Abstract
For a fairly general reductive group \({G_{/\mathbb{Q}_p}}\), we explicitly compute the space of locally algebraic vectors in the Breuil–Herzig construction \({\Pi(\rho)^{ord}}\), for a potentially semistable Borel-valued representation \({\rho}\) of \({Gal(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)}\). The point being we deal with the whole representation, not just its socle—and we go beyond \({GL_n(\mathbb{Q}_p)}\). In the case of \({GL_2(\mathbb{Q}_p)}\), this relation is one of the key properties of the \({p}\)-adic local Langlands correspondence. We give an application to \({p}\)-adic local-global compatibility for \({\Pi(\rho)^{ord}}\) for modular representations, but with no indecomposability assumptions.
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Gao, H., Sorensen, C. Locally algebraic vectors in the Breuil–Herzig ordinary part. manuscripta math. 151, 113–131 (2016). https://doi.org/10.1007/s00229-016-0831-5
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DOI: https://doi.org/10.1007/s00229-016-0831-5