AN ORBIT MODEL FOR THE SPECTRA OF NILPOTENT GELFAND PAIRS

Transformation Groups volume 25, pages859–886(2020)Cite this article

Abstract

Let N be a connected and simply connected nilpotent Lie group, and let K be a subgroup of the automorphism group of N. We say that the pair (K, N) is a nilpotent Gelfand pair if \( {L}_K^1(N) \) is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs (K, N) where the K-orbits in the center of N have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set Δ(K, N) of bounded K-spherical functions on N and the set \( \mathcal{A} \)(K, N) of K-orbits in the dual 𝔫* of the Lie algebra for N established in [BR08] is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for N a free group and N a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs in [BR08].

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Author information

Affiliations

  1. Department of Mathematics and Computer Science, Dickinson College, Carlisle, PA, 17013, USA

    HOLLEY FRIEDLANDER

  2. Department of Mathematics, St. Olaf College, Northfield, MN, 55057, USA

    WILLIAM GRODZICKI

  3. Department of Mathematics and Physics, Westminster College, Fulton, MO, 65251, USA

    WAYNE JOHNSON

  4. Department of Mathematics, East Carolina University, Greenville, NC, 27858, USA

    GAIL RATCLIFF

  5. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2042, Australia

    ANNA ROMANOV

  6. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    BENJAMIN STRASSER

  7. Department of Mathematics, St. Louis University, St. Louis, MO, 63103, USA

    BRENT WESSEL

Authors
  1. HOLLEY FRIEDLANDER
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  2. WILLIAM GRODZICKI
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  3. WAYNE JOHNSON
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  4. GAIL RATCLIFF
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  5. ANNA ROMANOV
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  6. BENJAMIN STRASSER
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  7. BRENT WESSEL
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Correspondence to ANNA ROMANOV.

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FRIEDLANDER, H., GRODZICKI, W., JOHNSON, W. et al. AN ORBIT MODEL FOR THE SPECTRA OF NILPOTENT GELFAND PAIRS. Transformation Groups 25, 859–886 (2020). https://doi.org/10.1007/s00031-019-09541-8

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