Isomorphy classes of involutions of \(\mathrm{SP}(2n,k)\), \(n>2\). (Q2791843)

Let \(G\) be an algebraic group defined over a field \(k\) of characteristic not \(2\). Let \(G_k\) be the set of \(k\)-rational points. Let \(\text{Inn}_A\) with \(A\in G\) be an automorphism of \(G_k\) such that \(\text{Inn}_A(G_k)=G_k\). If \(\text{Inn}_A^2\) is the identity but \(\text{Inn}_A\) is not, then \(\text{Inn}_A\) is called an inner involution of \(G_k\).NEWLINENEWLINE The central result that the authors prove is the following theorem: Suppose \(A\in\text{GL}(2n,\overline k)\), \(\overline G=\text{SP}(2n,\overline k)\) and \(G=\text{SP}(2n,k)\). Then 1. The inner automorphism \(\text{Inn}_A\) keeps \(\text{SP}(2n,\overline k)\) invariant if and only if \(A=pM\) for some \(p\in\overline k\) and \(M\in\text{SP}(2n,\overline k)\). 2. If \(A\in\text{SP}(2n,\overline k)\), then \(\text{Inn}_A\) keeps \(\text{SP}(2n,k)\) invariant if and only if \(A\in\text{SP}(2n,k(\sqrt\alpha))\) where each entry of \(A\) is a \(k\)-multiple of \(\sqrt\alpha\) for some \(\alpha\in k\).NEWLINENEWLINE They describe the four possible types of involutions given by the four combinations of: \(A\in\text{SP}(2n,k)\) or \(A\in\text{SP}(2n,k[\sqrt\alpha])\setminus\text{SP}(2n,k)\), and \(A^2=I\) or \(A^2=-I\). -- This classifies the involutions of \(\text{SP}(2n,k)\) for an algebraically closed field \(k\), the real numbers, or a finite field of characteristic not 2.