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Towards the finite slope part for \(\mathrm{GL}_n\). (English) Zbl 1476.11086

Let \(F/\mathbb{Q}\) be a quadratic imaginary extension in which the prime \(p\) splits (in the paper, \(\mathbb{Q}\) may be an arbitrary totally real number field \(F^+\)); choose a place \(\tilde{v}\) of \(F\) above \(p\). By considering completed cohomology for an appropriate definite unitary group and tame level one can associate to each irreducible automorphic Galois representation \(\rho : G_F \to \mathrm{GL}_n(E)\), with \(E/\mathbb{Q}_p\) a finite extension, a continuous representation \(\Pi^{\mathrm{glob}}(\rho)\) of \(\mathrm{GL}_n(\mathbb{Q}_p)\) and its locally \(\mathbb{Q}_p\)-analytic subrepresentation \(\Pi^{\mathrm{glob}}_{\mathrm{an}}(\rho)\). Let \(\rho_p = \rho\mid_{G_{\mathbb{Q}_p}}\). One expects that \(\Pi^{\mathrm{glob}}(\rho)\) depends only on \(\rho_p\).
When \(\rho_p\) is crystalline, the first author [Math. Ann. 361, No. 3–4, 741–785 (2015; Zbl 1378.11060)] has made a conjecture predicting the socle of \(\Pi^{\mathrm{glob}}_{\mathrm{an}}(\rho)\) in terms of \(\rho_p\). This was proved by him C. Breuil et al. [Publ. Math., Inst. Hautes Étud. Sci. 130, 299–412 (2019; Zbl 1454.14120)] under mild hypotheses. The current article defines a (non-semisimple) admissible locally analytic representation \(\Pi^{\mathrm{fs}}(\rho_p)\) of \(\mathrm{GL}_n(\mathbb{Q}_p)\) and prove that, if the socle conjecture holds – so, unconditionally under mild hypotheses – then \(\Pi^{\mathrm{fs}}(\rho_p)\) is a subrepresentation of \(\Pi^{\mathrm{glob}}_{\mathrm{an}}(\rho)\). In the introduction, the authors write: “Going beyond this subrepresentation [\(\Pi^{\mathrm{fs}}(\rho_p)\)] will almost certainly require (seriously) new ideas.”
Finally, in the ordinary case, when \(\rho\) is upper triangular and if \(p\) is split in \(F^+\), the authors [Duke Math. J. 164, No. 7, 1271–1352 (2015; Zbl 1321.22019)] have constructed a (non-semisimple) continuous representation \(\Pi^{\mathrm{ord}}(\rho_p)\) that should conjecturally be a subrepresentation of \(\Pi^{\mathrm{glob}}(\rho)\). In the present paper they prove this using their results on \(\Pi^{\mathrm{fs}}(\rho_p)\) (again, when the socle conjecture holds).

MSC:

11F80 Galois representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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