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We first discuss the relationship between the SL(2;R)/U(1) supercoset and N=2Liouville theory and make a precise correspondence between theirrepresentations. We shall show that the discrete unitary representations ofSL(2;R)/U(1) theory correspond exactly to those massless representations of N=2Liouville theory which are closed under modular transformations and studied inour previous work hep-th/0311141.  It is known that toroidal partition functions of SL(2;R)/U(1) theory (2DBlack Hole) contain two parts, continuous and discrete representations. Thecontribution of continuous representations is proportional to the space-timevolume and is divergent in the infinite-volume limit while the part of discreterepresentations is volume-independent.  In order to see clearly the contribution of discrete representations weconsider elliptic genus which projects out the contributions of continuousrepresentations: making use of the SL(2;R)/U(1), we compute elliptic genera forvarious non-compact space-times such as the conifold, ALE spaces, Calabi-Yau3-folds with A_n singularities etc. We find that these elliptic genera ingeneral have a complex modular property and are not Jacobi forms as opposed tothe cases of compact Calabi-Yau manifolds.Comment: 39 pages, no figure; v2 references added, minor corrections; v3 typos  corrected, to appear in JHEP; v4 typos corrected in eqs. (3.22) and (3.44

SL(2;R)/U(1) Supercoset and Elliptic Genera of Non-compact Calabi-Yau Manifolds