A note on Mellin transform, Eisenstein Series and distribution $dε_{it}$ on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$

Abstract

Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $dε_{it}$ to $SL(2,\mathbb{C})$. We calculate $(f,dε_{it})$ for $f$ a cuspidal form and for $f$ an incomplete Eisenstein series. We also establish asymptotic estimates when $t$ tends to $\infty$. We conjecture that a new positive distribution $dε_{it}^F$, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that $dε_{it}$. This would imply the quantum ergodicity for Eisenstein series on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$