  We study the semigroup generated by the hypoelliptic Laplacian on the circle
and the maximal bounded holomorphic extension of this semigroup. Using an
orthogonal decomposition into harmonic oscillators with complex shifts, we
describe the domain of this extension and we show that boundedness in a
half-plane corresponds to absolute convergence of the expansion of the
semigroup in eigenfunctions. This relies on a novel integral formula for the
spectral projections which also gives asymptotics for Laguerre polynomials in a
large-parameter regime.
