Hilbert modules and Kähler geometry

Abstract In the first part, we describe the recent work (Upmeier in Atti Accad Naz Lincei Cl Sci Fis Mat Nat 32:565–591, 2021; Upmeier in J Reine Angew Math (Crelle J) 799:155–187, 2023; Upmeier in Complex Anal Oper Theory 17:74, 2023) of the author concerning the eigenbundle of analytic Hilbert modules over bounded symmetric domains of arbitrary rank. In the second part, we develop in full detail the realization of the fibers of the eigenbundle in terms of Kähler geometry, as holomorphic section modules over certain flag manifolds. Here a new feature is to work directly on the complex strata (Kepler manifolds) instead of just the so-called tripotents. This makes the complex analytic properties more transparent.