Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

2021 ◽  
Vol 28 (02) ◽  
pp. 181-194
Author(s):  
Xin Hou ◽  
Shangzhi Li ◽  
Yucheng Yang
Keyword(s):  
Classical Group ◽  
Maximal Subgroups ◽  
Arbitrary Field ◽  
Classical Groups ◽  
Solvable Subgroup

Let [Formula: see text] be a classical group over an arbitrary field [Formula: see text], acting on an [Formula: see text]-dimensional [Formula: see text]-space [Formula: see text]. All those maximal subgroups of [Formula: see text] are classified each of which normalizes a solvable subgroup [Formula: see text] of [Formula: see text] not lying in [Formula: see text].

scholarly journals CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS

2019 ◽  
Vol 7 ◽  
Author(s):  
IAN LE
Keyword(s):  
Moduli Spaces ◽  
Classical Group ◽  
Teichmüller Space ◽  
Classical Groups ◽  
Local Systems ◽  
Cluster Ensemble ◽  
Teichmüller Theory ◽  
Cluster Varieties ◽  
Simply Connected ◽  
Higher Teichmüller Theory

Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$.

scholarly journals Endo-parameters for p-adic classical groups

2020 ◽  
Author(s):  
Robert Kurinczuk ◽  
Daniel Skodlerack ◽  
Shaun Stevens
Keyword(s):  
Local Field ◽  
Classical Group ◽  
Linear Groups ◽  
Closed Field ◽  
Classical Groups ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Langlands Parameters ◽  
Local Langlands Correspondence ◽  
Residual Characteristic

Abstract For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

scholarly journals Constructing Maximal Subgroups of Classical Groups

2005 ◽  
Vol 8 ◽  
pp. 46-79 ◽  
Author(s):  
Derek F. Holt ◽  
Colva M. Roney-Dougal
Keyword(s):  
Polynomial Time ◽  
Maximal Subgroups ◽  
Classical Groups ◽  
Natural Representation ◽  
Degree Polynomial ◽  
Low Degree ◽  
Finite Classical Groups

AbstractThe maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

The Steinberg Character of Finite Classical Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 163-168
Author(s):  
Xueling Song ◽  
Yanjun Liu
Keyword(s):  
Classical Group ◽  
Classical Groups ◽  
Characteristic P ◽  
Combinatorial Objects ◽  
Finite Classical Group ◽  
Finite Classical Groups ◽  
Steinberg Character

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol).

scholarly journals Cesàro kernels on classical groups

1997 ◽  
Vol 63 (3) ◽  
pp. 364-389
Author(s):  
Dashan Fan
Keyword(s):  
Continuous Function ◽  
Harmonic Analysis ◽  
Classical Group ◽  
Sufficient Condition ◽  
Classical Groups ◽  
Cesàro Operator ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Cesaro Operator

AbstractWe study the Cesàro operator on the classical group G and give a necessary and sufficient condition on the index α = α(G) for which the operator is convergent to f(U) for any continuous function f as N → ∞. The result in this paper solves a question posed by Gong in the book Harmonic analysis on classical groups.

Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators

2009 ◽  
Vol 61 (3) ◽  
pp. 691-707 ◽  
Author(s):  
Xiaoxiang Yu
Keyword(s):  
Parabolic Subgroup ◽  
Classical Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Intertwining Operators ◽  
Classical Groups ◽  
Intertwining Operator ◽  
Necessary And Sufficient ◽  
Tempered Representation

Abstract.Suppose that P = MN is amaximal parabolic subgroup of a quasisplit, connected, reductive classical group G defined over a non-Archimedean field and A is the standard intertwining operator attached to a tempered representation of G induced from M . In this paper we determine all the cases in which Lie(N ) is prehomogeneous under Ad(m) when N is non-abelian, and give necessary and sufficient conditions for A to have a pole at 0.

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