The n-cohomology of limits of discrete series
Let G be a connected noncompact real semisimple Lie group with finite center and \(\Gamma\) a cocompact discrete subgroup of G. We assume that G has a compact Cartan subgroup H. The results of this paper are concerned with a cohomological realization of the limits of discrete series representations appearing in \(L^ 2(\Gamma \setminus G)\), where \(L^ 2(\Gamma \setminus G)\) is the space of all square integrable functions on \(\Gamma\setminus G\) with respect to the canonical G-invariant measure. We state those results more precisely after the following notations. Let \({\mathfrak g}\) and \({\mathfrak h}\) be the Lie algebras of G and H, respectively. The root system of (\({\mathfrak g}_ C,{\mathfrak h}_ C)\) will be denoted by \(\Delta\) where \({\mathfrak g}_ C\), \({\mathfrak h}_ C\) are the complexifications of \({\mathfrak g}\), \({\mathfrak h}\). Fix a linear order in \(\Delta\), and denote one half the sum of all positive roots in \(\Delta\) by \(\delta\). Let \(b^*\) be a unitary character of H. Then there exists a unique integral form \(\lambda\) on \({\mathfrak h}_ C\) such that \(b^*(\exp X)=e^{\lambda (X)}\) for all X in \({\mathfrak h}\). We fix a character \(e^{\lambda}\) of H. Let \(\Theta_{\lambda +\delta}\) be Harish- Chandra's tempered invariant eigendistribution on G corresponding to the data \(\lambda +\delta\). If \(\Theta_{\lambda +\delta}\) is not identically zero on G, then \(\Theta_{\lambda +\delta}\) is a character, up to a constant factor, of an irreducible unitary representation \(\pi_{\lambda +\delta}\) of G. \(\pi_{\lambda +\delta}\) is called a discrete [resp. limit of discrete] series representation if \(\lambda +\delta\) is regular [resp. singular]. Let \({\mathcal L}_{\lambda}\to G/H\) be the canonical line bundle to the character \(e^{\lambda}\) of H and \(H_ 2(\Gamma \setminus {\mathcal L}_{\lambda})\) the \(L^ 2\)-cohomology space of the quotient bundle \(\Gamma\setminus {\mathcal L}_{\lambda}\). W. Schmid proved that dim \(H_ 2(\Gamma \setminus {\mathfrak L}_{\lambda})=m_{\pi_{\lambda +\delta}}(\Gamma)\) for a sufficiently regular integral form \(\lambda +\delta\), where \(m_{\pi_{\lambda +\delta}}(\Gamma)\) is the multiplicity of \(\pi_{\lambda +\delta}\) appearing in \(L^ 2(\Gamma \setminus G)\). For the remaining discrete series to them, the author proved the same dimension formula as above. Therefore all discrete series representations which appear in \(L^ 2(\Gamma \setminus G)\) are realized on the cohomological spaces \(H_ 2(\Gamma \setminus {\mathcal L}_{\lambda}).\) The author proves a vanishing theorem for a certain cohomology space. This result is a step to prove the cohomological realization for limits of discrete series representations appearing in \(L^ 2(\Gamma \setminus G)\). In particular, for the limits of holomorphic discrete series \(\pi_{\lambda +\delta}\), he also proves that dim \(H_ 2(\Gamma \setminus G)=m_{\pi_{\lambda +\delta}}(\Gamma)\).
limits of discrete series representations
scientific article; zbMATH DE number 4165191
scientific article