  We extend the definition of conical representations for Riemannian symmetric
spaces to a certain class of infinite-dimensional Riemannian symmetric spaces.
Using an infinite-dimensional version of Weyl's Unitary Trick, 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.
