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

Yury A. Neretin

doi:10.1007/s00041-017-9589-8

paper

The Fourier transform on the group $G L_{2} (R)$ and the action of the overalgebra $g l_{4}$

arXiv:1704.04018 · doi:10.1007/s00041-017-9589-8

Abstract

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

16pp

The Fourier transform on the group GL2​(R) and the action of the overalgebra gl4​

Abstract

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The Fourier transform on the group $G L_{2} (R)$ and the action of the overalgebra $g l_{4}$