Classification of generalized linear periods for all n

Classify, for every integer n>2, all irreducible smooth representations of \(\mathrm{GL}_n(\mathrm{D})\) that admit a generalized linear period with respect to \((\mathrm{H}_{1,n-1},\chi_s)\), and establish whether they are exactly the three representation families specified in Conjecture 1: \(\nu^{-2s}\times\tau\) with \(\tau\) infinite-dimensional and \(\mathrm{H}_{1,1}\)-distinguished, together with the two explicitly described exceptional induced families and their exceptional irreducible subrepresentations.

Background

The paper studies generalized linear periods for $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$, where $\mathrm{D}$ is a quaternion division algebra over a non-Archimedean local field, relative to $\mathrm{H}_{1,n-1}$ and the character $\chi_s$. The authors prove the proposed classification for $n=3$ and $n=4$, but leave the corresponding classification for arbitrary $n>2$ as a conjecture.

The conjectural classification consists of three families: representations induced from $\nu^{-2s}$ and an infinite-dimensional $\mathrm{H}_{1,1}$-distinguished representation of $\mathrm{G}_2$, and two families involving characters of $\mathrm{G}_1$ induced with characters of $\mathrm{G}_{n-1}$, with specified irreducible-subrepresentation replacements at exceptional values of $s$. The paper further derives a Langlands-parameter characterization only under the assumption that this conjecture holds.