On the integrality of locally algebraic representations of \(\mathrm{GL}_2(D)\). (English) Zbl 07786320
In [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 775–839 (2006; Zbl 1117.22008)], M. Emerton provided necessary conditions for the existence of integral structures in terms of the exponents of Jacquet modules of locally algebraic representations of the group of rational points of a connected reductive group over a non-archimedean local field.
Let \(F\) stand for a non-archimedean local field, and let \(G= \mathrm{GL}_2(D)\), where \(D\) stands for a central \(F\)-divison algebra. In the paper under the review, the authors prove that the integrality conditions of Emerton are also sufficient for the existence of an integral structure in a smooth tamely ramified principal series representation \(\mathrm{Ind}_B^{G}(\tau_1 \otimes \tau_2)\) of \(G\), where \(B\) stands for the minimal parabolic subgroup of \(G\).
Let \(F\) stand for a non-archimedean local field, and let \(G= \mathrm{GL}_2(D)\), where \(D\) stands for a central \(F\)-divison algebra. In the paper under the review, the authors prove that the integrality conditions of Emerton are also sufficient for the existence of an integral structure in a smooth tamely ramified principal series representation \(\mathrm{Ind}_B^{G}(\tau_1 \otimes \tau_2)\) of \(G\), where \(B\) stands for the minimal parabolic subgroup of \(G\).
Reviewer: Ivan Matić (Osijek)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Keywords:
locally algebraic representations; diagrams; Emerton’s locally analytic Jacquet modules; Speh representationsCitations:
Zbl 1117.22008References:
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