Projectors separating spectra for $L^2$ on symmetric spaces $GL(n,\text{C})\mathrm{/}GL(n,\mathbb{R})$

The Plancherel decomposition of $L^2$ on a pseudo-Riemannian symmetric space $GL(n,C)\mathrm{/}GL(n,R)$ has spectrum of $[n\mathrm{/}2]$ types. We write explicitly orthogonal projectors separating spectrum into uniform pieces