Harmonic analysis on \(\text{SO}(n,\mathbb C)/\text{SO}(n-1,\mathbb C)\), \(n \geq 3\) (Q2498370)

The goal of this paper is to compute explicitly the Plancherel measure for \(\text{SO}(n,\mathbb C)/\text{SO}(n-1,\mathbb C)\) where \(\text{SO}(n,\mathbb C)\) is the complex orthogonal group of \(n\) by \(n\) matrices with determinant \(1\). Inspired by \textit{E. van der Ban}'s approach [The Plancherel theorem for a reductive symmetric space, Lectures for the European School of Group Theory, August 14-26, 2000, SDU Odense University], the definition of the Plancherel measure is based on a specific intertwining operator. A large part of the paper consists of various harmonic analysis facts on \(\text{SO}(n, \mathbb C)/\text{SO}(n-1,\mathbb C)\), \(n\geq 3\), used by the author in order to reach the main goal. As an application, in the last section the Plancherel formula of \(\text{SL}(2,\mathbb C)\) is obtained.