Multiplicity theorems modulo p for GL2(Qp)

• Published in: Mathematische Zeitschrift 2014-02, Vol.276 (1-2), p.421-456
• Main Author: Morra, Stefano
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-013-1207-0

Let F be a non-archimedean local field, π an admissible irreducible GL 2 ( F ) -representation with complex coefficients. For a quadratic extension L / F and an L × -character χ a classical result of Tunnell and Saito establish a precise connection between the dimension of the Hom-space Hom L × ( π | L × , χ ) and the normalized local factor of the pair ( π , χ ) . The study of analogous Hom-spaces for complex valued representations has recently been generalized to GL n in Aizenbud et al. (Ann Math 172:1407–1434, 2010 ) and their connections with local factors have been established by work of Moeglin and Waldspurger (La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: le cas général, preprint. Available at http://www.math.jussieu.fr/moeglin/gp.pdf , 2009 ). In this paper we approach the analogous problem in the context of the p -modular Langlands correspondence for GL 2 ( Q p ) . We describe the restriction to Cartan subgroups of an irreducible p -modular representation π of GL 2 ( Q p ) and deduce generalized multiplicity results on the dimension of the Ext-spaces Ext O L × i ( π | O L × , χ ) where O L × is the ring of integers of a quadratic extension of Q p and χ a smooth character of O L × .