Universal module in characteristic \(l>0\) associated to a character of the Hecke algebra of \(\text{GL}(n)\) over a \(p\)-adic field with \(l\neq p\)
Let \(F\) be a non-archimedean local field with finite residue field and let \(G = \text{GL}(n,F)\). Let \(\chi\) be an unramified character of the diagonal subgroup of \(G\) and let \(\lambda\) denote the character of the Hecke algebra corresponding to \(\chi\) by the Satake map. We then have the principal series representation \(I_{\chi}\) of \(G\) and the universal module (or induced module) \(M_{\lambda}\) associated to \(\lambda\). Moreover, there is a natural map \(M_{\lambda} \rightarrow I_{\chi}\). This has been studied for characters with values in \(\mathbb C\) and representations over \(\mathbb C\). The present paper is concerned with the same situation with \(\mathbb C\) replaced by an algebraically closed field of characteristic \(l > 0\) and different from the residual characteristic of \(F\). It is proved that the \(G\)-modules \(M_{\lambda}\) and \(I_{\chi}\) have the same, finite, Jordan-Hölder sequence, under the assumption that \(l\) does not divide the number of elements of the reduced group \(\overline{G} = \text{GL}(n,\mathbb{F}_q)\). There is also a criterion for \(M_{\lambda} \rightarrow I_{\chi}\) to be an isomorphism, under the assumption that \(l\) does not divide \(q(q - 1)\). The main tool for the proofs is the functor \(V\mapsto V^B\) from representations of \(G\) to modules over the Hecke algebra of \(G\) with respect to an Iwahori subgroup \(B\). There are some inaccuracies in the text (e.g., \(\delta\) defined in 1.1 is the inverse of \(\delta\) used further on; Proposition 1.20 is not in concordance with the definition 1.8). Lemma 1.11(i) is false (the right hand side of the equality must be multiplied by an integral power of \(q\)).
scientific article; zbMATH DE number 1270869
scientific article