The Relative Lie Algebra Cohomology of the Weil Representation

Otherwise, all relative Lie algebra cohomology groups would vanish, see Proposition \ref{genuineprop}. In case the symplectic group is large relative to the orthogonal group ($k \geq pq$), the $E_0$ term is isomorphic to a Koszul complex defined by a regular sequence, see \ref{defofkoszulsection}. Thus, the cohomology vanishes except in top degree.

This result is obtained without calculating the space of cochains and hence without using any representation theory. On the other hand, in case $k < p$, we know the Koszul complex is not that of a regular sequence from the existence of the class $\varphi_{kq}$ of Kudla and Millson, see \cite{KM2}, a nonzero element of the relative Lie algebra cohomology of degree $kq$. For the case of $\SO_0(p,1)$ we compute the cohomology groups in these remaining cases, namely $k < p$.

We do this by first computing a basis for the relative Lie algebra cochains and then splitting the complex into a sum of two complexes, each of whose $E_0$ term is then isomorphic to a Koszul complex defined by a regular sequence. This thesis is adapted from the paper, \cite{BMR}, this author wrote with his advisor John Millson and Nicolas Bergeron of the University of Paris.