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Modular representations of $$GL_ 2$$ of a local field: The ordinary, unramified case. (English) Zbl 0841.11026
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 $$\lambda \neq 0$$.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields 11F85 $$p$$-adic theory, local fields
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