Appears in collection : Journée de géométrie arithmétique en l’honneur de Michel Gros

Let $F/\mathbb{Q}_p$ be a finite field extension with ring of integers $\mathcal{O}_F$ and residue field $k_F$. Let $\mathbf{G}$ be a split connected reductive group over $\mathcal{O}_F$, and $D(G)$ be the derived category of smooth representations of $G:=\mathbf{G}(F)$ over a fixed field extension of $k_F$. For a Borel $\mathbf{B}=\mathbf{T}\mathbf{U}\subset\mathbf{G}$, the $t$-exact parabolic induction functor ${\rm Ind}_B^G:D(T)\rightarrow D(G)$ admits a left adjoint $L(U,-)$, as proved by Heyer. We study the functor $L(U,-)$ on algebraic weights $L(\lambda)$ which are $p$-small. When $F$ is unramified, we show that $L(U,ind_{G(\mathcal{O}_F)}^G(L(\lambda)))\in D(T)$ splits completely (under some mild modular assumptions). This is joint work with Karol Koziol.