On the convergence of the Hochschild-Serre spectral sequence for the continuous cohomology of parabolic discrete subgroups
Let \(G\) be a connected semisimple algebraic group defined over \(\mathbb{Q}\), with \(\text{rank}_{\mathbb{Q}} G>0\). Given an arithmetic torsion free subgroup \(\Gamma\) of \(G(\mathbb{Q})\), the associate quotient \(\Gamma\setminus X\) (\(X=G(\mathbb{R})/K\), \(K\subset G(\mathbb{R})\) maximal compact) may be viewed as the interior of a manifold with corners, glued together out of faces \(e'(P)\), one for each \(\Gamma\)-conjugacy class of parabolic \(\mathbb{Q}\)- subgroups of \(G\). A single face \(e'(P)\), \(P\neq G\), is described by a fibration \(\Gamma\cap N(\mathbb{R})\setminus N(\mathbb{R})\to e'(P)\to\Gamma_ M\setminus Z_ M\) (induced from the projection \(P\to P/N\), \(N\) the unipotent radical of \(P\)) over the locally symmetric space attached to the Levi subgroup of \(P\) with compact fibers. This fibration gives rise to a spectral sequence in cohomology (with a coefficient system given by a finite dimensional representation of \(G(\mathbb{R})\)) which degenerates at \(E_ 2\) [see 2.7. in \textit{J. Schwermer}, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen (Lect. Notes Math. 988) (1983; Zbl 0506.22015)]. This extends the result of \textit{G. Harder} obtained for groups of \(\mathbb{Q}\)-rank one [Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Conf. Bombay 1973, 129-160 (1975; Zbl 0317.57022)] to the general case. The paper under review takes up Harder's approach and extends it to the square integrable cohomology groups, with coefficients given by a smooth representation of \(G\) in a Hilbert space with finite spectrum. The changes are not significant.
scientific article; zbMATH DE number 35218