Representations distinguished by pairs of exceptional representations and a conjecture of Savin (Q2797803)

Let \(F\) be a local non-Archimedean field with \(2\not=0\). Let \(\tau\) be an admissible representation of \(G\)=GL(\(n,F\)). Let \(\theta\), \(\theta'\) be a pair of exceptional representations -- in the sense of Kazhdan and Patterson -- of the metaplectic double cover \(\widetilde{G}\) of \(G\). The representation \(\tau\) is called distinguished if there is a nonzero \(G\)-invariant trilinear form on the space of \(\tau\times\theta\times\theta'\). Equivalently, Hom\(_{G}(\theta\otimes\theta',\tau^{\vee})\not=0\).NEWLINENEWLINEHere \(\tau^{\vee}\) is the representation contragredient to \(\tau\). The main result of the paper under review is the following combinatorial characterization of irreducible distinguished principal series representations. Say that a character \(\eta=\eta_1\otimes\dots\otimes\eta_n\) of the diagonal torus satisfies condition (*) if, up to a permutation of the characters \(\eta_j\), there is \(0\leq k\leq [n/2]\) such that (1) \(\eta_{2i}=\eta_{2i-1}^{-1}\) for \(1\leq i\leq k\), (2) \(\eta_i^2=1\) for \(2k+1\leq i\leq n\). Theorem 1.1. Let \(\tau\) be a principal series representation of \(G\), induced from the character \(\eta\). If \(\tau\) is distinguished, \(\eta\) satisfies (*). Conversely, if (*) holds and \(\tau\) is irreducible, then \(\tau\) is distinguished. The main corollary of this theorem is the validity of the ``only if'' part of the following conjecture of Savin. Theorem 1.2. Let \(\tau\) be a spherical representation of \(G\), that is, an irreducible unramified quotient of some principal series representation, with a trivial central character. Then \(\tau\) is distinguished if and only if \(\tau\) is the lift of a representation of (split) SO(\(2[n/2],F\)) if \(n\) is even or Sp(\(2[n/2],F\)) if \(n\) is odd. Further discussion of related work appears in the latter part of the introduction.