Abstract
In this paper we make another step in the direction of proving a certain conjecture on the spherical transform of a nilpotent spherical pair (N,K). We obtain a result of an independent interest which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. Nilpotent Gelfand pairs here are assumed to satisfy Vinberg’s condition, meaning that K acts irreducibly on the quotient of n = Lie N by its derived subalgebra.
Dedicated to Joe Wolf
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
F. Astengo, B. Di Blasio, F. Ricci, Gelfand transforms of polyradial Schwartz functions on the Heisenberg group, J. Funct. Anal., 251 (2007), 772–791.
F. Astengo, B. Di Blasio, F. Ricci, Gelfand pairs on the Heisenberg group and Schwartz functions, J. Funct. Anal., 256 (2009), 1565–1587.
C. Benson, J. Jenkins, G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc., 321 (1990), 85–116.
C. Benson, G. Ratcliff, Rationality of the generalized binomial coefficients for a multiplicity free action, J. Austral. Math. Soc., 68 (2000), 387–410.
G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. 7 (1987), 1091–1105.
F. Ferrari Ruffino, The topology of the spectrum for Gelfand pairs on Lie groups, Boll. Un. Mat. Ital. 10 (2007), 569–579.
V. Fischer, F. Ricci, Gelfand transforms of S O(3)-invariant Schwartz functions on the free nilpotent group N 3, 2, Ann. Inst. Fourier Gren., 59 (2009), no. 6, 2143–2168.
V. Fischer, F. Ricci, O. Yakimova, Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I. Rank-one actions on the centre, Math. Zeitschrift, 271 (2012), no.1-2, 221–255.
V. Fischer, F. Ricci, O. Yakimova, Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg’s condition, arxiv:1210.7962v1[math.FA].
W. Fulton, J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, Vol. 129. Readings in Mathematics. Springer-Verlag, New York, 1991.
R. Howe, T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann., 290 (1991) 565–619.
F. Knop, Some remarks on multiplicity free spaces, in: Broer, A. (ed.) et al., Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal (Canada); Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 301–317 (1998).
B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), no. 3, 753–809.
P. Littelmann, On spherical double cones, J. Algebra, 166 (1994), 142–157.
J.N. Mather, Differentiable invariants, Topology 16 (1977), 145–155.
G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.
E.B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys, 56 (2001), 1–60.
J. Wolf, Harmonic Analysis on Commutative Spaces, Math. Surveys and Monographs 142, Amer. Math. Soc., 2007.
O. Yakimova, Gelfand pairs, Dissertation, Rheinischen Friedrich-Wilhelms-Universität Bonn, 2004; Bonner Mathematische Schriften 374 (2005).
O. Yakimova, Principal Gelfand pairs, Transform. Groups, 11 (2006), 305–335.
Acknowledgments
Parts of this work were carried out during the third author’s stay at the Max-Planck-Institut für Mathematik (Bonn) and Centro di Ricerca Matematica Ennio De Giorgi (SNS, Pisa). She would like to thank these institutions for warm hospitality and support.
The first author acknowledges the support of the London Mathematical Society via the Grace Chisholm Fellowship held at King’s College London in 2011.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fischer, V., Ricci, F., Yakimova, O. (2013). Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions II: Taylor Expansions on Singular Sets. In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_5
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7193-6_5
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4614-7192-9
Online ISBN: 978-1-4614-7193-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)