On the \(({\mathfrak g},K)\)-cohomology of certain theta lifts. (Q1858133)

The authors consider those representations \(\theta ^ {p,q}\) of \(Sp(4,\mathbb R)\) which are theta ``lifts'' from the trivial representation of \(SO(p,q)\) with \(p+q\) even. In [Trans. Am. Math. Soc. 350, No. 12, 5017--5046 (1998; Zbl 0901.22008)] and [Isr. J. Math. 100, 29--59 (1997; Zbl 0890.22008)] the authors determined the structure of these representations completely. In this paper they use this information to compute all the cohomology groups \(H ^{i}(\mathfrak{g},K;\theta ^ {p,q} \otimes F)\) where \(F\) is a finite dimensional representation.