  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).
