[Paper Review] The SL(2)-type and Base Change

Omer Offen, Eitan Sayag|ArXiv.org|Aug 25, 2008

Advanced Algebra and Geometry7 references3 citations

TL;DR

This paper extends Venkatesh's notion of SL(2)-type from unitarizable to all smooth, irreducible representations of GL_n over a p-adic field using the local Langlands reciprocity map. It proves that base change preserves the SL(2)-type, and as a consequence, Klyachko models (including symplectic distinction) are also preserved under base change, generalizing known results on symplectic distinction to arbitrary finite extensions.

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 base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation πof GL(n) depends only on the SL(2)-type of π. As a consequence we observe that the Klyachko model of πand of its base-change are of the same type.

Motivation & Objective

• To extend the definition of the SL(2)-type from unitarizable to all smooth, irreducible representations of GL_n(F) over a p-adic field F.
• To establish that base change with respect to any finite extension E/F preserves the SL(2)-type of a smooth, irreducible, and unitarizable representation π of GL_n(F).
• To show that the Klyachko model type of π, which depends only on its SL(2)-type, is preserved under base change.
• To re-interpret prior results on Klyachko models—particularly symplectic distinction—using the SL(2)-type framework.
• To demonstrate that the distinction of π by Sp_{2n}(F) is equivalent to the distinction of its base change by Sp_{2n}(E), for any finite extension E/F.

Proposed method

• Use the local Langlands reciprocity map to lift representations of GL_n(F) to Weil-Deligne representations, enabling a natural extension of the SL(2)-type beyond unitarizable representations.
• Define the SL(2)-type of a representation π as the partition associated to the Jordan decomposition of the monodromy operator in the Weil-Deligne representation corresponding to π via the reciprocity map.
• Apply the reciprocity map to define base change as the 'mirror image' of restriction on Weil-Deligne representations: bc_{E/F}(π) corresponds to restricting the Weil-Deligne representation of π to W_E.
• Prove that the SL(2)-type is preserved under base change by showing that the partition P_F(rec_F(π)) equals P_E(rec_E(bc_{E/F}(π))), using compatibility of the reciprocity map with restriction.
• Use Tadic’s classification of the unitary dual to express representations in terms of segments and derive formulas for the SL(2)-type and Klyachko type.
• Leverage the fact that the Klyachko type (r(π), 2k(π)) is determined by the number of odd parts in the SL(2)-type, via r(π) = odd(𝒱(π)), to show invariance under base change.

Experimental results

Research questions

• RQ1 Can the SL(2)-type, originally defined only for unitarizable representations, be naturally extended to all smooth, irreducible representations of GL_n(F)?
• RQ2 Does base change with respect to a finite extension E/F preserve the SL(2)-type of a smooth, irreducible, and unitarizable representation π of GL_n(F)?
• RQ3 To what extent do Klyachko models—particularly symplectic distinction—depend on the SL(2)-type of a representation?
• RQ4 Is the distinction of a representation π by Sp_{2n}(F) equivalent to the distinction of its base change bc_{E/F}(π) by Sp_{2n}(E)?
• RQ5 Can prior results on Klyachko models be reinterpreted and generalized using the SL(2)-type framework?

Key findings

• The SL(2)-type of any smooth, irreducible representation π of GL_n(F) is extended via the local Langlands reciprocity map, assigning to π the partition associated with the monodromy operator in its Weil-Deligne realization.
• For any finite extension E/F, the base change bc_{E/F}(π) of a smooth, irreducible, and unitarizable representation π of GL_n(F) has the same SL(2)-type as π.
• The Klyachko type (r(π), 2k(π)) of π is preserved under base change, with r(π) = odd(𝒱(π)) and 𝒱(π) being the SL(2)-type.
• A smooth, irreducible, and unitarizable representation π of GL_{2n}(F) is Sp_{2n}(F)-distinguished if and only if its base change bc_{E/F}(π) is Sp_{2n}(E)-distinguished, for any finite extension E/F.
• The proof relies on showing that the partition of the monodromy operator is invariant under restriction of Weil-Deligne representations, which implies invariance of the SL(2)-type under base change.
• The result generalizes earlier results on symplectic distinction, showing that such distinction is preserved under base change due to the invariance of the underlying SL(2)-type.