Geometric realization of discrete series for semisimple symmetric spaces (Q1825976)

Let G be a real semisimple Lie group, \(\sigma\) an involution of G and H the identity component of the group of \(\sigma\)-fixed points in G. Let K be a \(\sigma\)-stable maximal compact subgroup of G and assume that \(rank(H\setminus G)=rank(K\cap H\setminus K)\); then \(H\setminus G\) has countably many discrete series representations [cf. \textit{M. Flensted- Jensen}, Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006) and \textit{T. Oshima} and \textit{T. Matsuki}, Adv. Stud. Pure Math. 4, 331-390 (1984; Zbl 0577.22012)]. Let \(\Gamma\) \(\setminus {\mathcal D}\) be a locally symmetric space, where \({\mathcal D}=G/K\), and where \(\Gamma\) is a cocompact discrete subgroup of G such that \(\Gamma\) \(\cap H\setminus H\) is compact. The authors consider a suitable subset \(\Lambda\) of the set of discrete series parameters (basically the restriction comes down to sufficient regularity). For \(\lambda\in \Lambda\) the associated discrete series representation \({\mathcal V}_{\lambda}\subset L^ 2(H\setminus G)\) occurs with multiplicity one. It has a unique minimal K-type: the associated isotopic component of \({\mathcal V}_{\lambda}\) contains Flensted-Jensen's generating function \(\psi_{\lambda}\). From \(\psi_{\lambda}\) the authors construct a harmonic form \({\tilde \omega}{}_{\lambda}\) on G with values in a finite dimensional G-module V,\(\tau\) (of lowest weight \(\rho\)-\(\lambda)\). This form is left H-invariant and the closed span of its right G-translates realizes the contragredient of \({\mathcal V}_{\lambda}\). Moreover, let E be the (locally constant) vector bundle associated with the principal fiber bundle \(\Gamma\) \(\setminus G\to \Gamma \setminus {\mathcal D}\) and \(\tau\) \(| K\), and let \(p^*E\) denote its pull back under the canonical projection p: \({\mathcal D}\to \Gamma \setminus {\mathcal D}\). Then by construction \({\tilde \omega}{}_{\lambda}\) is the pull-back of a \(p^*E\)-valued closed form \(\omega_{\lambda}\) on \({\mathcal D}\). Moreover, \[ {\hat \omega}_{\lambda}=\sum_{\gamma \in \Gamma \cap H\setminus \Gamma}\gamma^*\omega_{\lambda} \] defines a cohomology class [\({\hat \omega}{}_{\lambda}]\in H^ R(\Gamma \setminus {\mathcal D},E)\); where \(R=\dim (G/K)-\dim (H/H\cap K)\). On the other hand, the authors use the finite G-module V to construct a section s of E over the cycle \(\Gamma\) \(\cap H\setminus H/H\cap K\). This geometric object is then related to \({\hat \omega}{}_{\lambda}\) by a duality.