Conical Representations for Direct Limits of Symmetric Spaces

TL;DR

This paper extends the concept of conical representations to infinite-dimensional symmetric spaces, establishing a classification and decomposition framework, and revealing new non-smooth unitary representations absent in finite dimensions.

Contribution

It introduces a classification of all smooth conical representations in the infinite-dimensional setting and identifies new non-smooth unitary representations unique to this context.

Findings

Classified all smooth conical representations that are unitary on the compact side.

Discovered a new class of non-smooth unitary conical representations with no finite-dimensional analogue.

Demonstrated how to decompose these representations into direct integrals of irreducibles.

Abstract

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.