and positive energy representations
Let $\mathfrak{g}$ be a simple (complex) Lie algebra. Given an invariant bilinear form $\kappa : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$, we can form the central extension $\hat{\mathfrak{g}}...
- 3,019
3
votes
2
answers
423
views
Quotient rule, differential operator on a localization is well-defined, underlying geometry?
Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, $\...
- 39
3
votes
1
answer
255
views
Projective representation of diffeomorphism group of $S^2$ [closed]
We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class.
Then do we have a classification of the ...
- 4,826
3
votes
1
answer
485
views
Slodowy slices for two-block nilpotents
My question is about an isomorphism between two varieties of the form discussed in this thread; it is also related to an earlier question of mine.
Let $z_n$ be the standard nilpotent of type $(n,n)$. ...
- 2,216
3
votes
1
answer
901
views
Behaviour of Hilbert functions
Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...
- 31
3
votes
0
answers
131
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
3
votes
0
answers
125
views
Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
3
votes
0
answers
166
views
Extending relative Langlands duality to more singular varieties
Recent work has studied two examples of pairs $(X,X^{\vee})$ of singular varieties attached to dual reductive groups $(G,G^{\vee})$. For these pairs, identities of the following form are proved:
$$\...
user528837
3
votes
0
answers
203
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.9k
3
votes
0
answers
105
views
When can we lift transitivity of an action from geometric points to a flat cover?
Let $G$ a nice group scheme (say, over $S$), $X$ a smooth $G$-scheme over $S$, that is, $\pi : X \to S$ a smooth, $G$-invariant morphism. Assume that the action is transitive on algebraically closed ...
- 605
3
votes
0
answers
503
views
The definition of a homogeneous vector bundle
For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
- 157
3
votes
0
answers
413
views
Understanding the proof of the Springer correspondence
Let $G$ be a connected reductive group over an algebraically closed field $k$ with Weyl group $W$.
Let
$$
\mathcal{S} = R\pi_*\mathbb{Q}_\ell[\dim \mathcal{N}]
$$
be the Springer sheaf, where $\...
- 31
3
votes
0
answers
116
views
The “Kunneth-type” morphism in equivariant $K$-theory
Suppose that one has two algebraic varieties with action of a reductive group $G$: say, $X$ and $Y$.
There is an evident Kunneth-type morphism
$K_G(X) \otimes K_G(Y) \to K_G(X \times Y)$,
where the ...
- 153
3
votes
0
answers
249
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
3
votes
0
answers
71
views
Explicit correspondence between quasi-coherent sheaves on [U/G] and representations
I have a small gap when proving the titled proposition, namely, given a finite constant group scheme $G$ over a field $k$, and let $U:=Spec(A)$ be an affine $k$ scheme which $G$ act on, then quasi-...
- 455
3
votes
0
answers
183
views
Representability of $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2)$
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ ...
- 2,907
3
votes
0
answers
111
views
Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits
Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
- 839
3
votes
0
answers
235
views
Auslander-Reiten translate of toric vector bundles
By the result of A. Klyachko, there is categorical equivalence of toric vector bundles i.e. t-equivariant vector bundles on toric varieties and filtered vector spaces. For instance, on a toral surface,...
- 31
3
votes
0
answers
203
views
A quantity computed from weights of representations -- Have you seen it?
The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
- 3,079
3
votes
0
answers
133
views
Classification of semisimple algebraic groups which act transitively on a projective space
Let $k$ be an algebraically closed field of characteristic 0, and $V$ be a vector space on $k$ of dimension $>1$.
In this situation, is there a classification of connected semisimple groups (up to ...
user163485
3
votes
0
answers
178
views
When is a quotient of an affine scheme by a group scheme action a uniform categorical quotient?
We work over a ring $k$ (not necessarily a field. Let $G$ be a group scheme acting on an affine scheme $X = \text{Spec }R$. This is a naive question - what are the properties satisfied by $X/G := \...
- 7,527
3
votes
0
answers
145
views
In Deligne-Lusztig theory which degrees do irreps show up in?
In Deligne-Lusztig theory we take an alternating sum over cohomology in all degrees. Given an irrep of a finite group of Lie type can we trace back which degree it shows up in?
- 81
3
votes
0
answers
145
views
For G a connective reductive algebraic group, can I find H semisimple and a closed embedding G into H such that Bun_G -> Bun_H is an immersion?
Let $G$ be a connected reductive algebraic group over an algebraically closed field and let $X$ be a smooth projective curve. I want to find a semisimple algebraic group $H$ and a closed embedding $G\...
- 415
3
votes
0
answers
108
views
Structure of fibers of (complex) moment map of hypertoric variety
I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an ...
- 71
3
votes
0
answers
248
views
Representation of Levi subgroup $L\subset P \subset G$
Let $G$ be a split connected reductive group over a finite field extension of $\mathbb{Q}_p$ with split maximal torus $T$ of rank $d$ and simple roots $\Delta$. Furthermore associated to $I\subset \...
- 473
3
votes
0
answers
102
views
Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?
Suppose that $\mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $\mathbb Z$-linear map $p:M\to M^*$ by
$$p(x)=\sum_{u\in R^*}(u,x)u$$
and proves many ...
- 3,137
3
votes
0
answers
276
views
Closed form for Jacobi sum $\sum_{a\in \mathbb{F}_{p^2}}\chi(a)\chi(1-a)$
Let $\mathbb{F}_{p^2}$ be a field with $p^2$ elements and $\chi:\mathbb{F}_{p^2}^*\to\mathbb{C}^{*}$
be a multiplicative and non-trivial character on the multiplicative group $\mathbb{F}_{p^2}^*$ (...
3
votes
0
answers
126
views
Modular representations of GL(n,q)
I wonder what is a good source to read about Modular representations of GL(n,q).
The specific question I am interested in is $GL(n,q)$ acts on $X=F_q^n$ in a natural way. If say q is prime and $q>n$...
- 2,179
3
votes
0
answers
265
views
Computation of character sheaf
I would ask this question in a comment but I don't have enough reputation to comment yet. So I am studying the paper of Mirkovic and Vilonen "Characteristic Varieties of character sheaves" and I am ...
- 203
3
votes
0
answers
229
views
Spherical perverse sheaves on the affine Grassmannian and critically twisted $D$-modules
Let $G$ be a reductive algebraic group and let $Gr_G=G((z))/G[[z]]$ be its affine Grassmannian. Define $\mathcal{D}(Gr_G)_{crit}-mod$ to be the category of right $D$-modules on $Gr_G$ twisted by the ...
- 3,019
3
votes
0
answers
104
views
A "Dynkin diagram locality" property of flag varieties
For $n\ge 2$ consider the set of Plücker variables $X_{i_1,\dots,i_k}$ with $1\le k\le n-1$ and $1\le i_1<\dots<i_k\le n$ and the ring $R$ of polynomials in these variables (with complex ...
- 3,513
3
votes
0
answers
194
views
Relative position on flag variety
Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...
- 677
3
votes
0
answers
166
views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
3
votes
0
answers
596
views
Shimura varieties of Hodge type
I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...
- 273
3
votes
0
answers
234
views
Moduli space of nilpotent Lie algebras
Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration.
I'm interested in some ...
- 4,600
3
votes
0
answers
134
views
Two notions of a "nilpotent orbit"
I am wondering about the equivalence of two notions of a "nilpotent orbit".
The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of ...
- 937