  We study real-time finite-temperature correlators for free scalars of any
mass in a $dS_{d+1}$ static patch in any dimension. We show that whenever the
inverse temperature is a rational multiple of the inverse de Sitter
temperature, certain Matsubara poles of the symmetric Wightman function
disappear. At the de Sitter temperature, we explicitly show how the Lorentzian
thermal correlators can all be obtained by analytic continuations from the
round $S^{d+1}$. We establish the precise relation between the Harish-Chandra
character for $SO(1,d+1)$ and the integrated spectral function, providing a
novel dynamical perspective on the former and enabling generalizations.
Furthermore, we study scalars with exceptional non-positive masses. We provide
a physical picture for the distinctive structures of their characters. For the
massless case, we perform a consistent static patch quantization, and find the
unique $S^{d+1}$ correlator that analytically continues to the correlators in
the quantum theory.
