Kernels in measurable cohomology for transitive actions

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 $$H^*_m(G \curvearrowright G/P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>↷</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$L&lt;P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> 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 $$H^p_m(G \curvearrowright G/L)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>↷</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> maps surjectively onto $$H^p_m(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with a kernel isomorphic to $$H^{p-1}_m(L)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 $$\textrm{SL}(2,\mathbb {K})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SL</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for $$\mathbb {K}=\mathbb {R}, \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> .