The non-generic local Gan–Gross–Prasad conjecture for general linear groups

Let $F$ be a field with Weil group $W_F$. For $r\in\mathbb{Z}_{>0}$, let an Arthur parameter for $GL_r(F)$ be a completely reducible representation

$$\phi=\bigoplus_{i=1}^m\psi_i\otimes V_{a_i}\otimes V_{b_i},$$

where each $\psi_i$ is an irreducible representation of $W_F$ with bounded image, and $V_d$ is the unique isomorphism class of a $d$-dimensional irreducible algebraic representation of $SL_2(\mathbb{C})$. Let $\pi(\phi)$ denote the irreducible representation of $GL_r(F)$ attached to $\phi$ by local Langlands reciprocity. Suppose that

$$\phi_1=\bigoplus_{i=1}^k\psi_i\otimes V_{a_i}\otimes V_{b_i},\qquad \phi_2=\bigoplus_{i=1}^l\psi'_i\otimes V_{a'_i}\otimes V_{b'_i}$$

are Arthur parameters for $GL_n(F)$ and $GL_{n-1}(F)$, respectively. The non-generic local Gan–Gross–Prasad conjecture. One has

$$\operatorname{Hom}_{GL_{n-1}(F)}\left(\pi(\phi_1)|_{GL_{n-1}(F)},\,\pi(\phi_2)\right)\ne 0$$

if and only if there are disjoint partitions

$$\{1,\ldots,k\}=I_1\cup I_2\cup I_3,\qquad \{1,\ldots,l\}=J_1\cup J_2\cup J_3,$$

and bijections $u:I_1\to J_2$ and $d:I_2\to J_1$ satisfying

$$(a'_{u(i)},b'_{u(i)})=(a_i,b_i+1),\qquad \psi'_{u(i)}\cong\psi_i\quad\forall i\in I_1,$$ $$(a'_{d(i)},b'_{d(i)})=(a_i,b_i-1),\qquad \psi'_{d(i)}\cong\psi_i\quad\forall i\in I_2,$$

with $b_i=1$ for every $i\in I_3$ and $b'_j=1$ for every $j\in J_3$. This conjecture proposes an explicit combinatorial criterion for nonzero branching multiplicity in the restriction from $GL_n(F)$ to $GL_{n-1}(F)$ beyond the generic case; its status is not resolved in the supplied source context.