Distinguished principal series representations for GLn over a p− adic field

For K/F a quadratic extension of p-adic fields, let σ be the conjugation relative to this extension, and ηK/F be the character of F ∗ with kernel norms of K. If π is a smooth irreducible representation of GL(n,K), and χ a character of F , the dimension of the space of linear forms on its space, which transform by χ under GL(n, F ) (with respect to the action [(L, g) 7→ L ◦ π(g)]) , is known to be at most one (Proposition 11, [F]). One says that π is χ-distinguished if this dimension is one, and says that π is distinguished if it is 1-distinguished. In this article, we give a description of distinguished principal series representations of GL(n,K). The result (Theorem 3.2) is that the the irreducible distinguished representations of the principal series of GL(n,K) are (up to isomorphism) those unitarily induced from a character χ = (χ1, ..., χn) of the maximal torus of diagonal matrices, such that there exists r ≤ n/2, for which χi+1 = χi −1 for i = 1, 3, .., 2r−1, and χi|F∗ = 1 for i > 2r. For the quadratic extension C/R, it is known (cf.[P]) that the analogous result is true for tempered representations. For n ≥ 3, this gives a counter-example (Corollary 3.1) to a conjecture of Jacquet (Conjecture 1 in [A]). This conjecture states that an irreducible representation π of GL(n,K) with central character trivial on F ∗ is isomorphic to π̌ if and only if it is distinguished or ηK/F -distinguished (where ηK/F is the character of order 2 of F , attached by local class field theory to the extension K/F ). For discrete series representations, the conjecture is verified, it was proved in [K]. Unitary irreducible distinguished principal series representations of GL(2,K) were described in [H], and the general case of distinguished irreducible principal series representations of GL(2,K) was treated in [F-H]. We use this occasion to give a different proof of the result for GL(2,K) than the one in [F-H]. To do this, in Theorems 4.1 and 4.2, we extend a criterion of Hakim (th.4.1, [H]) characterising smooth unitary irreducible distinguished representations of GL(2,K) in terms of γ factors at 1/2, to all smooth irreducible distinguished representations of GL(2,K).