### Abstract

This paper is a continuation of [8] in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N, K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum Σ_{D}, appropriately embedded in a Euclidean space. We prove a result, of independent interest for the representation-theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as v ⊕ [n,n] with v irreducible under K.

Original language | English |
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Title of host publication | Lie Groups |

Subtitle of host publication | Structure, Actions, and Representations: In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday |

Editors | A. Huckleberry, I. Penkov, G. Zuckerman |

Place of Publication | New York, U. S. A. |

Publisher | Birkhauser Boston |

Pages | 81-112 |

Number of pages | 32 |

ISBN (Electronic) | 9781461471936 |

ISBN (Print) | 9781461471929 |

DOIs | |

Publication status | Published - 1 Jan 2013 |

### Publication series

Name | Progress in Mathematics |
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Volume | 306 |

### Fingerprint

### Keywords

- Gelfand pairs
- Invariants
- Schwartz functions
- Spherical transform

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Lie Groups: Structure, Actions, and Representations: In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday*(pp. 81-112). (Progress in Mathematics; Vol. 306). New York, U. S. A.: Birkhauser Boston. https://doi.org/10.1007/978-1-4614-7193-6_5

**Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II : Taylor expansions on singular sets.** / Fischer, Véronique; Ricci, Fulvio; Yakimova, Oksana.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lie Groups: Structure, Actions, and Representations: In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday.*Progress in Mathematics, vol. 306, Birkhauser Boston, New York, U. S. A., pp. 81-112. https://doi.org/10.1007/978-1-4614-7193-6_5

}

TY - CHAP

T1 - Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II

T2 - Taylor expansions on singular sets

AU - Fischer, Véronique

AU - Ricci, Fulvio

AU - Yakimova, Oksana

PY - 2013/1/1

Y1 - 2013/1/1

N2 - This paper is a continuation of [8] in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N, K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum ΣD, appropriately embedded in a Euclidean space. We prove a result, of independent interest for the representation-theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as v ⊕ [n,n] with v irreducible under K.

AB - This paper is a continuation of [8] in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N, K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum ΣD, appropriately embedded in a Euclidean space. We prove a result, of independent interest for the representation-theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as v ⊕ [n,n] with v irreducible under K.

KW - Gelfand pairs

KW - Invariants

KW - Schwartz functions

KW - Spherical transform

UR - http://www.scopus.com/inward/record.url?scp=84924941350&partnerID=8YFLogxK

U2 - 10.1007/978-1-4614-7193-6_5

DO - 10.1007/978-1-4614-7193-6_5

M3 - Chapter

SN - 9781461471929

T3 - Progress in Mathematics

SP - 81

EP - 112

BT - Lie Groups

A2 - Huckleberry, A.

A2 - Penkov, I.

A2 - Zuckerman, G.

PB - Birkhauser Boston

CY - New York, U. S. A.

ER -