Parabolic induction and Hecke modules in characteristic \(p\) for \(p\)-adic GL\(_n\)
Let \(F\) be a local non-archimedean field of residual characteristic \(p\), let \(k\) be an algebraic closure of \(\mathbb F_p\), and let \(G:=\text{GL}_n(F)\). The author studies \(k\)-representations of \(G\). A key tool is the functor \(V\mapsto V^{I(1)}\) that associates to a smooth representation \(V\) of \(G\) its subspace of invariant vectors under the pro-\(p\)-Iwahori subgroup \(I(1)\) of \(G\), that is, \(I(1)\) is the pro-\(p\)-radical of the standard Iwahori subgroup \(I\) of \(G\). The space \(V^{I(1)}\) is a module over the \textit{pro-\(p\)-Iwahori-Hecke algebra} \(\mathcal H=\mathcal H(G)\) of compactly supported functions \(f: I(1)\backslash G/I(1)\to\mathbb Z\). The structure of \(\mathcal H\) has been studied by Vignéras, \(\mathcal H\) a finite extension of an affine Hecke algebra \(\mathcal H_{\text{aff}}\) and it admits a Bernstein-type presentation with only positive powers of the cardinal of the residue field of \(F\). The author has obtained previously the classification of the simple modules of \(\mathcal H\) in the case \(n=3\). Three families of \(\mathcal H\)-modules appear: the \textit{regular}, \textit{singular}, and \textit{supersingular} ones. In the article under review the author proves in particular a conjecture stated by Vignéras in 2005, which says that any non-zero simple supersingular module contains a character of \(\mathcal H_{\text{aff}}\). It implies the following numerical Langlands correspondence on the side of \(\mathcal H\)-modules: The number of \(n\)-dimensional simple supersingular \(\mathcal H\)-modules (with fixed action of the uniformizer) is equal to the number of smooth irreducible \(n\)-dimensional mod \(p\) representations of the absolute Galois group of \(F\) (with fixed determinant of a Frobenius). On the other hand, the author investigates the non-supersingular \(\mathcal H\)-modules by defining a process of induction from the pro-\(p\)-Iwahori-Hecke algebra \(\mathcal H(L)\) of a Levi subgroup \(L\) of \(G\), and by relating it to the parabolic induction on the side of the representations of \(G\). Unlike the characteristic zero case, one cannot embed the entire algebra \(\mathcal H(L)\) naturally in \(\mathcal H\) but only a certain subalgebra \(\mathcal H(L^+)\) (the \textit{positive part}) of \(\mathcal H(L)\), which corresponds to the \textit{\(L\)-positive} elements of the extended affine Weyl group of \(L\). Let \(\mathcal M\) be a right \(\mathcal H(L)\)-module with scalar action of the uniformizers. The \(\mathcal H\)-module \textit{induced} from \(\mathcal M\) is defined to be the tensor product \[ \text{Ind}(\mathcal M):=\mathcal M\otimes_{\mathcal H(L^+)}\mathcal H. \] This process of induction defines an exact functor from the category of \(\mathcal H(L)\)-modules with scalar action of the uniformizers into the category of right \(\mathcal H\)-modules. It is used in the article to characterize which simple \(\mathcal H\)-modules are supersingular or not.
mod p representation of Hecke algebras and p-adic groups
scientific article; zbMATH DE number 5809196