We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using a certain infinite-dimensional version of Weyl’s Unitary Trick, we show that there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.
Publication Source (Journal or Book title)
Dawson, M., & Ólafsson, G. (2017). Conical representations for direct limits of symmetric spaces. Mathematische Zeitschrift, 286 (3-4), 1375-1419. https://doi.org/10.1007/s00209-016-1806-7