Legendrian tori and the semi-classical limit

Let X   be CPnCPn or a compact smooth quotient of the n  -dimensional complex hyperbolic space, n>1n>1. Let L be a hermitian holomorphic line bundle (with hermitian connection) on X   chosen as follows: if X=CPnX=CPn then L is the hyperplane bundle, and in the second case L   is chosen so that L⊗(n+1)=KX⊗EL⊗(n+1)=KX⊗E, where KXKX is the canonical line bundle and E is a flat line bundle. The unit circle bundle P   in L∗L∗ is a contact manifold. Let k′k′ be a fixed positive integer. We construct certain Legendrian tori in P   (the construction depends, in particular, on the choice of k′k′) and sequences {uk}{uk}, k=k′mk=k′m, m=1,2,…, of holomorphic sections of L⊗kL⊗k associated to these tori. We study asymptotics of the norms ‖uk‖k‖uk‖k as m→+∞m→+∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n  -dimensional complex hyperbolic space. We obtain an n>1n>1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n=1n=1).