Abstract
The spectrum of a Gelfand pair of the form \({(K\ltimes N,K)}\), where N is a nilpotent group, can be embedded in a Euclidean space \({{\mathbb R}^d}\). The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on \({{\mathbb R}^d}\) has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 (Astengo et al. in J Funct Anal 251:772–791, 2007; Astengo et al. in J Funct Anal 256:1565–1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143–2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra \({{\mathfrak n}}\) of N for all Gelfand pairs \({(K\ltimes N,K)}\) in Vinberg’s list (Vinberg in Trans Moscow Math Soc 64:47–80, 2003; Yakimova in Transform Groups 11:305–335, 2006).
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References
Adamovich, O.M., Golovina, E.O.: Simple linear Lie groups having a free algebra of invariants. Selecta Math. Sov. 3, 183–220 (1984); originally published in Voprosy teorii grupp i gomologicheskoi algebry, Yaroslavl, 3–41 (1979, in Russian)
Astengo F., Di Blasio B., Ricci F.: Gelfand transforms of polyradial Schwartz functions on the Heisenberg group. J. Funct. Anal. 251, 772–791 (2007)
Astengo F., Di Blasio B., Ricci F.: Gelfand pairs on the Heisenberg group and Schwartz functions. J. Funct. Anal. 256, 1565–1587 (2009)
Benson C., Jenkins J., Ratcliff G.: On Gelfand pairs associated with solvable Lie groups. Trans. Am. Math. Soc. 321, 85–116 (1990)
Benson C., Jenkins J., Ratcliff G.: The spherical transform of a Schwartz function on the Heisenberg group. J. Funct. Anal. 154, 379–423 (1998)
Benson C., Ratcliff G.: Rationality of the generalized binomial coefficients for a multiplicity free action. J. Aust. Math. Soc. 68, 387–410 (2000)
Benson C., Ratcliff G.: The space of bounded spherical functions on the free 2-step nilpotent Lie group. Transform. Groups 13, 243–281 (2008)
Carcano G.: A commutativity condition for algebras of invariant functions. Boll. Un. Mat. Ital. 7, 1091–1105 (1987)
Damek E., Ricci F.: Harmonic analysis on solvable extensions of H-type groups. J. Geom. Anal. 2, 213–248 (1992)
Ferrari Ruffino F.: The topology of the spectrum for Gelfand pairs on Lie groups. Boll. Un. Mat. Ital. 10, 569–579 (2007)
Fischer V., Ricci F.: Gelfand transforms of SO(3)-invariant Schwartz functions on the free nilpotent group N 3,2. Ann. Inst. Fourier Gren. 59, 2143–2168 (2009)
Folland G., Stein E.M.: Hardy Spaces on Homogeneous Groups. Princeton Univ. Press, Princeton (1982)
Geller D.: Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 36, 205–254 (1980)
Knop, F.: Some remarks on multiplicity free spaces. In: Broer, A. et al. (eds.) Representation Theories and Algebraic Geometry, NATO ASI Ser., Ser. C, Math. Phys. Sci., vol. 514, pp. 301–317. Kluwer, Dordrecht (1998)
Kostant, B.: On the existence and irreducibility of certain series of representations. In: Gelfand, I.M. Lie Groups and their Representations, Wiley, New York (1975)
Ludwig J.: Polynomial growth and ideals in group algebras. Man. Math. 30, 215–221 (1980)
Mather J.: Differentiable invariants. Topology 16, 145–155 (1977)
Moore C., Wolf J.: Square integrable representations of nilpotent groups. Trans. Am. Math. Soc. 185, 445–462 (1973)
Onishchik A.L., Vinberg E.B.: Lie groups and algebraic groups. Springer, Berlin (1990)
Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry IV, Encyclopaedia Math. Sci., vol. 55, pp. 123–284. Springer, Berlin (1994)
Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975)
Schwarz G.W.: Representations of simple Lie groups with regular rings of invariants. Invent. Math. 49, 167–191 (1978)
Vinberg E.B.: Commutative homogeneous spaces and coisotropic actions. Russian Math. Surveys 56, 1–60 (2001)
Vinberg E.B.: Commutative homogeneous spaces of Heisenberg type. Trans. Moscow Math. Soc. 64, 47–80 (2003)
Wolf, J.: Harmonic Analysis on Commutative Spaces, Math. Surveys and Monographs vol. 142, Am. Math. Soc. (2007)
Yakimova, O.: Gelfand pairs. Dissertation, Rheinischen Friedrich-Wilhelms-Universität Bonn, 2004, Bonner Mathematische Schriften, vol. 374 (2005)
Yakimova O.: Principal Gelfand pairs. Transform. Groups 11, 305–335 (2006)
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Fischer, V., Ricci, F. & Yakimova, O. Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: rank-one actions on the centre. Math. Z. 271, 221–255 (2012). https://doi.org/10.1007/s00209-011-0861-3
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DOI: https://doi.org/10.1007/s00209-011-0861-3