Kirillov models and integral structures in \(p\)-adic smooth representations of \(\mathrm{GL}_2(F)\) (Q441400)

Let \(F\) denote a local field of residual characteristic \(p\) and let \(C\) be a fixed algebraic closure of the field \(\mathbb{Q}_{p}\) or its completion. Further, let \(\mathcal{O}_{C}\) be the ring of integers in \(C\). The authors study irreducible smooth representations of the group \(\mathrm{GL}_{2}(F)\), realized over \(C\), where a smooth representation \((V, \rho)\) of the group \(G\) over \(C\) is a \(C\)-vector space \(V\) equipped with a homomorphism \(\rho : G \rightarrow \mathrm{GL}(V)\), for which the stabilizer of every \(v \in V\) is open in \(G\).NEWLINENEWLINEStudying the Kirillov model of an irreducible smooth representation \(\rho\) of \(\mathrm{GL}_{2}(F)\), the authors obtain a necessary and sufficient criterion for the existence of an integral structure in \((V, \rho)\), where the integral structure in \((V, \rho)\) is an \(\mathcal{O}_{C}\)-submodule \(V_{0}\) of \(V\), stable under \(\rho(\mathrm{GL}_{2}(F))\), spanning \(V\) over \(C\), which contains no \(C\)-line.NEWLINENEWLINEAlso, the obtained criterion has been applied to tamely ramified principal series of \(\mathrm{GL}_{2}(F)\).