The \(\ell \)-modular local Langlands correspondence and local constants. (English) Zbl 1490.22011
The \(l\)-adic local Langlands correspondence has an important property that Rankin-Selberg local constants of a pair of irreducible \(l\)-adic representations of \(\mathrm{GL}_n(F)\) and \(\mathrm{GL}_m(F)\), and the Artin-Deligne local constants of the corresponding tensor product of Deligne representations of \(W_F\) are equal.
Furthermore, Vignéras developed the theory of \(l\)-modular representations of reductive \(p\)-adic group with the culmination in the \(l\)-modular local Langlands correspondence for general linear groups. In contrast to the \(l\)-adic correspondence, the generalized Rankin-Selberg local constants for \(l\)-modular generic representations of general linear groups are not equal to the factors of Artin-Deligne.
In this paper the authors classify the \(l\)-modular indecomposable \(W_F\)-semisimple Deligne representations, extend the definitions of Artin-Deligne factors to this setting, and define an \(l\)-modular local Langlands correspondence where in the generic case, the Rankin-Selberg factors of representations on one side equal the Artin-Deligne factors of the corresponding representations on the other.
Furthermore, Vignéras developed the theory of \(l\)-modular representations of reductive \(p\)-adic group with the culmination in the \(l\)-modular local Langlands correspondence for general linear groups. In contrast to the \(l\)-adic correspondence, the generalized Rankin-Selberg local constants for \(l\)-modular generic representations of general linear groups are not equal to the factors of Artin-Deligne.
In this paper the authors classify the \(l\)-modular indecomposable \(W_F\)-semisimple Deligne representations, extend the definitions of Artin-Deligne factors to this setting, and define an \(l\)-modular local Langlands correspondence where in the generic case, the Rankin-Selberg factors of representations on one side equal the Artin-Deligne factors of the corresponding representations on the other.
Reviewer: Barbara Bošnjak (Zagreb)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
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