  We study modular properties of the AdS3 WZNW model. Although the Euclidean
partition function is modular invariant, the characters on the Euclidean torus
are ill-defined and their modular transformations are unknown. We reconsider
the characters defined on the Lorentzian torus, focusing on their structure as
distributions. We find a generalized S matrix, depending on the sign of the
real modular parameters, which has two diagonal blocks and one off-diagonal
block, mixing discrete and continuous representations, that we fully determine.
We then explore the relations among the modular transformations, the fusion
algebra and the boundary states. We explicitly construct Ishibashi states for
the maximally symmetric D-branes and show that the generalized S matrix defines
the one-point functions associated to point-like and H2 branes as well as the
fusion rules of the degenerate representations of SL(2,R) appearing in the open
string spectrum of the point-like D-branes, through a generalized Verlinde
theorem.
