Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q)

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q).