All Questions
828 questions
9
votes
0
answers
295
views
Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
- 1,666
2
votes
1
answer
207
views
Subschemes in group action
Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $...
- 21
4
votes
1
answer
1k
views
Complexification of compact Lie groups and complex algebraic linear reductive groups
I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie ...
- 103
9
votes
1
answer
833
views
Endomorphism ring of simple ordinary abelian variety
Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
- 3,076
3
votes
1
answer
328
views
Contragredient of a cuspidal representation
Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.
A ...
- 726
3
votes
1
answer
78
views
Functions in the induced space compactly supported in $PN^-$ modulo $P$
Let $P_0$ be a minimal parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field $k$. Let $P$ be a parabolic subgroup containing $P_0$ with Levi decomposition $P = MN$. Let $N^-$ ...
- 6,180
3
votes
1
answer
317
views
Is there a "big open cell" analogue for parabolic subgroups?
Let $G$ be a reductive group over a $p$-adic field. Let $P$ be a parabolic subgroup of $G$ containing a minimal parabolic $P_0$. Let $S$ be a maximal split torus of $P_0$, and let $\Delta$ be a set ...
- 6,180
1
vote
1
answer
264
views
Is the Borel subgroup the only closed double coset?
Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined ...
- 6,180
2
votes
0
answers
81
views
Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
- 5,434
6
votes
1
answer
313
views
Making sense out of intertwining operators defined by a vector valued integral
Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal ...
- 6,180
4
votes
1
answer
342
views
For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?
Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ ...
- 6,180
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
6
votes
0
answers
217
views
Dimension of space of K-fixed vectors
If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...
- 91
8
votes
1
answer
320
views
Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...
- 6,180
3
votes
0
answers
154
views
Are these statements correct for all reductive groups, or just for $\operatorname{GL}_2$?
I'm reading the following notes on Brian Conrad's website. There are a couple of statements there which seem too good to be true. There is Proposition 3.6, which states:
(Bernstein): Every ...
- 6,180
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
4
votes
0
answers
289
views
Meaning of a highly ramified character for reductive groups
Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in ...
- 6,180
1
vote
1
answer
206
views
Reduced decomposition for Weyl group elements which support a Bessel function
Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to ...
- 6,180
5
votes
1
answer
433
views
Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$
Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and ...
- 6,180
14
votes
2
answers
876
views
Split rank of inner forms
Let $G$ be a (connected) reductive group over some ground field $F$ and $G^*$ its unique quasi-split inner form. Denote by $\operatorname{rank}_F G$ the split rank of $G$, i.e. the dimension of a ...
- 1,856
7
votes
1
answer
1k
views
Rosenlicht's theorem and rationality questions
Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\...
- 6,180
2
votes
1
answer
271
views
Discrete decomposability of unitary representation
[INTRODUCTION]
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...
- 951
7
votes
3
answers
927
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 305
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 43
11
votes
2
answers
1k
views
Cotangent complex of perfect algebra over a perfect field
Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
- 2,011
3
votes
0
answers
156
views
Holomorphic homomorphism of complex reductive groups is algebraic
Could you hint (or give a reference) for: A holomorphic group homomorphism between complex reductive algebraic groups is algebraic.
Thank you!
- 5,562
3
votes
0
answers
231
views
Kottwitz's vertical map
I'm looking at the map $w_H$ defined by Kottwitz in "Isocrystals with additional structure II" in section 7. It is a surjective group homomorphism defined for all reductive groups $H$ from $H$ to $X^{*...
- 2,429
0
votes
1
answer
374
views
Existence of a model over S-integers
Let $G$ is a connected reductive group over a number field $F$. Let $S$ be a set of places containing the archimedean places for which $G$ has a model over the ring of $S$-integers in $F$. Is it true ...
5
votes
0
answers
243
views
Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
8
votes
1
answer
603
views
absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
- 4,070
1
vote
1
answer
280
views
Norms of elements in Artin-Schreier extensions
The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
- 428
17
votes
2
answers
1k
views
A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
- 3,362
2
votes
0
answers
304
views
Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 203
2
votes
0
answers
121
views
Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
3
votes
0
answers
307
views
Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
11
votes
1
answer
334
views
An identity in Lie algebras over fields of positive characteristic
Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
- 630
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
0
votes
0
answers
459
views
Clarification on the definition of a smooth affine scheme over an integral domain
$\DeclareMathOperator{\Spec}{Spec}$
The following is from Bruhat and Tits article Groupes Reductifs sur un Corps Locale II. $A$ is an integral domain. Here $A$-scheme means "affine $A$-scheme," and $...
- 6,180
12
votes
2
answers
314
views
Automorphic quotients for inner forms or $GSp(4)$
For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups:
\begin{equation}
\mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star
\left(
\begin{array}{cc}
& 1 \\
1 &...
- 5,434
3
votes
0
answers
108
views
Special fibers of parahorics
Tits' Corvallis article introduces a map on special fibers of group schemes associated to the elements fixing sets pointwise in a building from $\bar{\mathcal{P}_\Omega}$ to $\bar{\mathcal{P}_{\Omega'}...
- 2,429
2
votes
0
answers
382
views
Integral smooth model of unramified reductive groups
My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$...
- 193
5
votes
0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595
14
votes
0
answers
1k
views
A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
8
votes
0
answers
471
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
9
votes
1
answer
1k
views
deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
- 528
11
votes
0
answers
316
views
Mysterious "raison d'être" of filtrations of congruence subgroups
I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$.
Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
- 5,434
7
votes
1
answer
608
views
$GSp(4)$ vs $PSp(4)$
After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, ...
- 5,434
7
votes
1
answer
165
views
Which groups can have $GSp(4)$ as local component?
In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G_v$ (where $v$ are the places of $F$) are well known. Obviously a group ...
- 5,434
5
votes
2
answers
738
views
What condition makes unitary reductive group unramified?
I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...
- 1,759
2
votes
0
answers
948
views
Description of the center of a reductive group using absolute and relative roots
Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...
- 6,180