On the absolute convergence of the spectral side of the twisted trace formula

We establish the absolute convergence of the spectral side of the twisted trace formula for reductive algebraic groups. More precisely, we extend the absolute convergence theorem of the spectral side due to Finis–Lapid–Müller to the twisted setting. This paper provides a detailed proof of the absolute convergence theorem, whose proof is known to experts but does not seem to have appeared in the literature. The resulting formulas are motivated by applications to limit multiplicity problems and to Weyl laws for self-dual automorphic representations of ${\mathrm{G}\mathrm{L}}_n$.