Alternating forms and self-adjoint operators
The authors study the action of the sympletic group over a field \(k\) on the set of alternating forms on \(k^{2n},\; n\geq 1.\) They prove that if there are two alternating forms on a finite-dimensional vector space \(V\), at least one of which is non-degenerate, the space \(V\) has a decomposition \(V=X_1\otimes X_2\) such that each subspace \(X_i\) is isotropic with respect to each of the forms. It is obtained by interpretation of the action on pairs of forms in terms of the conjugation action on self-adjoint operators. As a consequence they get an explicit bijection between double cosets of Sp\(_{2n}(k)\) in GL\(_{2n}(k)\) and conjugation classes in GL\(_{2n}(k),\) which implies that (GL\(_{2n}(k),\text{Sp}_{2n}(k))\) is a Gelfand pair if \(k\) is a finite field. Among other results of the paper it is worth mentionig determination of the smallest base size of the action of PGL\(_{2n}(k)\) on the set \(X_{2n}(k)\) of invertible skew-symmetric matrices modulo scalars, where the action is given by \(g\cdot x=gxg^t\), and description of the intersection Sp\((J)\cap \text{Sp}(K)\) of two symplectic groups, where \(J\) and \(K\) are skew-symmetric \(2n\times 2n\) matrices.
scientific article; zbMATH DE number 5129722