Distinguished principal series representations of GL(n) over a p-adic field

Nadir Matringe

Vol. 239 (2009), No. 1, 53–63

DOI: 10.2140/pjm.2009.239.53

Abstract

Let K∕F be a quadratic extension of p-adic fields, and χ a character of F^∗. A representation (π,V ) of GL(n,K) is said to be χ-distinguished if there is a nonzero linear form L on V such that L(π(h)v) = χ ∘ det(h)L(v) for h ∈ GL(n,F) and v ∈ V . We classify here distinguished principal series representations of GL(n,K). Call η_K∕F the nontrivial character of F^∗ that is trivial on the norms of K^∗, and σ the nontrivial element of the Galois group of K over F. A conjecture attributed to Jacquet asserts that admissible irreducible representations π of GL(n,K) are such that the smooth dual π^∨ is isomorphic to π ∘ σ if and only if it is 1-distinguished or η_K∕F-distinguished. Our classification gives a counterexample for n ≥ 3.

Keywords

distinguished representations, Jacquet’s conjecture

Mathematical Subject Classification 2000

Primary: 22E50

Secondary: 22E35

Milestones

Received: 2 August 2008

Accepted: 1 October 2008

Published: 1 January 2009