JournalsrlmVol. 32, No. 1pp. 79–96

On the Schwartz correspondence for Gelfand pairs of polynomial growth

  • Francesca Astengo

    Università di Genova, Italy
  • Bianca Di Blasio

    Università degli Studi di Milano-Bicocca, Italy
  • Fulvio Ricci

    Scuola Normale Superiore, Pisa, Italy
On the Schwartz correspondence for Gelfand pairs of polynomial growth cover
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Abstract

Let (G,K)(G,K) be a Gelfand pair, with GG a Lie group of polynomial growth, and let ΣR\Sigma\subset\mathbb R^\ell be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system D1,,DD_1,\dots,D_\ell of GG-invariant differential operators on G/KG/K and associating to a bounded spherical function ϕ\phi the \ell-tuple of its eigenvalues under the action of the DjD_j's.

We say that property (S) holds for (G,K)(G,K) if the spherical transform maps the bi-KK-invariant Schwartz space S(K\G/K)\mathcal S(K\backslash G/K) isomorphically onto S(Σ)\mathcal S(\Sigma), the space of restrictions to Σ\Sigma of the Schwartz functions on R\mathbb R^\ell. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where G=KNG=K\ltimes N, with NN 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.

Cite this article

Francesca Astengo, Bianca Di Blasio, Fulvio Ricci, On the Schwartz correspondence for Gelfand pairs of polynomial growth. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 79–96

DOI 10.4171/RLM/927