Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

Wang, Zhenqi Jenny

doi:10.1007/s00039-015-0351-6

Cited by 11 publications

(14 citation statements)

References 29 publications

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“…Let H be a unitary representation of (SL(2, R) R 2 ) R 3 . We write the representations H σ in the decomposition 2 as representations H t,r of two parameters, which is realized by a unitarily equivalent model K t,r , see Section 4.3 of [29] for a discussion. We provide it here for the convenience of the reader.…”

Section: 4mentioning

confidence: 99%

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Damjanović¹,

Tanis²,

Wang³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We define globally hypoelliptic smooth R k actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When k = 1, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous R k actions (k ≥ 2) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic R 2 actions on homogeneous spaces G/Γ, with at least one quasi-unipotent generator, where G = SL(n, R). We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.2010 Mathematics Subject Classification. Primary: 37C15, 37C85, 37D20.

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Section: 4mentioning

confidence: 99%

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Damjanović¹,

Tanis²,

Wang³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…The description of representations of SL(2, R) ⋉ R 2 appears in [42]. Here we just briefly quote the results.…”

Section: Explicit Calculations Based On Mackey Theorymentioning

confidence: 99%

“…For example, the space of obstructions to a smooth solution of the cohomological equation of horocycle maps has infinite countable dimension in each irreducible component of P SL(2, R), see [38], as opposed to being at most two dimensional in each component for the horocycle flow, see [8]. Moreover, Sobolev estimates for solutions of the cohomological equation of horocycle maps are not tame in [38], [39] and [9], as opposed to the tame estimates obtained for the horocycle flow in [8] and parabolic flow in [42]. Because tame estimates for solutions of the cohomological equation lays the groundwork for proving smooth action rigidity, see [5] and [6], not having them complicates this effort.…”

mentioning

confidence: 94%

“…Finally, we comment on the proofs of Theorems 2.1 and 2.2. Regarding Theorem 2.1, the analogous theorem for parabolic flows was proven in [42], where Sobolev estimates of the solution were obtained from estimates of lower rank subgroups of G: SL(2, R)×R and SL(2, R)⋉R 2 . Trying the same approach for the map causes a non-tame Sobolev loss of regularity in every direction that does not commute with the unipotent flow direction.…”

mentioning

confidence: 98%

“…In general, the unitary dual of many higher rank almost-simple algebraic groups is not completely classified, and even when the classification is known, it is too complicated to apply. To handle these cases, a new method was introduced in [42] that is based on an analysis of the unitary dual of various subgroups in G rather than that of G itself. Mackey theory was used to find models for the representations of these groups that appeared in a restricted non-trivial representation of G. After explicit computations in irreducible models, global properties of the solution came with having sufficiently many semidirect product groups that contain the one-parameter root subgroup in the cohomological equation.…”

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confidence: 99%

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Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Tanis¹,

Wang²

2018

Preprint

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Let G denote a higher rank R-split simple Lie group of the following type: SL(n, R), SOo(m, m), E 6( 6) , E 7( 7) and E 8(8) , where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete parabolic actions on G via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case G = SL(n, R) are sharp up to finite loss of regularity. Moreover, we prove that for general G the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that G = SL(n, R), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, R) ⋉ R 2 and SL(2, R) ⋉ R 4 . A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

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Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL $${(3,\mathbb{R})}$$ ( 3 , R )

Ravotti

2019

Commun. Math. Phys.

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We consider a family of smooth perturbations of unipotent flows on compact quotients of SL(3, R) which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in SL(3, R).

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Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

Cited by 11 publications

References 29 publications

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL $${(3,\mathbb{R})}$$ ( 3 , R )

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