Convexity theorem for infinite-dimensional isoparametric submanifolds (Q690201)

The main purpose of this paper is to prove an analogue for infinite- dimensional isoparametric submanifolds in Hilbert space of the Kostant convexity theorem for generalized complex and real flag manifolds which can be stated as follows: Let \(M\) be an isoparametric submanifold of a Hilbert space \(V\), and \(W\) the associated affine Weyl group of \(M\), \(q \in M\), and \(\nu = q + \nu(M)_ q\). Assume that 0 is a vertex of some Weyl chamber of \(W\) in \(\nu\). Let \(P\) be the orthogonal projection onto \(\nu\), and \(\mu: M \to \nu \times R,\quad \mu(x) = (P(x),\;\| x\|^ 2).\) Then (i) \(\mu(M) = cvx((W \cdot q))\), (ii) if \(M_ v\) is a parallel submanifold of \(M\) with respect to some parallel normal field \(v\), then \[ \mu(M_ v) = cvx(\mu(W \cdot (q+v(q)))). \]