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Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$.

Let K/Q p be a finite extension with ring of integers o, let G be a connected reductive split Q p -group of Borel subgroup P = T N and let α be a simple root of T in N . We associate to a finitely generated module D over the Fontaine ring over o endowed with a semilinearétale action of the monoid T + (acting on the Fontaine ring via α), a G(Q p )equivariant sheaf of o-modules on the compact space G(Q p )/P (Q p ). Our construction generalizes the representation D ⊠ P 1 of GL(2, Q p ) associated by Colmez to a (ϕ, Γ)module D endowed with a character of Q * p .

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