Some Remarks on Representations of Quaternion Division Algebras

For the quaternion division algebra D over a non-Archimedean local field k, and π an irreducible finite dimensional representation of D×, say with trivial central character, we prove the existence of a quadratic extension K of k such that the trivial character of K× appears in π, as well as the existence of a quadratic extension L of k such that the trivial character of L× does not appear in π. Consequences for theta lifts, and variation of local root numbers under quadratic twists, are made. The aim of this paper is to make some remarks on representations of D× restricted to tori where D is a quaternion division algebra over a non-Archimedean local field k, and deduce some consequences for variation of local root numbers under quadratic twists using theorems of Saito and Tunnell. By the Jacquet-Langlands correspondence, these remarks can also be made for discrete series representations of GL2(k). Our first result, motivated by some applications to theta correspondence to which we come to later, should be contrasted with the situation for SU2(R) where for irreducible representations π1 and π2 of SU2(R) with the same central characters, any character of the maximal torus S ↪→ SU2(R) that appears in π1 also appears in π2 if dim π1 ≤ dim π2. We prove that this does not happen for non-Archimedean local fields. It may be of interest to investigate a similar question for a general compact connected Lie group G with a maximal torus T : what are irreducible representations π1 and π2 of G such that each character of T appearing in π1 also appears in π2 (counted with multiplicity, so that π1|T ⊂ π2|T ). Proposition 0.1. Let π1 and π2 be two irreducible representations of D × where D is the quaternion division algebra over a non-Archimedean local field k of odd residue characteristic with the same central characters. Assume that dim(π1) ≤ dim(π2). Then there exists a quadratic extension K of k, and a character χ : K× → C× which appears in π1 but not in π2 unless π1 = π2. The proof of this proposition will be based on a few preliminary results. The next proposition is about certain multiplicity one results for which we refer to [P]. Proposition 0.2. (1) Let π be an irreducible representation of D×, and K a quadratic extension of k. Then any character χ : K× → C× appears in π with multiplicity at most 1. (2) Let π1, π2, π3 be three irreducible representations of D ×, then the space of D×-invariant linear forms ` : π1 ⊗ π2 ⊗ π3 → C has dimension at most 1. Before we state the next proposition, we introduce some notation. Let O be the maximal compact subring of k, OD the maximal compact subring of D, πD an element of OD such that πDOD is its unique maximal ideal with OD/πDOD a finite field Fq2 1