Reflection Positivity pp 69-78 | Cite as

# Integration of Lie Algebra Representations

## Abstract

A central problem in the context of reflection positive representations of a symmetric Lie group \((G,\tau )\) on a reflection positive Hilbert space \((\mathscr {E},\mathscr {E}_+,\theta )\) is to construct on the associated Hilbert space \(\widehat{\mathscr {E}}\) a unitary representations of the 1-connected Lie group \(G^c\) with Lie algebra \({\mathfrak {g}}^c = {\mathfrak {h}}+ i {\mathfrak {q}}\). As we have seen in Remark 3.3.9, the main point is to “integrate” a unitary representation of the Lie algebra \({\mathfrak {g}}^c\) on a pre-Hilbert space. In general this problem need not have a solution, but we shall see below that in the reflection positive contexts, where the Hilbert spaces are mostly constructed from *G*-invariant positive definite kernels or positive definite *G*-invariant distributions, there are natural assumptions that apply in all cases that we consider.