# Abstract Florian Herzig

**Florian Herzig**will speak on

**Towards the "finite slope part" for p-adic GL_n
**

**Abstract:** Suppose that rho is an irreducible automorphic n-dimensional global p-adic Galois representation that is upper-triangular locally at p. In previous work with Breuil we constructed a unitary representation of GL_n(Q_p) on a p-adic Banach space (depending only on rho locally at p) that is an extension of finitely many principal series, and we conjectured that this representation occurs globally in a space of p-adic automorphic forms cut out by rho. We prove many new cases of this conjecture, assuming that rho is moreover crystalline. In addition, when rho is just generic crystabelian and no longer necessarily upper-triangular, we construct a locally analytic representation of finite length (built out of locally analytic principal series) and again prove in many cases that it occurs globally. This is joint work with C. Breuil.