Homogeneity of infinite dimensional anti-Kaehler isoparametric   submanifolds II
  In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is an orbit of the action of a Banach Lie group generated by one-parameter transformation groups induced by holomorphic Killing vector fields defined entirely on the ambient Hilbert space. 
Naoyuki Koike
math.DG