Let p be a prime number and G a connected split reductive algebraic group over ℚp such that both G and its dual Gˆ have connected center. Let ρ:Gal(ℚ‾‾p/ℚp)→Gˆ(E) be a continuous group homomorphism where E is a finite extension of ℚp. The aim of the p-adic Langlands program is to associate to (the conjugacy class of) ρ some hypothetical p-adic Banach space(s) Π(ρ) over E endowed with a unitary continuous action of G(ℚp) such that Π(ρ) occurs in some completed cohomology groups when ρ comes from some (pro)modular representation of a global Galois group.

Assume that ρ takes values in a Borel subgroup Bˆ(E)⊂Gˆ(E). I will explain how one can associate to such a (sufficiently generic) ρ a Banach space Π(ρ)ord endowed with a unitary continuous action of G(ℚp) and which is expected to be a closed subrepresentation of Π(ρ), namely its maximal closed subrepresentation where all irreducible constituents are subquotients of unitary continuous principal series. The representation Π(ρ)ord decomposes as Π(ρ)ord=⊕w∈W(ρ)Π(ρ)ordw where W(ρ) is a subset of the Weyl group W. One important point is that its construction is directly inspired by the study of the ``ordinary part'' of the tensor product of the fundamental algebraic representations of Gˆ(E) (composed with ρ).

One can extend the construction of Π(ρ)ord in characteristic p and associate to a (sufficiently generic) ρ‾:Gal(ℚ‾‾p/ℚp)→Bˆ(kE)⊂Gˆ(kE) a smooth representation:

Π(ρ‾)ord=⊕w∈W(ρ‾)Π(ρ‾)ordw

of G(ℚp) over kE where kE is a finite extension of 𝔽p. When G=GLn and ρ‾ comes from some modular Galois representation, I will explain how one can use recent results of Gee and Geraghty on ordinary Serre weights to prove that all GLn(ℚp)-representations Π(ρ‾)ordw really do occur in spaces of automorphic forms modulo p for definite unitary groups which are outer forms of GLn.

The first lecture will be largely introductory, in particular I will recall the situation for G=GL2 and ρ reducible as above. The second lecture will be devoted to the construction of Π(ρ)ord and I will stress the parallel with the restriction to subgroups of Bˆ(E) of the tensor product of the fundamental algebraic representations of Gˆ(E). The last lecture will be devoted to the local-global compatibility result in characteristic p mentioned above.

This is joint work with Florian Herzig.