We use geometrical methods adapted from Borel-Weil-Bott theory to compute the character of every finite-dimensional simple module over a basic classical Lie superalgebra. The answer is given by a combinatorial algorithm in terms of weight and cap diagrams, which were defined by Brundan and Stroppel for the general linear superalgebra.We thank Laurent Gruson for hospitality and stimulating discussions and Ian Musson for explaining the method of weight diagrams and fruitful discussions on supergeometry.

We prove a BGG type reciprocity law for the category of finite dimensional modules over algebraic supergroups satisfying certain conditions. The equivalent of a standard module in this case is a virtual module called Euler characteristic due to its geometric interpretation. In the orthosymplectic case, we also describe indecomposable projective modules in terms of those Euler characteristics.

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