Subgroups of isotropic orthogonal groups containing the centralizer of a maximal split torus

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

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Subgroups of a spinor group containing a maximal split torus. I

Journal of Soviet Mathematics ◽

10.1007/bf01097976 ◽

1993 ◽

Vol 63 (6) ◽

pp. 638-652 ◽

Cited By ~ 1

Author(s):

N. A. Vavilov

Keyword(s):

Spinor Group ◽

Maximal Split ◽

Split Torus

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On a Certain Residual Spectrum of Sp8

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-008-0 ◽

2004 ◽

Vol 56 (1) ◽

pp. 168-193

Author(s):

James Todd Pogge

Keyword(s):

Eisenstein Series ◽

Fundamental Problem ◽

Regular Representation ◽

Automorphic Representation ◽

Levi Subgroup ◽

Parabolic Subgroups ◽

Residual Spectrum ◽

The Right ◽

Maximal Split ◽

Split Torus

AbstractLet G = Sp2n be the symplectic group defined over a number field F. Let 𝔸 be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of G(𝔸) acting on the Hilbert space L2 (G(F) \ G(𝔸)). Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: , where (M, π) is a Levi subgroup with a cuspidal automorphic representation π taken modulo conjugacy. (Here we normalize π so that the action of the maximal split torus in the center of G at the archimedean places is trivial.) and is a space of residues of Eisenstein series associated to (M, π). In this paper, we will completely determine the space , when M ≃ GL2 × GL2. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than GLn.

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Low-dimensional representations of finite orthogonal groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000037 ◽

2021 ◽

pp. 1-22

Author(s):

KAY MAGAARD ◽

GUNTER MALLE

Keyword(s):

Orthogonal Groups ◽

Brauer Characters ◽

Low Dimensional

Abstract We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

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STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000177 ◽

2021 ◽

pp. 1-48

Author(s):

Federico Scavia

Keyword(s):

Finite Groups ◽

Reductive Groups ◽

Orthogonal Groups ◽

De Rham Cohomology ◽

Algebraic Stacks ◽

Steenrod Operations ◽

De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

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Higher pullbacks of modular forms on orthogonal groups

Forum Mathematicum ◽

10.1515/forum-2020-0066 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Brandon Williams

Keyword(s):

Modular Forms ◽

Differential Operators ◽

Jacobi Form ◽

Siegel Modular Forms ◽

Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

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