Year: 2025
Type: article
Abstract: Given a connected semisimple Lie group G, Monod (Trans. Amer. Math. Soc. B 144–159, 2022) has recently proved that the measurable cohomology of the G-action $$H^*_m(G \curvearrowright G/P)$$ H m ∗ ( G ↷ G / P ) on the Furstenberg boundary G/P, where P is a minimal parabolic subgroup, maps surjectively on the measurable cohomology of G through the evaluation on a fixed basepoint. Additionally, the kernel of this map depends entirely on the invariant cohomology of a maximal split torus. In this paper we show a similar result for a fixed subgroup $$L L < P such that the stabilizer of almost every pair of points in G/L is compact. More precisely, we show that the cohomology of the G-action $$H^p_m(G \curvearrowright G/L)$$ H m p ( G ↷ G / L ) maps surjectively onto $$H^p_m(G)$$ H m p ( G ) with a kernel isomorphic to $$H^{p-1}_m(L)$$ H m p - 1 ( L ) . Examples of such groups are given either by any term of the derived series of the unipotent radical N of P or by a maximal split torus A. We conclude the paper by computing explicitly some cocycles on quotients of $$\textrm{SL}(2,\mathbb {K})$$ SL ( 2 , K ) for $$\mathbb {K}=\mathbb {R}, \mathbb {C}$$ K = R , C .
Source: Geometriae Dedicata
Authors: Michelle Bucher, Alessio Savini
Institutions: University of Geneva, University of Milano-Bicocca
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,090
OpenAlex ID: W4409765595