scholarly journals CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

2018 ◽  
Vol 19 (4) ◽  
pp. 1031-1091
Author(s):  
Thierry Stulemeijer
Keyword(s):  
Algebraic Group ◽  
Closed Subset ◽  
Algebraic Groups ◽  
Local Fields ◽  
Locally Finite ◽  
Simple Algebraic Group ◽  
Groups Acting On Trees ◽  
Residue Characteristic

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

FIXED VECTORS FOR ELEMENTS IN MODULES FOR ALGEBRAIC GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1249-1261 ◽  
Author(s):  
IRINA D. SUPRUNENKO ◽  
ALEXANDRE E. ZALESSKI
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Chevalley Groups ◽  
Simple Algebraic Group

We provide conditions guaranteeing that a given element of a simple algebraic group G has a fixed vector in every nonzero G-module, and deduce similar results for finite Chevalley groups.

On the topology of geometric and rational orbits for algebraic group actions over valued fields

2020 ◽  
Vol 23 (6) ◽  
pp. 965-981
Author(s):  
Phuong Bac Dao
Keyword(s):  
Algebraic Group ◽  
Local Field ◽  
Algebraic Groups ◽  
Local Fields ◽  
Unipotent Group ◽  
Affine Variety ◽  
Local Function ◽  
Relative Orbit ◽  
Completely Reducible ◽  
The Relationship

AbstractIn this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types {D_{4}}, {E_{6}}, {E_{7}}, {E_{8}}, {G_{2}} on an affine variety defined over a local function field k, and {v\in V(k)} such that the geometric orbit {G.v} is Zariski closed although the corresponding relative orbit {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

scholarly journals On Algebraic Groups and Discontinuous Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 279-322 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Discrete Subgroup ◽  
Simple Algebraic Group ◽  
Discontinuous Groups

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

scholarly journals Unipotent elements forcing irreducibility in linear algebraic groups

2018 ◽  
Vol 21 (3) ◽  
pp. 365-396 ◽  
Author(s):  
Mikko Korhonen
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Algebraic Groups ◽  
Closed Field ◽  
Unipotent Element ◽  
Tilting Modules ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Highest Weight

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

The P-radical classes in simple algebraic groups and finite groups of Lie type

2012 ◽  
Vol 15 (5) ◽  
Author(s):  
R. Lawther
Keyword(s):  
Finite Group ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Algebraic Groups ◽  
Unipotent Radical ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract.Given either a simple algebraic group or a finite group of Lie type, of rank at least 2, and a maximal parabolic subgroup, we determine which non-trivial unipotent classes have the property that their intersection with the parabolic subgroup is contained within its unipotent radical. Such classes are rare; listing them provides a basis for inductive arguments.

scholarly journals FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF -ADIC TORI

2015 ◽  
Vol 17 (1) ◽  
pp. 1-37
Author(s):  
Clifton Cunningham ◽  
David Roe
Keyword(s):  
Cohomology Group ◽  
Monoidal Category ◽  
Algebraic Groups ◽  
Group Scheme ◽  
Local Fields ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Algebraic Tori ◽  
Component Group ◽  
Character Sheaves

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$, and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
Author(s):  
Tanmay Deshpande
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Groups ◽  
Unipotent Group ◽  
Connected Component ◽  
Unipotent Groups ◽  
Character Sheaves ◽  
Treat All ◽  
Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

scholarly journals The Modality of a Borel Subgroup in a Simple Algebraic Group of Type E8

2018 ◽  
Vol 29 (3) ◽  
pp. 326-327
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Simple Algebraic Group
Sign in / Sign up

Export Citation Format

Share Document