  We establish linearized well-posedness of the Triple-Deck system in
Gevrey-$\frac32$ regularity in the tangential variable, under concavity
assumptions on the background flow. Due to the recent result \cite{DietertGV},
one cannot expect a generic improvement of the result of \cite{IyerVicol} to a
weaker regularity class than real analyticity. Our approach exploits two
ingredients, through an analysis of space-time modes on the Fourier-Laplace
side: i) stability estimates at the vorticity level, that involve the concavity
assumption and a subtle iterative scheme adapted from \cite{GVMM} ii) smoothing
properties of the Benjamin-Ono like equation satisfied by the Triple-Deck flow
at infinity. Interestingly, our treatment of the vorticity equation also adapts
to the so-called hydrostatic Navier-Stokes equations: we show for this system a
similar Gevrey-$\frac32$ linear well-posedness result for concave data,
improving at the linear level the recent work \cite{MR4149066}.
