Skip to main content

All Questions

Filter by
Sorted by
Tagged with
3 votes
0 answers
74 views

Density of the Mellin transform inside the direct integral of induced representations

I'm trying to better understand the continuous spectrum of $G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, which is the direct integral of induced representations $\mathbf H(s) = \operatorname{Ind}_{...
D_S's user avatar
  • 6,180
3 votes
1 answer
160 views

Action of split torus on positive root spaces

Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi_k(S,G)$. Let $\Phi^+$ ...
Laura's user avatar
  • 33
7 votes
1 answer
400 views

Does perfect fraction field imply perfect residue field?

Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect? Thoughts: If $A$ is ...
user2831784's user avatar
9 votes
2 answers
630 views

How to read the paper of Arthur on trace formula on general reductive groups

My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily. I would like ...
Yang Yang's user avatar
3 votes
3 answers
582 views

Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
LAGC's user avatar
  • 143
3 votes
0 answers
40 views

Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected

Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...
Hadi's user avatar
  • 741
5 votes
1 answer
277 views

Signs in Chevalley systems for reductive groups

Let $G$ be a pinned split reductive group. There exists a Chevalley system: For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding ...
Elle's user avatar
  • 51
3 votes
0 answers
103 views

compactly induction of smooth modules over Hecke algebras

Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
M masa's user avatar
  • 479
4 votes
2 answers
332 views

Ring of invariants for $n$-tuples of Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...
skeptic's user avatar
  • 142
4 votes
2 answers
120 views

When representations of reductive Lie group in a Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
asv's user avatar
  • 21.8k
1 vote
1 answer
275 views

The norm of the principal series intertwining operator for $\operatorname{GL}_2$

Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$? Background: For a character $\chi = (\chi_1,\chi_2)$ of the ...
D_S's user avatar
  • 6,180
4 votes
1 answer
638 views

perfect fields in positive characteristic

Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
JNS's user avatar
  • 168
4 votes
1 answer
311 views

Maximal torus of linear algebraic group over a ring

Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field. Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$. Assume that for every $k$-point ...
Joaquín Moraga's user avatar
9 votes
1 answer
356 views

Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$

Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
tyrese's user avatar
  • 239
19 votes
1 answer
2k views

Perfectoid approach to resolution of singularities in char $p$

Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
Arna's user avatar
  • 199
8 votes
0 answers
300 views

Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
Andy Jiang's user avatar
  • 2,356
1 vote
0 answers
114 views

Maximal compact subgroups of p-adic orthogonal groups

Let F be a non-archimedean local field, and G be split orthogonal groups of odd degrees $\geq$ 3. In this setting, my question is; Is there explicit descriptions of maximal compact subgroups of G? I ...
user avatar
3 votes
1 answer
183 views

Calculating the residue of Eisenstein series from the residue of the intertwining operator

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232). The ...
D_S's user avatar
  • 6,180
4 votes
0 answers
130 views

Castelnuovo–Mumford regularity and wedge powers in positive characterisitc

A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if $$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$. It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
Andy Jiang's user avatar
  • 2,356
4 votes
0 answers
79 views

How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...
D_S's user avatar
  • 6,180
9 votes
1 answer
225 views

Why are characters orthogonal to cusp forms?

Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...
D_S's user avatar
  • 6,180
3 votes
0 answers
137 views

Span of parabolic inductions of discrete series representations

Let $G$ be the $\mathbf{Q}_p$-points in a $p$-adic reductive group, and let $R(G)$ be the Grothendieck group of the category $\mathrm{Rep}(G)$ of finite-length admissible smooth complex ...
David Hansen's user avatar
  • 13.1k
3 votes
0 answers
135 views

Localizations of smooth spherical varieties at simple roots

Setup Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup $B \subset G$ and a maximal torus $T \subset B$. Let $\lambda: \mathbb{G}_m \to T$ be a ...
Michael Christianson's user avatar
5 votes
0 answers
122 views

Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
pbarron's user avatar
  • 71
3 votes
1 answer
88 views

Image of square map on reductive group

$\DeclareMathOperator\ss{ss}$Let $G$ be a reductive group over a field $F$ of characteristic 0. (Here not necessarily $F=\overline{F}$.) Consider the square map $$ G(F)\longrightarrow G(F), \quad g\...
Windi's user avatar
  • 833
1 vote
0 answers
59 views

Distributivity property for smooth parabolic induction

Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...
KKD's user avatar
  • 473
1 vote
0 answers
92 views

The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...
D_S's user avatar
  • 6,180
8 votes
0 answers
688 views

An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich

Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$. In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map $$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
Joshua Mundinger's user avatar
1 vote
0 answers
88 views

$H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$

I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...
D_S's user avatar
  • 6,180
3 votes
1 answer
284 views

Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
Filip's user avatar
  • 1,687
2 votes
2 answers
273 views

Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and ...
D_S's user avatar
  • 6,180
4 votes
1 answer
428 views

p-torsion in the Picard group of a regular projective curve

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$. If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
Arno Fehm's user avatar
  • 2,051
3 votes
0 answers
191 views

Harmonic analysis on reductive groups over $\mathbb{R}$

A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
Chris's user avatar
  • 131
8 votes
1 answer
943 views

Automorphisms over finite field that do not lift to an automorphism in characteristic zero

My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
Jérémy Blanc's user avatar
3 votes
0 answers
197 views

How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?

Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers. Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ? This ...
lefuneste's user avatar
  • 417
4 votes
1 answer
703 views

Centralizers of semisimple subgroups

$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....
unknownymous's user avatar
9 votes
1 answer
331 views

Tempered Iwahori-spherical representations

Consider a local field $F$ of characteristic 0 and $G=GL_n(F)$. It is well known (for example Cartiers article in Corvallis) that an admissible, irreducible representation $\pi$ of $G$ has a ...
Julien's user avatar
  • 163
8 votes
1 answer
427 views

Branching laws for smooth representations

Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL_2(E)$. Let $B$ be the ...
David Loeffler's user avatar
2 votes
0 answers
190 views

Conjugacy classes in centralizers

Let $G$ be a complex reductive group, let $g$ be an element, and let $C$ be the connected component of its centralizer. I'm curious about what is known about the intersection of conjugacy classes in $...
unknownymous's user avatar
4 votes
0 answers
98 views

Is the union of conic orbits for a reductive group Zariski closed?

Let $G$ be a reductive group over an algebraically closed field $k$ of characteristic $p>0$. If $V$ is a rational $G$-module then we can define the Hilbert nullcone $\mathcal{N}(V)$ to be the zero ...
Lewis Topley's user avatar
11 votes
4 answers
1k views

Explicit large finite fields in characteristic $2$

Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$. For small degree, a simple algorithm gives a way to find $P$. Is ...
Jérémy Blanc's user avatar
5 votes
0 answers
148 views

algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
Somatic Custard's user avatar
1 vote
0 answers
281 views

Relation between reductive homogeneous spaces and reductive groups

To start, I would like to note that my background on Lie algebras is quite basic, so this question might be trivial when seen from a Lie algebra perspective, which I lack. We have the concept of a ...
Lezkus's user avatar
  • 177
3 votes
3 answers
423 views

Are there characteristic-dependent Betti numbers in characteristic not equal to two?

Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
Aaron Dall's user avatar
1 vote
0 answers
51 views

Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?

Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
Stabilo's user avatar
  • 1,479
16 votes
2 answers
2k views

Why are coroots needed for the classification of reductive groups?

As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots? Let's break it down to two questions:...
Andrew NC's user avatar
  • 2,081
3 votes
0 answers
218 views

Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?

By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
Kim's user avatar
  • 4,164
3 votes
0 answers
249 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 \...
KKD's user avatar
  • 473
6 votes
1 answer
298 views

Real forms of complex reductive groups

I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in ...
Marc Besson's user avatar
3 votes
1 answer
202 views

Globalising tori and weak approximation

Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a ...
Pol van Hoften's user avatar

1
3 4
5
6 7
17