On a semigroup of measures with irregular densities

We study the densities of the semigroup generated by the operator ${\displaystyle -X^2+\left|Y\right|}$ on the 3-dimensional Heisenberg group. We show that the 7th derivatives of the densities have a jump discontinuity. Outside the plane x=0 the densities are ${\displaystyle C^{\infty }}$. We give explicit spectral decomposition of images of ${\displaystyle -X^2+\left|Y\right|}$ in representations.