Identify the distinguished constituent in every L-packet

Determine, for every irreducible smooth representation \(\bm{\pi}^{\#}\) of \(M^{\#}=GL_2^3(F)\) whose central character is trivial on the diagonal center and whose restriction \(\bm{\pi}^{\#}|_M=\bigoplus_i\bm{\sigma}_i\) is the associated multiplicity-free \(L\)-packet, which constituent \(\bm{\sigma}_i\) has a nonzero \(G^\Delta\)-invariant functional when \(\dim\operatorname{Hom}_{G^\Delta}(\bm{\pi}^{\#},\mathbf{1})=1\).

Background

The paper considers $M=(GL_2^3)^{\det}$ and its diagonal subgroup $G^\Delta\simeq GL_2(F)$. Restricting an irreducible representation $\bm{\pi}^{\#}$ of $M^{\#}=GL_2^3(F)$ to $M$ produces a finite, multiplicity-free $L$-packet of irreducible constituents $\bm{\sigma}_i$.

Prasad’s multiplicity-one theorem gives $$\dim\operatorname{Hom}_{G^\Delta}(\bm{\pi}^{\#},\mathbf{1})\leq 1$$, so when this dimension equals one, exactly one constituent of the restriction can be distinguished by $G^\Delta$. The paper explicitly poses the problem of identifying that constituent in general, and then addresses it for unitary principal-series representations.