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

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.

9p

References in corpus (3)

• Restriction of representations of $GL(n+1,C)$ to $GL(n,C)$ and action of the Lie overalgebra
• The Fourier transform on the group $G{L}_2(R)$ and the action of the overalgebra $g{l}_4$
• After Plancherel formula