A fourier-type transform on translation-invariant valuations on convex sets

Semyon Alesker

doi:10.1007/s11856-011-0008-6

A fourier-type transform on translation-invariant valuations on convex sets

Semyon Alesker

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Let V be a finite-dimensional real vector space. Let V al^sm(V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism F_V: Val^sm(V)→^̃Val^sm(V*) ⊗ Dens(V) such that F_V commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

Original language	English
Pages (from-to)	189-294
Number of pages	106
Journal	Israel Journal of Mathematics
Volume	181
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0008-6
State	Published - Mar 2011

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s11856-011-0008-6

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title = "A fourier-type transform on translation-invariant valuations on convex sets",

abstract = "Let V be a finite-dimensional real vector space. Let V alsm(V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism FV: Valsm(V){\~→}Valsm(V*) ⊗ Dens(V) such that FV commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.",

author = "Semyon Alesker",

note = "Funding Information: 0.1. An overview of the main results. . . . . . . . . . . 191 0.2. Organization of the article. . . . . . . . . . . . . . 195 0.3. Notation. . . . . . . . . . . . . . . . . . . . . . . . 196 1. Background from representation theory . . . . . . 197 1.1. Some structure theory of reductive groups. . . . . 197 1.2. Admissible and tempered growth representations and a theorem of Casselman–Wallach. . . . . . . . . . 198 1.3. Induced representations. . . . . . . . . . . . . . . . 200 ∗ Partially supported by ISF grant 1369/04. Received January 7, 2009",

year = "2011",

month = mar,

doi = "10.1007/s11856-011-0008-6",

language = "الإنجليزيّة",

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AU - Alesker, Semyon

N1 - Funding Information: 0.1. An overview of the main results. . . . . . . . . . . 191 0.2. Organization of the article. . . . . . . . . . . . . . 195 0.3. Notation. . . . . . . . . . . . . . . . . . . . . . . . 196 1. Background from representation theory . . . . . . 197 1.1. Some structure theory of reductive groups. . . . . 197 1.2. Admissible and tempered growth representations and a theorem of Casselman–Wallach. . . . . . . . . . 198 1.3. Induced representations. . . . . . . . . . . . . . . . 200 ∗ Partially supported by ISF grant 1369/04. Received January 7, 2009

PY - 2011/3

Y1 - 2011/3

N2 - Let V be a finite-dimensional real vector space. Let V alsm(V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism FV: Valsm(V)→̃Valsm(V*) ⊗ Dens(V) such that FV commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

AB - Let V be a finite-dimensional real vector space. Let V alsm(V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism FV: Valsm(V)→̃Valsm(V*) ⊗ Dens(V) such that FV commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

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U2 - 10.1007/s11856-011-0008-6

DO - 10.1007/s11856-011-0008-6

M3 - مقالة

SN - 0021-2172

VL - 181

SP - 189

EP - 294

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -

A fourier-type transform on translation-invariant valuations on convex sets

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