Notes on MVW-extensions (Q2900292)

Let \(k\) be a field of characteristic zero. It is well-known that, for any \(x\in \operatorname{GL}(n,k)\), \(x\) and its transpose \(x^t\) are conjugate. This article presents a unified generalization of this fact for a wide range of classical groups and their extensions. Let \(A\) be a \(k\)-algebra and \(\tau\) a \(k\)-automorphism of \(A\) of order dividing two. Let \(\varepsilon=\pm 1\), and let \(E\) be a finitely generated \(A\)-module with a non-degenerate Hermitian or skew-Hermitian (depending on the sign of \(\epsilon\)) form \(E\times E\rightarrow A\). Such \(E\) are referred to as \(\epsilon\)-Hermitian \(A\)-modules. Let \(U(E)\) be the group of \(A\)-module automorphism of \(E\) preserving the form. If \(A\) is simple then, depending on \(\varepsilon\) and the action of \(\tau\) on \(A\), \(U(E)\) can be an orthogonal, symplectic, unitary, or general linear group. In [\textit{C. Moeglin} et al., Howe correspondences on a \(p\)-adic field. Berlin: Springer (1987; Zbl 0642.22002)], \(U(E)\) was extended to a group \(\breve U(E)\) which contains it with index two. Moreover, it is shown that any \(x\in U(E)\) is conjugate to \(x^{-1}\) via an element of \({\breve U}(E)\setminus U(E)\).NEWLINENEWLINEThe author of the present article gives a uniform proof of this result and analogous results for special orthogonal groups, Jacobi groups, and metaplectic groups involving \(\mathfrak{sl}_2\)-actions on \(E\) and Harish-Chandra descent. Moreover, when \(k\) is a local field, he gives a series of related analytic statements concerning invariant distributions on these groups. In particular, for a group \(G\) of one of the above types and an appropriate subgroup \(H\subset G\) of one of these types, if \(\breve g\not\in H\) belongs to the extended group \(\breve H\), then \(f({\operatorname{Ad}({\breve g}} )(x)) = f(x^{-1})\) for distributions \(f\) on \(G\) invariant under the adjoint action of \(H\). Such a result implies that \((G,H)\) is a multiplicity-one pair. This phenomenon has been studied extensively by the present author and many others. When \(G=H\) and \(k\) is non-Archimedean, this result leads to a concrete realization of the contragredient of an irreducible smooth representation \(\pi\) of \(G\) on the underlying space of \(\pi\).NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].