Toward symmetric spaces of affine Kac-Moody type

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces (G) over cap/(K) over cap where (G) over cap is an affine Kac-Moody group and (K) over cap the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1: 1 correspondence with hyperpolar actions on compact simple Lie groups.