Distinguished representations for $SL(2)$

Abstract

Abstract. Let E/F be a quadratic extension of p-adic fields. We compute the multiplicity of the space of SL 2 (F )-invariant linear forms on a representation of SL 2 (E). This multiplicity varies inside an L-packet similar in spirit to the multiplicity formula for automorphic representations due to Labesse and Langlands.

Theta correspondence and the Prasad conjecture for SL(2)

“…The map φ τ → φ τ | W DE is called the base change map. Prasad's conjecture for SL(2) predicts the following result, which was shown in [AP03].…”

A local-global question in automorphic forms

“…From these theorems due to Labesse-Langlands [LL79], and the theorems due to the authors in [AP03], [AP06] we deduce the following:…”

“…Our work in the previous two sections for the pair (SL 2 (E), SL 2 (F )) was a consequence of this characterization of distinction for GL 2 (E) representations in terms of Asai Lfunction, and an input on distinction for the pair (SL 2 (E), SL 2 (F )) in terms of Whittaker model with respect to a character of E trivial on F which was proved in [AP03] in the local case, and [AP06] in the global case.…”

“…According to the notation introduced in [AP03], [AP06], and the proof of the previous theorem, we have,…”

The SL1(D)-distinction problem

“…Remark 3.4. For the SL 2 (F )-distinction problem, the set of the multiplicities in the L-packet Π φτ is {4, 0, 0, 0} in this case, see [AP03,Lu18].…”