Let $D$ be the transverse signature operator constructed by Connes and Moscovici in the paper "Local index formula in Noncommutative Geometry":this is first order hypoelliptic pseudodifferential operator $D$ defined by the equality $D|D|=Q$ where $Q=(d_Vd_V^*-d_V^*d_V) \oplus (d_H+d_H^*)$ where $d_V,d_H$ are vertical and horizontal exterior derivative. It acts on the sections of the bundle $\Lambda(V^*) \otimes \Lambda(p^*(T^*M))$ over $P:=GL^+(M)/SO(n)$ (the bundle of euclidean metrics) where $V=\ker{p_*}$ where $p:P \to M$ is a projection map and $H$ is defined using a connection via $V \oplus H=TP$.
How the spectrum of $D$ looks like?
