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 in GL(n,F) and v in 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

Primary: 22E50

Secondary: 22E35

Authors