Stability Theorem for \( \mathbb {Z}_2^n \) -Lie Supergroups

We show that every pre-representation of a \( \mathbb {Z}_{2}^{n} \) -Lie supergroup has a unique extension to a unitary representation. By a \( \mathbb {Z}_{2}^{n} \) -Lie supergroup we mean a Harish-Chandra pair \( (G_0, \mathfrak {g}_{\mathbb {C}}) \) where \( G_0 \) is a common finite-dimensional Lie Group and \( \mathfrak {g}_{\mathbb {C}} \) is a \( \mathbb {Z}_{2}^{n} \) -graded Lie superalgebra such that there exists an action \(Ad:G_0 \times \mathfrak {g_{\mathbb {C}}} \rightarrow \mathfrak {g_{\mathbb {C}}} \) preserves the \(\mathbb {Z}_{2}^{n}\) -grading and \( \mathrm {Ad}|_{\mathfrak {g_{0}}}:G_{0} \times \mathfrak {g_{0}} \rightarrow \mathfrak {g_{0}} \) is the adjoint action of \(G_0\) on \(\mathfrak {g_{0}} \cong Lie(G_{0})\) .