# The Fourier transform on the group $GL_2(R)$ and the action of the overalgebra $gl_4$

## TL;DR

This paper develops an operational calculus for the group $GL_2(R)$ by leveraging its embedding in the Grassmannian and explores the action of the Lie algebra $gl_4$ on its Plancherel decomposition, involving differential-difference operators.

## Contribution

It introduces a novel approach to analyze $GL_2(R)$ using the action of $gl_4$ on its harmonic analysis decomposition, connecting group actions with differential-difference operators.

## Key findings

- Derived explicit formulas for $gl_4$ action on $GL_2(R)$
- Expressed the Lie algebra action as differential-difference operators
- Extended the analysis to $GL_2(C)$ with $gl_4 imes gl_4$ action

## Abstract

We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$. We transfer the corresponding action of the Lie algebra $gl_4$ to the Plancherel decomposition of $GL_2(R)$, the Lie algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of $gl_4\oplus gl_4$ in the Plancherel decomposition of $GL_2(C)$