Geodesic
Operational Calculus for the Fourier Transform on the Group $\operatorname{GL}(2,\mathbb{R})$ and the Problem about the Action of an Overalgebra in the Plancherel Decomposition
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 42-52

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The Fourier transform on the group $\operatorname{GL}(2,\mathbb{R})$ of real $2\times2$ matrices is considered. It is shown that the Fourier images of polynomial differential operators on $\operatorname{GL}(2,\mathbb{R})$ are differential-difference 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.
Mots-clés : Fourier transform on groups, semisimple Lie group
Keywords: differential-difference operator, Weil representation, principal series of representations, operational calculus, unitary representation, Heisenberg algebra. 
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     author = {Yu. A. Neretin},
     title = {Operational {Calculus} for the {Fourier} {Transform} on the {Group} $\operatorname{GL}(2,\mathbb{R})$ and the {Problem} about the {Action} of an {Overalgebra} in the {Plancherel} {Decomposition}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {42--52},
     publisher = {mathdoc},
     volume = {52},
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Yu. A. Neretin. Operational Calculus for the Fourier Transform on the Group $\operatorname{GL}(2,\mathbb{R})$ and the Problem about the Action of an Overalgebra in the Plancherel Decomposition. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 42-52. http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a3/