Operational calculus for Fourier transform on the group $GL(2,R)$

TL;DR

This paper develops an operational calculus for the Fourier transform on the group $GL(2,R)$, describing how polynomial differential operators transform into differential-difference operators with meromorphic coefficients, including explicit formulas for key operators.

Contribution

It introduces a novel operational calculus for Fourier transforms on $GL(2,R)$, detailing the structure of transformed differential operators with explicit formulas.

Findings

Fourier images of polynomial differential operators are differential-difference operators.

Operators involve shifts in imaginary directions related to the representation parameters.

Explicit formulas for images of partial derivatives and coordinate multiplications are provided.

Abstract

Consider the Fourier transform on the group $GL(2,R)$ of real $2\times 2$-matrices. We show that Fourier-images of polynomial differential operators on $GL(2,R)$ are differential-difference operators with coefficients meromorphic in parameters of representations. Expressions for operators contain shifts in imaginary direction with respect to the integration contour in the Plancherel formula. We present explicit formulas for images of partial derivations and multiplications by coordinates.