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Operational Calculus for the Fourier Transform on the Group GL(2,ℝ) and the Problem about the Action of an Overalgebra in the Plancherel Decomposition

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Abstract

The Fourier transform on the group GL(2,ℝ) of real 2 × 2 matrices is considered. It is shown that the Fourier images of polynomial differential operators on GL(2,ℝ) are differentialdifference operators with coefficients meromorphic in the parameters of representations. Expressions for operators contain shifts in the imaginary direction with respect to the integration contour in the Plancherel formula. Explicit formulas for the images of partial derivations and multiplications by coordinates are presented.

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Authors and Affiliations

  1. Mathematical Department, University of Vienna, Vienna, Austria

    Yu. A. Neretin

  2. Institute for Theoretical and Experimental Physics, Moscow, Russia

    Yu. A. Neretin

  3. Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    Yu. A. Neretin

  4. Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia

    Yu. A. Neretin

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  1. Yu. A. Neretin
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Correspondence to Yu. A. Neretin.

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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 52, No. 3, pp. 42–52, 2018 Original Russian Text Copyright © by Yu. A. Neretin

Supported by FWF grant P28421.

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Neretin, Y.A. Operational Calculus for the Fourier Transform on the Group GL(2,ℝ) and the Problem about the Action of an Overalgebra in the Plancherel Decomposition. Funct Anal Its Appl 52, 194–202 (2018). https://doi.org/10.1007/s10688-018-0228-1

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  • DOI: https://doi.org/10.1007/s10688-018-0228-1

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