All Questions
828 questions
2
votes
0
answers
45
views
Automorphic forms annihilated by $I_1$ but not $I_2$ for finite codim ideals $I_1 \subsetneq I_2$
Suppose $G$ is a connected real reductive Lie group, $\mathfrak{g} = \text{Lie}(G)$, and $\mathcal{Z} = \mathcal{Z}[U(\mathfrak{g})]$ the center of the universal enveloping algebra of $\mathfrak{g}$.
...
- 233
3
votes
0
answers
145
views
Correspondence between dual center and linear characters of finite reductive group
Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...
- 309
5
votes
0
answers
257
views
Symplectic leaves in positive characteristic
I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
- 599
8
votes
2
answers
1k
views
Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...
- 685
14
votes
1
answer
1k
views
Definition of discrete spectrum and continuous and basic properties
I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...
- 787
7
votes
2
answers
611
views
Relation between unipotent cuspidal representations and cuspidal local systems
This could well be a question for reading suggestion. Hope it's not too bad and thanks a lot.
So the question is as in the title. What are the relations between the notion of unipotent cuspidal ...
- 1,856
8
votes
1
answer
167
views
Symmetries of the flag variety
Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma (T\...
- 43.2k
4
votes
1
answer
349
views
Existence of lattices in reductive Lie groups
What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups ...
- 43
12
votes
0
answers
729
views
Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
18
votes
1
answer
2k
views
Independence of $\ell$ of Betti numbers
When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. ...
- 28.8k
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
3
votes
2
answers
282
views
Replacement for Lie-algebra complements
All groups are linear algebraic over some fixed field $k$.
I believe that it is true that, in characteristic $0$, if $G'$ is a reductive subgroup of $G$, then there is a $G'$-invariant complement to $...
- 13k
6
votes
1
answer
1k
views
Generic Smoothness Type of Results in Positive Characteristic
Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...
- 313
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
1
vote
0
answers
166
views
Conjugacy scheme, fppf versus GIT
I would be glad to have some guidance in the following.
Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
- 5,562
2
votes
0
answers
975
views
Complete reducibility of representations of reductive algebraic groups
I need a reasonably detailed reference for the proof of the fact that, in characteristic 0, any linear representation of a reductive algebric group is completely reducible. I looked in Humphries and ...
- 21
0
votes
1
answer
97
views
Automorphisms of Lie algebra of type $A_5$ modulo its center in characteristic 2
Let $L$ be classical Lie algebra of type $A_5$ over field of characteristic 2; let $M$ be the quotient $L/Z(L)$ modulo its center $Z(L)$.
What about the group of automorphisms of M?
Does anybody ...
- 11
2
votes
1
answer
123
views
F-points of product of closed subgroups vs. product of F-points, F a local field, reference?
Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
- 41
7
votes
2
answers
827
views
$p$-torsion of an abelian variety of $p$-rank $0$
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
- 2,663
10
votes
3
answers
2k
views
Pullback along Frobenius morphism
Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
- 63.1k
52
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,434
4
votes
1
answer
358
views
Examples of perfect pseudo algebraically closed fields in positive characteristic
Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
- 2,323
4
votes
0
answers
285
views
Application of Frobenius splitting in characteristic $0$
In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
- 849
27
votes
2
answers
3k
views
Reference for de Rham cohomology in positive characteristic
It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
- 28.8k
8
votes
1
answer
808
views
Automorphisms of curves in positive characteristic
It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...
user68440
1
vote
0
answers
231
views
Product of standard parabolic subgroups
Consider a quasi-split reductive group $G$ over a field $k$. Let $B$ be a Borel subgroup of $G$, and let $P, Q$ be two parabolic subgroups of $G$ that contain $B$. Is the product set $PQ = \{xy| x \in ...
- 11
4
votes
1
answer
887
views
Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$
I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of ...
- 685
4
votes
1
answer
272
views
How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic
Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.
If $k= \mathbb C$, there is a natural injective morphism of vector spaces
$$H^0(X,\...
- 49
3
votes
2
answers
358
views
degeneration of reductive group
If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...
- 163
5
votes
1
answer
514
views
Lifting torsors in characteristic $p$ to characteristic zero
Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
- 151
1
vote
1
answer
136
views
Reference requests for complex duals of connected reductive groups
In Kottwitz's paper "Stable trace formula: cuspidal tempered terms", it is said that if $$1 \rightarrow G_1 \rightarrow G_2 \rightarrow G_3 \rightarrow 1$$ is an exact sequence of connected reductive ...
4
votes
1
answer
200
views
Braid relations $n_\alpha n_\beta n_\alpha \ldots = n_\beta n_\alpha n_\beta \ldots $ in arbitrary reductive groups
I'm currently trying to prove or disprove the following claim. First let me set up some notation.
Let $G$ be a connected reductive group over a field $K$, let $S \leq Z \leq N \leq G$ be respectively ...
- 1,704
3
votes
2
answers
472
views
Representations of complex semi-simple algebraic group "defined over $\mathbf{Z}$"?
If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over $\mathrm{Spec}(\...
- 163
11
votes
1
answer
928
views
Non-algebraic K3 surfaces in characteristic $p$
I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
- 21.4k
14
votes
2
answers
2k
views
Primer on Eisenstein series
My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.
In my continuing journey of ...
- 2,516
7
votes
1
answer
540
views
Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?
The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
- 5,422
7
votes
0
answers
355
views
Are curves over imperfect fields defined over a smaller field?
Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
- 149k
3
votes
1
answer
343
views
Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism
MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
- 5,422
3
votes
1
answer
235
views
Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve
Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...
- 2,663
2
votes
1
answer
153
views
A question about decomposing mod 2 modular forms of level p^2
Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
- 5,422
1
vote
1
answer
361
views
Subgroups generated by opposite root groups
Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...
- 1,704
0
votes
1
answer
223
views
Unipotent orbit in adjoint group over finite field
[Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...
- 1,856
8
votes
0
answers
381
views
Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
- 149k
4
votes
1
answer
502
views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
- 16.9k
8
votes
2
answers
726
views
Is every connected reductive group over a local field already defined over a global field?
Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...
- 113
2
votes
1
answer
281
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
2
votes
2
answers
663
views
Regular embeddings of reductive groups
A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where $G'$...
- 309
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 53
4
votes
1
answer
321
views
Torsors in the analytic topology versus torsors in the etale topology
Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over $S$...
20
votes
3
answers
2k
views
Is there a scheme parametrizing the closed subgroups of an algebraic group?
In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...
- 4,902