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We study isoparametric submanifolds of rank at least two in a separableHilbert space, which are known to be homogeneous by a result of Heintze andLiu, and associate to such a submanifold M and a point x in M a canonicalhomogeneous structure (a certain bilinear map on the tangent space). We provethat the homogeneous structure together with the second fundamental formencodes all the information about M, and deduce from this the rigidity resultthat M is completely determined by the second fundamental form and itscovariant derivative, thereby making such submanifolds accessible toclassification. As an essential step, we show that the one-parameter groups ofisometries constructed by Heintze and Liu to prove their homogeneity inducesmooth and hence everywhere defined Killing fields, implying the continuity ofthe homogeneous structure. Here an important tool is the introduction of affineroot systems of isoparametric submanifolds.

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space