All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
3
votes
0
answers
130
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
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
4
votes
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 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
1
vote
0
answers
88
views
Equivariant resolution of singularity making a pullback of a line bundle admit a root
I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
- 323
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
1
vote
0
answers
78
views
Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
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
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
1
vote
0
answers
125
views
When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
- 159
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
0
votes
0
answers
97
views
Weight space decomposition of smooth representation of complex algebraic torus
Question: Let $T=(\mathbb{C}^{*})^{n}$ and $\pi:\mathcal{E}\to \mathbb{C}^{n}$ a smooth complex $T$-equivariant vector bundle (i.e. $\pi$ is $T$-equivariant and $T$ acts on the fibers linearly). The ...
- 101
2
votes
0
answers
136
views
Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field
Let $V$ be an $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true:
Absolutely irreducible subgroups $H$ of $\...
- 455
1
vote
0
answers
68
views
Uniqueness of a canonical homography decomposition
Consider a multi-camera system with $n \geq 3$ calibrated cameras, each represented by a projection matrix $P_i \in \mathbb{R}^{3 \times 4}$ for $i=1, \dots, n$. We first want to detect and track ...
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
1
vote
0
answers
92
views
Counting conjugacy classes of completely reducible subgroups in general linear groups [closed]
Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\...
1
vote
0
answers
84
views
Describing monoidal categories of positive-weight representations geometrically
Let $G=\mathbb{G}_m.$ The monoidal category $\mathcal{C}=\text{Rep}(G)$ of $G$-representations (also known as the category $\text{Gr}$ of graded vector spaces) can be written geometrically as $\...
- 423
2
votes
0
answers
118
views
Growth of invariants in mod-$p$ representations of $\mathrm{GL}_n$
Let $G$ be smooth admissible mod-$p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$. Also suppose $\pi$ is an irreducible infinite-dimensional smooth admissible representation of $G$ over $\mathbb{F}...
1
vote
0
answers
128
views
Induced map of a GIT quotient map
Let $X$ be a smooth variety, $G$ a connected reductive group, and there is a $G$-linearlisatioin of a line bundle $L$.
Let us consider $p:X^{ss}(L)\rightarrow X/\!\!/_LG$, which is the GIT quotient ...
- 11
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
4
votes
1
answer
215
views
Proper morphism
Maybe this could be a silly question, but I am considering the following problem.
Let $G$ be a connected reductive group and $X$ be an affine $G$-variety, i.e., $G$ acts on $X$. (You can assume the ...
- 147
6
votes
1
answer
139
views
Quiver variety, generically symplectic
Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
- 985
13
votes
3
answers
1k
views
$\ell$- vs. $p$-adic and de Rham vs. Betti in the geometric Langlands correspondence
In Emerton–Gee–Hellmann’s IHÉS notes on the categorical $p$-adic local Langlands programme, one finds the following remark:
The differences between the $\ell$-adic and $p$-adic settings are ...
- 607
2
votes
1
answer
177
views
Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
The action of the orthogonal group $O(3,\mathbb{R})$ on the conic
$C= \{ x^2+y^2+z^2=0 \}$ in $\mathbb{P}^2$ must be well-understood, but I could not find any reference.
Is it doubly transitive?
- 14k
5
votes
0
answers
140
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
7
votes
1
answer
613
views
Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?
Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence:
$$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
- 2,907
6
votes
0
answers
230
views
Fourier transform for perverse sheaves
I am interested in studying the Fourier transform for perverse sheaves on "nice" spaces, say affine complex space stratified by the action of an algebraic group into finitely many orbits.
In ...
- 71
9
votes
1
answer
506
views
Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
3
votes
0
answers
166
views
Extending relative Langlands duality to more singular varieties
Recent work has studied two examples of pairs $(X,X^{\vee})$ of singular varieties attached to dual reductive groups $(G,G^{\vee})$. For these pairs, identities of the following form are proved:
$$\...
user528837
4
votes
0
answers
129
views
How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
4
votes
0
answers
186
views
Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic
I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...
- 2,964
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
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
21
votes
3
answers
808
views
Examples when quantum $q$ equals to arithmetic $q$
First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...
- 1,391
4
votes
3
answers
266
views
References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 741
1
vote
0
answers
75
views
Affine Springer fibers for symmetric spaces
Springer fibers are defined to be the varieties of "isotropic" full flags which are fixed by a certain element in the symmetric space. In a similar manner, affine Springer fibers can be ...
- 93
4
votes
1
answer
252
views
Symplectic structure of Higgs branch
I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
- 323
6
votes
1
answer
536
views
How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,527
4
votes
0
answers
129
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Identify the universal centralizer of $G$ as moduli space of 'flat connections' on lagrangians in cotangent bundle of $G^{\vee}$
Note that $\mathbb{C}^*$ can be interpreted as the space of flat $\mathbb{C}^*$-connections on the dual of $\mathbb{C}^*$. Our goal is to find a similar construction for $\mathbb{C}$, particularly in ...
- 303
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
8
votes
2
answers
577
views
Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
- 541
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
1
vote
0
answers
50
views
Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
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
2
votes
0
answers
147
views
Abstract definition of hypertoric varieties
I'm reading Proudfoot's survey on hypertoric varieties. In Section 1.4 he mentioned such a conjecture:
Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its ...
- 341
5
votes
0
answers
293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
2
votes
0
answers
143
views
Comparison of IC sheaves on Schubert varieties on two settings (l-adic vs. complex)
This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...
- 323