The wave front set of the Fourier transform of algebraic measures

Aizenbud, Avraham; Drinfeld, Vladimir

doi:10.1007/s11856-015-1181-9

Cited by 8 publications

(39 citation statements)

References 18 publications

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“…Recall that

N_{Y}^{X}

y \in Y

is the quotient of the tangent space to

X

y

by the tangent space to

Y

y

, and that the co‐normal bundle

C N_{Y}^{X}

is the dual bundle of

N_{Y}^{X}

. For a smooth algebraic variety

scriptX

over

K

and a locally closed smooth subvariety

Y \subset X

one defines the tangent bundle

T X

, the cotangent bundle

T^{*} X

, the normal bundle

N_{Y}^{X}

and the co‐normal bundle

C N_{Y}^{X}

as usual, see, for example, [, Section 2]. For a strict

C^{1}

morphism

f : X \to Y

between strict

C^{1}

submanifolds of

K^{n}

, respectively,

K^{m}

and

x_{0} \in X

, we write

D f (x_{0})

for the linear map from the tangent space

T_{x_{0}} X

X

x_{0}

to the tangent space

T_{f (x_{0})} Y

Y

f (x_{0})

, and

…”

Section: Some Additions On Wave Front Sets In the Non‐archimedean Casementioning

confidence: 99%

Distributions and wave front sets in the uniform non‐archimedean setting

Cluckers

Halupczok²,

Loeser

et al. 2018

Trans. London Math. Soc.

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We study some constructions on distributions in a uniform p‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of scriptC exp ‐class and which is based on the notion of scriptC exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a scriptC exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.

show abstract

“…Recall that

N_{Y}^{X}

y \in Y

is the quotient of the tangent space to

X

y

by the tangent space to

Y

y

, and that the co‐normal bundle

C N_{Y}^{X}

is the dual bundle of

N_{Y}^{X}

. For a smooth algebraic variety

scriptX

over

K

and a locally closed smooth subvariety

Y \subset X

one defines the tangent bundle

T X

, the cotangent bundle

T^{*} X

, the normal bundle

N_{Y}^{X}

and the co‐normal bundle

C N_{Y}^{X}

as usual, see, for example, [, Section 2]. For a strict

C^{1}

morphism

f : X \to Y

between strict

C^{1}

submanifolds of

K^{n}

, respectively,

K^{m}

and

x_{0} \in X

, we write

D f (x_{0})

for the linear map from the tangent space

T_{x_{0}} X

X

x_{0}

to the tangent space

T_{f (x_{0})} Y

Y

f (x_{0})

, and

…”

Section: Some Additions On Wave Front Sets In the Non‐archimedean Casementioning

confidence: 99%

Distributions and wave front sets in the uniform non‐archimedean setting

Cluckers

Halupczok²,

Loeser

et al. 2018

Trans. London Math. Soc.

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show abstract

“…Note that Theorems B and 1.9(i) imply that the dimension of (the Zariski closure of) the wave front set of a distribution that satisfies (1-3) does not exceed dim G. In many ways the wave front set replaces the singular support, in absence of the theory of differential operators (see, e.g., [2,3,7,8]). Thus, in order to prove Conjecture G, it is left to prove analogs of Theorems 1.7 and 3.13 for the integral system of equations (1-3).…”

Section: This Dimension Is Uniformly Bounded When λ Variesmentioning

confidence: 99%

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

Aizenbud

Gourevitch

Minchenko

2016

Sel. Math. New Ser.

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We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group G defined over R, the space of G(R)-equivariant distributions on the manifold of real points of any algebraic G-manifold X is finite-dimensional if G has finitely many orbits on X . Mathematics Subject Classification

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“…Sometimes however, arbitrary ramification and even positive characteristic local fields can be allowed, for example in situations with some smoothness or smooth models, see e.g. [23], [30], [25], [26], and in situations where variants of Hironaka's resolution can be used over Q like for Theorem E of [1] about wave front sets and for the rationality result from the 1970s by Igusa, see Theorem 8.2.1 of [22] or Theorem (1.3.2) of [16].…”

mentioning

confidence: 99%

“…This yields several kinds of new uniformities for the behaviour of p-adic integrals and for bad (or exceptional) loci. In the aforementioned Theorem 8.2.1 of [22], it is the set of candidate poles and the form of the denominator that is completely uniform over all local fields of characteristic zero; in Theorem E of [1] it is the wave front which is included in a Zariski closed set of controlled dimension which is completely uniform over all local fields of characteristic zero. These two phenomena should now find a common ground in the uniform treatment of this paper, see Sections 4.4 and 4.5.…”

mentioning

confidence: 99%

“…First, we deduce that certain bad or exceptional loci are small; see for example Theorem 4.4.3. Roughly, the idea is that loci of several kinds of bad behaviour are contained in proper Zariski closed subsets, uniformly in all local fields of characteristic zero, roughly as in Theorem E of [1]. Many such results are already known for large enough residue field characteristic (or assuming bounds on the ramification), so that the new point is again to be completely uniform in all local fields of characteristic zero.…”

mentioning

confidence: 99%

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