On the Restriction to ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$ of Representations of $p$ -adic $G{{L}_{2}}(\mathcal{D})$

Raghuram, A.

Canadian journal of mathematics, Tome 59 (2007) no. 5, pp. 1050-1068

Résumé

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi $ of $G{{L}_{2}}(\mathcal{D})$ , 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.

Détail

DOI : 10.4153/CJM-2007-045-8

Mots-clés : 22E50, 22E35, 11S37

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