Schur algebras of reductive \(p\)-adic groups. I
This paper concerns representations of the \(p\)-adic group \(G=GL(n,F)\) over an algebraically closed field \(R\) of characteristic not equal to \(p\). It gives a link between the representations of the \(p\)-affine Schur algebra of \(G\) over \(R\) and the smooth \(R\)-representations of \(G\). In the earlier work [\textit{M.-F. Vigneras}, Invent. Math. 144, 177-223 (2001; Zbl 1031.11068)], the author gave the local Langlands \(R\)-correspondence: a parametrization of irreducible \(R\)-representations of \(G\). In this paper, she considers the category of all \(R\)-representations. Let \(\text{B}_{R,\tau}\) be the block parametrized by the parameter \(\tau\). The author considers the unipotent block \(\text{B}_{R,1}\). Let \(I\) be an Iwahori subgroup of \(G\) and let \(\text{ J}_R\) be the annihilator in the Hecke \(R\)-algebra of \(G\) of the \(R\)-representation of \(G\), \(R[I\setminus G]\). The main theorem is the following: There exists an integer \(N\) such that \(\text{ J}^N_R\) annihilates the unipotent block. The category of \(R\)-representations of \(G\) annihilated by \(\text{ J}_R\) is equivalent to the category of modules of the affine Schur algebra of \(G\). The main theorem is interesting because it creates a bridge between the \(p\)-adic groups and the affine quantum groups.
scientific article; zbMATH DE number 1911422
scientific article