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On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$

  Published:2007-10-01
 Printed: Oct 2007
  • A. Raghuram
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Abstract

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we describe its restriction to the diagonal subgroup $\mathcal{D}^* \times \mathcal{D}^*$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi$. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if $\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic case.
MSC Classifications: 22E50, 22E35, 11S37 show english descriptions Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Analysis on $p$-adic Lie groups
Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]
22E35 - Analysis on $p$-adic Lie groups
11S37 - Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
 

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