We prove that any polar action on a separable Hilbert space by a connected Hilbert Lie group does not have exceptional orbits. This generalizes a result of Berndt, Console and Olmos in the finite dimensional Euclidean case. As an application, we give an alternative proof of the fact that any hyperpolar action on a simply connected compact Riemannian symmetric space by a connected Lie group does not have exceptional orbits.

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.2010 Mathematics Subject Classification. 53C40.

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.2010 Mathematics Subject Classification. 53C40.

Let G be a connected compact Lie group. We study the heat operator of a transversally elliptic operator. After we review the spectral properties of a transversally elliptic operator, we define the character, that is a distribution on G generalizing the trace of the heat operator to the G-equivariant case. The main theorem of this paper gives the estimate of f α (t), which essentially determines the convergence of the character.

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