Reversible quaternionic hyperbolic isometries (Q2174518)

This is a study of properties of cetain isometry groups related to \(n\)-dimensional quaternionic hyperbolic spaces. With \({\mathbb H}\) denoting the quaternions, \(V:= {\mathbb H}^{n,1} ={\mathbb H}^{n+1}\) is an \((n + 1)\)-dimensional right vector space over \({\mathbb H}\), equipped with an \({\mathbb H}\)-Hermitian form \(\Phi(z, w) = -\overline{z_0}w_0 + \overline{z_1}w_1 +\ldots + \overline{z_n}w_n\), where \(z = (z_0, z_1,\dots, z_n)\), \(w = (w_0, w_1,\dots, w_n) \in {\mathbb H}^{n+1}\). The matrix representation of \(\Phi\) with respect to the standard basis \({e_0, e_1,\dots, e_n}\) of \({\mathbb H}^{n+1}\) is \(J = \mathrm{diag}(-1, 1,\dots,1)\). The symplectic group of signature \((n, 1)\) is \(\mathrm{Sp}(n, 1) = \{g \in \mathrm{GL}(n + 1, {\mathbb H})\mid ^t\overline{g}Jg = J\}\). Also, let \(\mathrm{Sp}(n) := \{g \in \mathrm{GL}(n, H) \mid ^t\overline{g}g = I_n \}\). The group of isometries of the \(n\)-dimensional hyperbolic space over \({\mathbb H}\), \(\mathrm{PSp}(n,1)\) is \(\mathrm{Sp}(n,1)/Z(\mathrm{Sp}(n,1)\), where \(Z(\mathrm{Sp}(n,1))=\{\pm I_n\}\) is the center of \(\mathrm{Sp}(n,1)\). For any group \(G\), an element \(g \in G\) is called reversible if \(g^{-1} = xgx^{-1}\) for some \(x\in G\). It is called strongly reversible if the \(x\) above is such that \(x^2=1\). This paper's main results are: (i) Every element of \(\mathrm{Sp}(n, 1)\) is reversible; (ii) An element \(g \in \mathrm{Sp}(n)\) is strongly reversible if and only if every eigenvalue class of \(g\) is either \(\pm 1\) or of even multiplicity; (iii) For \(n > 1\), every element in \(\mathrm{Sp}(n)\) can be expressed as a product of four (respectively, five) involutions if \(n\) is even (resectively, \(n\) is odd); (iv) Every element of \(\mathrm{PSp}(n, 1)\) is a product of two involutions (i.e., \(\mathrm{PSp}(n, 1)\) is bireflectional) .