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Higher \(\mathcal{L} \)-invariants for \(\operatorname{GL}_3 (\mathbb{Q}_p)\) and local-global compatibility. (English) Zbl 1469.11454

Summary: Let \(\rho_p\) be a 3-dimensional semi-stable representation of \(\mathrm{Gal} (\overline{\mathbb{Q}_p} / \mathbb{Q}_p)\) with Hodge-Tate weights \((0, 1, 2)\) (up to shift) and such that \(N^2 \neq 0\) on \(D_{\mathrm{st}} (\rho_p)\). When \(\rho_p\) comes from an automorphic representation \(\pi\) of \(G(\mathbb{A}_{F^+})\) (for a unitary group \(G\) over a totally real field \(F^+\) which is compact at infinite places and \(\mathrm{GL}_3\) at \(p\)-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of \(p\)-adic automorphic forms on \(G(\mathbb{A}^{\infty}_{F^+})\) of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of \(\mathrm{GL}_3 (\mathbb{Q}_p)\) of the form considered in [the first author, Am. J. Math. 141, No. 3, 611–703 (2019; Zbl 1469.11154)] which only depends on and completely determines \(\rho_p\).

MSC:

11S23 Integral representations
22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 1469.11154
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