C1-triangulations of semialgebraic sets

Ohmoto, Toru; Shiota, Masahiro

doi:10.1112/topo.12024

Cited by 12 publications

(18 citation statements)

References 13 publications

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“…Consider the one point compactification of g, which is a sphere S. Let C(O λ0 , q) be the closure of C(O λ0 , q) in S. With respect to a standard metric on the sphere S, its compact semialgebraic subset C(O λ0 , q) has finite volume (see e.g. [OS17]). By comparing the standard metric on S and the Euclidean metric on g, this can be restated as (5.8) for N ≥ 4n.…”

Section: Reduction To Quantizations Of Semisimple Orbitsmentioning

confidence: 99%

On the asymptotic support of Plancherel measures for homogeneous spaces

Harris¹,

Oshima²

2022

Preprint

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Let G be a real linear reductive group and let H be a unimodular, locally algebraic subgroup. Let supp L 2 (G/H) be the set of irreducible unitary representations of G contributing to the decomposition of L 2 (G/H), namely the support of the Plancherel measure. In this paper, we will relate supp L 2 (G/H) with the image of moment map from the cotangent bundleFor the homogeneous space X = G/H, we attach a complex Levi subgroup L X of the complexification of G and we show that in some sense "most" of representations in supp L 2 (G/H) are obtained as quantizations of coadjoint orbits O such that O ≃ G/L and that the complexification of L is conjugate to L X . Moreover, the union of such coadjoint orbits O coincides asymptotically with the moment map image. As a corollary, we show that L 2 (G/H) has a discrete series if the moment map image contains a nonempty subset of elliptic elements.2010 Mathematics Subject Classification. 22E46.

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Section: Reduction To Quantizations Of Semisimple Orbitsmentioning

confidence: 99%

On the asymptotic support of Plancherel measures for homogeneous spaces

Harris¹,

Oshima²

2022

Preprint

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“…All arguments hinge on the existence of semi-algebraic or more generally definable triangulations. These exist in the C 0 -setting (this seems standard), but also in the C 1 -setting, see [OS17] and [CP18]. However, the questions of C p or even C ∞ -triangulations seems to be open.…”

Section: Generalisationsmentioning

confidence: 99%

Semi-algebraic motives

Huber

2020

Preprint

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“…Recently Ohmoto and Shiota proved a remarkable global S 1 version of the latter theorem. We state their result in the compact case only, see [OS,Thm.1.1].…”

Section: Amentioning

confidence: 99%

“…It is not known if the semialgebraic homeomorphism Ψ can be chosen of class C 2 (see [OS,Sect.1]). However, for a locally S ν polyhedral semialgebraic set we have in addition the following result, see [Sh2,Prop.I.3.13 & Rmk.I.3.22].…”

Section: Amentioning

confidence: 99%

Differentiable approximation of continuous semialgebraic maps

Fernando

Ghiloni

2019

Sel. Math. New Ser.

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In this work we approach the problem of approximating uniformly continuous semialgebraic maps f : S Ñ T from a compact semialgebraic set S to an arbitrary semialgebraic set T by semialgebraic maps g : S Ñ T that are differentiable of class C ν for a fixed integer ν ě 1. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For ν " 1 we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For ν ě 2 we obtain density results in the two following relevant situations: either T is compact and locally C ν semialgebraically equivalent to a polyhedron, for instance when T is a compact polyhedron; or T is an open semialgebraic subset of a Nash set, for instance when T is a Nash set. Our density results are based on a recent C 1 -triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.

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C1-triangulations of semialgebraic sets

Cited by 12 publications

References 13 publications

On the asymptotic support of Plancherel measures for homogeneous spaces

On the asymptotic support of Plancherel measures for homogeneous spaces

Semi-algebraic motives

Differentiable approximation of continuous semialgebraic maps

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