Modular representations of \(GL_ 2\) of a local field: The ordinary, unramified case (Q1903497)

Let \(F\) be a local field of residue characteristic \(p\), and let \(G= PGL_2 (F)\). Let \(E\) be an algebraically closed field of characteristic \(p\), and let \(V\) be an \(E\)-vector space on which \(G\) acts with \(V^K\neq 0\). If \(V\) has a trivial central character, the authors prove that there exists an eigenvector in \(V^K\) for an appropriate Hecke operator \(T\) and classify them by the eigenvalues \(\lambda\) of \(T\) to one-dimensional \((\lambda= \pm 1)\), principal series \((\lambda\neq 0,\pm1)\), or supersingular \((\lambda =0)\). They also show that any \(\lambda\) can appear and prove the uniqueness for \(\lambda \neq 0\).