Let K be a finite extension of Qp. It is believed that one can attach a smooth Fp -representation of GLn (K) (or a packet of such representations) to a continuous Galois representation Gal(bar Qp /K) ? GLn (Fp) in a natural way, that is called modp Langlands program for GLn (K).
This conjecture is known only for GL2 (Qp) : one of the main difficulties is that there is no classification of such smooth representations of GLn (K) unless K = Qp and n = 2.
However, for a given continuous Galois representation ?0 : Gal(bar Qp/Qp) ? GLn (Fp), one can define a smooth Fp -representation ?0 of GLn (Qp) by a space of mod p automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to ?0 for modp Langlands correspondence in the spirit of Emerton.
The structure of ?0 is very mysterious as a representation of GLn (Qp), but it is conjectured that ?0 determine ?0 .
In this talk, we discuss that ?0 determines ?0 , provided that ?0 is ordinary and generic. More precisely, we prove that the tamely ramified part of ?0 is determined by the Serre weights attached to ?0 , and the wildly ramified part of ?0 is obtained in terms of refined Hecke actions on ?0 .
This is a joint work with Chol Park.