学术报告（刘克峰 2025.2.25）

Abstract: I will present my recent joint works with Yang Shen on geometry and representation theory aspects of non-classical flag domains. 

We prove various conjectures of Griffiths and Green-Griffiths-Kerr about the automorphic cohomology on compact quotients of non-classical flag domains.

We first .construct new  complex structures on non-classical flag domains $D=G_\R/V$ with $G_\R$ of Hermitian type and their compact quotients. As applications, we give new examples of compact smooth manifolds on which there are  two complex structures with very different geometric properties.

Building on these works, we then construct Penrose transformations of the cohomology groups of homogeneous line bundles on flag domains $D = G_\R / T$ and identify conditions under which the Penrose transformation of the automorphic cohomology groups on compact quotients of flag domains is an isomorphism.As applications we prove that the higher automorphic cohomology groups of certain homogeneous line bundles are isomorphic to the groups of automorphic forms on the Hermitian symmetric domain.