In this paper, we characterize unitary representations of $\pi:=\piS$ whose generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions $u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space $\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.

Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define two sequences of algebras with involution $\mathcal{R}_p$, $\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that $T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$ are $*$-asymptotically equivalent for suitable $p,q$.

A $\mathbb{Z}_2$-action with ``maximal number of isolated fixed points'' ({\it i.e.}, with only isolated fixed points such that $\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$ on a $3$-dimensional, closed manifold determines a binary self-dual code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines the cohomology algebra $H^*(M;k)$ and the equivariant cohomology $H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual codes one gets information about the cohomology type of $3$-manifolds which admit involutions with maximal number of isolated fixed points. In particular, ``most'' cohomology types of closed $3$-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, {\it e.g.}, one gets that ``most'' cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do not admit a non-trivial involution.