# SCNTD Claus Sorensen

### Abstract: Ordinary representations and local-global compatibility

The elusive p-adic local Langlands correspondence for GL_n(F) is expected to somehow encode an n-dimensional p-adic representation rho of Gal(\bar{F}/F) in a p-adic unitary Banach space representation Pi(\rho) of GL_n(F). Here F/\Q_p is a finite extension. For GL_2(\Q_p) this is all in good shape, but for other groups much remains do be done. A couple of years ago, Breuil and Herzig gave a purely local construction of a representation Pi(\rho)^{ord} when F=\Q_p and rho takes values in a Borel subgroup; it is believed to be the largest sub of Pi(\rho) built from continuous principal series. They also give a precise conjecture as to how their construction intervenes with p-adic modular forms on definite unitary groups, when \rho arises from such. In the talk we will present a slight variant of Pi(\rho)^{ord}, which behaves better with respect to reduction mod p, and whose socle is easier to interpolate in families. At the end we hope to hint at how this variant occurs in spaces of p-adic modular forms, and as a result provide new cases of the Breuil-Herzig conjecture. This is joint work in progress with Przemyslaw Chojecki.