### Abstract

Let Θ denote an involution for a simply connected compact Lie group U let K denote the fixed point set and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding φ :U/K → U:u → uu ^{-}Θ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for φ*μ relative to a compatible triangular decomposition of G the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation involving a Poisson structure on the dual symmetric space G_{0}/K discovered by Evens and Lu.

Original language | English (US) |
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Pages (from-to) | 705-724 |

Number of pages | 20 |

Journal | Transformation Groups |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The diagonal distribution for the invariant measure of a unitary type symmetric space.** / Pickrell, Douglas M.

Research output: Contribution to journal › Article

*Transformation Groups*, vol. 11, no. 4, pp. 705-724. https://doi.org/10.1007/s00031-005-1126-1

}

TY - JOUR

T1 - The diagonal distribution for the invariant measure of a unitary type symmetric space

AU - Pickrell, Douglas M

PY - 2006/12

Y1 - 2006/12

N2 - Let Θ denote an involution for a simply connected compact Lie group U let K denote the fixed point set and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding φ :U/K → U:u → uu -Θ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for φ*μ relative to a compatible triangular decomposition of G the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation involving a Poisson structure on the dual symmetric space G0/K discovered by Evens and Lu.

AB - Let Θ denote an involution for a simply connected compact Lie group U let K denote the fixed point set and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding φ :U/K → U:u → uu -Θ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for φ*μ relative to a compatible triangular decomposition of G the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation involving a Poisson structure on the dual symmetric space G0/K discovered by Evens and Lu.

UR - http://www.scopus.com/inward/record.url?scp=33847636141&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847636141&partnerID=8YFLogxK

U2 - 10.1007/s00031-005-1126-1

DO - 10.1007/s00031-005-1126-1

M3 - Article

AN - SCOPUS:33847636141

VL - 11

SP - 705

EP - 724

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

IS - 4

ER -