Oscillator semigroup over a non-archimedean field
Let \(\mathbf F\) be a non-Archimedean local field and \(\mathbf O\) the ring of integers in \(\mathbf F\). For each \(n \in \mathbb{N}\) consider a \(2n\)-dimensional symplectic \(\mathbf F\)-vector space \(V(n)\) and the \(\mathbf O\)-span \(r(n)\) of a symplectic basis. Then one can normalize a Haar measure on \(V(n)\) such that \(r(n)\) has measure one. Let \({\mathcal C} (n)\) be the subspace of \(L^2(V (n))\) consisting of functions \(\varphi\) satisfying \(\varphi (v) = \chi(B(u,v)) \varphi(u)\) for \(u -v \in r(n)\), where \(B\) is the symplectic form and \(\chi\) is a fixed additive character on \(\mathbf F\) vanishing on \(\mathbf O\). Then the symplectic group \(\text{Sp}(n) = \text{Sp}(n, {\mathbf F})\) admits a projective representation on \({\mathcal C} (n)\) called the oscillator representation which is given by explicit integral operators. Extending symplectic bases one can view \(\text{Sp}(n - 1, {\mathbf F})\) as a subgroup of \(\text{Sp}(n, \mathbf F)\) and obtains invariant embeddings \({\mathcal C}(n-1) \to {\mathcal C}(n)\). Taking inductive limits one finds an infinite dimensional symplectic group \(\text{Sp}(\infty, \mathbf F)\) acting projectively on a space \({\mathcal C} (\infty)\). The author gives a detailed description of the weak closure of the image of \(\text{Sp}(\infty, {\mathbf F})\) under this representation. In particular he shows that it can be viewed as an inductive limit of semigroups which are constructed in analogy to Howe's oscillator semigroups.
scientific article; zbMATH DE number 764014