The continuous cohomology of period domains over local fields (Q2466960)

Let \(L\) be an algebraically closed field of characteristic \(p>0\), let \(K_0=\text{ Frac}(W(L))\). To a triple \((G,b,\{\mu\})\) consisting of a quasi-split reductive group \(G\) defined over \({\mathbb Q}\), an element \(b\in G(K_0)\) and a conjugacy class \(\{\mu\}\) of one parameter subgroups of \(G\), \textit{M. Rapoport} and \textit{T. Zink} [``Period spaces for \(p\)-divisible groups'', Ann. Math. Stud.. 141 (1996; Zbl 0873.14039)] associated a period domain \({\mathcal F}_b^{wa}\), a rigid analytic space which is defined over on explicit finite extension \(E_s\) of \({\mathbb Q}_p\). It is characterized as the weak admissibility locus of the space of all filtrations corresponding to the class \(\{\mu\}\), on the \(F\)-isocrystal with \(G\)-structure corresponding to \(b\). Let \(J\) denote the automorphism group of this isocrystal. In this paper, the \({\ell}\)-adic continuous cohomology (\(\ell\neq p\)) of \({\mathcal F}_b^{wa}\) is determined as a representation of \(J({\mathbb Q}_p)\times \text{ Gal}(\overline{E}_s/E_s)\) --- in the case where \(J\) is an inner form of \(G\) (the so called `basic' case). The overall strategy is very similar to that developed by the author in an earlier paper [Invent. Math. 162, No. 3, 523--549 (2005; Zbl 1093.14065)] where the étale cohomology with torsion coefficients had been considered.