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Schwartz correspondence for real motion groups in low dimensions

Francesca Astengo, Bianca Di Blasio, Fulvio Ricci

Abstract

For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$ a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto the space ${\mathcal S}(Σ_{\mathcal D})$, where $Σ_{\mathcal D}$ is an embedded copy of the Gelfand spectrum in ${\mathbb R}^\ell$, canonically associated to a generating system ${\mathcal D}$ of $G$-invariant differential operators on $G/K$, and ${\mathcal S}(Σ_{\mathcal D})$ consists of restrictions to $Σ_{\mathcal D}$ of Schwartz functions on ${\mathbb R}^\ell$. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair $(M_n,SO_n)$ with $n=3,4$. The rather trivial case $n=2$ is included in previous work by the same authors.

Schwartz correspondence for real motion groups in low dimensions

Abstract

For a Gelfand pair with a Lie group of polynomial growth and a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi--invariant Schwartz space isomorphically onto the space , where is an embedded copy of the Gelfand spectrum in , canonically associated to a generating system of -invariant differential operators on , and consists of restrictions to of Schwartz functions on . Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair with . The rather trivial case is included in previous work by the same authors.
Paper Structure (21 sections, 23 theorems, 189 equations)

This paper contains 21 sections, 23 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Spherical transforms for strong Gelfand pairs
  3. The $\tau$-spherical transform and its inverse.
  4. ${\rm End}(\mathcal{V}_\tau)$-valued Schwartz functions
  5. $\tau$-spherical analysis on the euclidean motion group ${SO}_n\ltimes\mathbb R^n$
  6. The euclidean motion group
  7. Invariant differential operators
  8. $K$-type subalgebras and ${\rm End}(\mathcal{V}_\tau)$-valued spherical analysis
  9. Differential operators in $(\mathbb D(\mathbb R^n)\otimes {\rm End}(\mathcal{V}_\tau))^{K}$
  10. Equivariant polynomials and the Gelfand spectrum $\Sigma^\tau_\mathcal{D}$ when $n=3,4$
  11. The case where $n=3$
  12. The case where $n=4$
  13. Polynomials in $B_\tau$
  14. Equivariant homogeneous polynomials
  15. ${\rm End}(\mathcal{V}_\tau)$-valued spherical functions as derivatives of scalar spherical functions
  16. ...and 6 more sections

Key Result

Theorem 1.1

The strong Gelfand pairs $(M_n(\mathbb R),SO_n)$, with $n=3,4$, satisfy property (S), i.e., the spherical transform maps $\mathcal{S}(M_n(\mathbb R))^{{\rm Int}(SO_n)}$ isomorphically onto $\mathcal{S}(\Sigma_\mathcal{D})$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2: ADR3
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 31 more