On the Schwartz correspondence for Gelfand pairs of polynomial growth
| Autor: | Fulvio Ricci, B Di Blasio, Francesca Astengo |
|---|---|
| Přispěvatelé: | Astengo, F, Di Blasio, B, Ricci, F |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Polynomial (hyperelastic model)
Gelfand pairs Groups of polynomial growth Schwartz space Spherical transform General Mathematics Image (category theory) Spectrum (functional analysis) Zonal spherical function Lie group Gelfand pair Functional Analysis (math.FA) Combinatorics Mathematics - Functional Analysis Nilpotent FOS: Mathematics 43A85 43A90 Mathematics::Representation Theory Mathematics |
| Popis: | Let $(G,K)$ be a Gelfand pair, with $G$ a Lie group of polynomial growth, and let $\Sigma\subset{\mathbb R}^\ell$ be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system $D_1,\dots,D_\ell$ of $G$-invariant differential operators on $G/K$ and associating to a bounded spherical function $\varphi$ the $\ell$-tuple of its eigenvalues under the action of the $D_j$'s. We say that property (S) holds for $(G,K)$ if the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto ${\mathcal S}(\Sigma)$, the space of restrictions to $\Sigma$ of the Schwartz functions on ${\mathbb R}^\ell$. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where $G=K\ltimes N$, with $N$ nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs. |
| Databáze: | OpenAIRE |
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