Functional Hecke algebras and simple Bernstein blocks of a p-adic GL_n in non-defining characteristic

2014/07/17 by Guiraud, David-Alexandre
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1407.4595

Abstract

Let G_n=GL_n(F), where F is a non-archimedean local field with residue characteristic p and where n=2k is even. In this article, we investigate a question occurring in the decomposition of the category of ℓ-modular smooth representations of G_n into Bernstein blocks (where ℓ≠ p). The easiest block not investigated in \citeguiraud is the one defined by the standard parabolic subgroup with Levi factor M=\GL_k(F) × \GL_k(F) and by an M-representation of the form π₀ \boxtimes π₀ with π₀ a supercuspidal \GL_k(F)-representation. This block is Morita equivalent to a Hecke algebra which we can describe as a twisted tensor product of a finite Hecke algebra (i. e. a Hecke algebra occurring in the representation theory of the finite group \GL_k(p^α) in non-defining characteristic ℓ) and the group ring of ℤ². This enables us to describe how a conjectured connection between finite Hecke algebras (which is similar to a connection postulated by Broué in \citeBroue) would lead to an equivalence between the described block and the unipotent block of GL₂(F^k), where F^k is the unramified extension of degree k over F.

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