Let \(S\) be a local Lie subgroup of a Lie group \(G\) with an involutive automorphism \(\sigma\) of \(G\) such that \(\sigma(S)=S\). Suppose that \(S\) acts as a local transformation group on a manifold \(X\). One defines a local multiplier action, \(\tau(g)f(x)=m(x,g)f(g^{-1}x)\), of \(S\) on the space \({\mathcal D}(X)\) of \(C^ \infty\) functions on \(X\) with compact support. Let \(\theta\) be a positive semi-definite Hermitian form on \({\mathcal D}(X)\times{\mathcal D}(X)\) such that \(\theta(\tau(g)f,h)=\theta(f,\tau(\sigma(g^{-1})h))\). Under additional assumptions, one proves that \(\tau\) gives rise to a local representation \(\rho\) of \(S\) on the Hilbert subspace \({\mathcal H}(\theta)\subset{\mathcal D}'(X)\) associated to \(\theta\). The Lie algebra \({\mathcal L}\) of \(G\) decomposes into the eigenspaces of \(\sigma\), \({\mathcal L}={\mathcal L}_ +\oplus{\mathcal L}_ -\). Let \({\mathcal L}^*={\mathcal L}_ +\oplus i{\mathcal L}_ -\), and let \(G^*\) be the connected and simply connected Lie group with \({\mathcal L}^*\) as Lie algebra. One proves that the local representation \(\rho\) gives rise to a unitary representation of \(G^*\) on \({\mathcal H}(\theta)\). Several concrete examples are given.