Explore advanced mathematical concepts in this research lecture focusing on finite field extensions and parabolic induction functors. Delve into the study of split connected reductive groups over rings of integers, examining how the t-exact parabolic induction functor admits a left adjoint. Learn about the behavior of the L(U,-) functor on algebraic weights that are p-small, with particular attention to cases where F is unramified. Understand the conditions under which L(U,ind_{G(O_F)}^G(L(λ))) splits completely in D(T), based on joint research with Karol Koziol. Presented by Cédric Pépin from Université Sorbonne Paris-Nord, this mathematical investigation contributes to the understanding of derived categories and smooth representations.