The 𝑆𝐿(2)-type and base change

Offen, Omer; Sayag, Eitan

doi:10.1090/S1088-4165-09-00353-7

The $SL(2)$-type and base change
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by Omer Offen and Eitan Sayag

Represent. Theory 13 (2009), 228-235

DOI: https://doi.org/10.1090/S1088-4165-09-00353-7

Published electronically: June 23, 2009

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Abstract:

The $SL(2)$-type of any smooth, irreducible and unitarizable representation of $GL_n$ over a $p$-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the $SL(2)$-type of a representation is preserved under the base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation $\pi$ of $GL_n$ depends only on the $SL(2)$-type of $\pi$. As a consequence we observe that the Klyachko model of $\pi$ and of its base change are of the same type.

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