The non-tempered Gan–Gross–Prasad conjecture for Arthur type representations of general linear groups

Let $F$ be a local field. For an Arthur type representation, let its associated Arthur parameter be a finite representation of $WD_F\times \mathrm{SL}_2(\mathbb C)$; two such parameters $M_A$ and $N_A$ are relevant if there exist $WD_F$-representations $M_0^+,\ldots,M_r^+,M_0^-,\ldots,M_s^-$ corresponding to tempered representations such that

$$M_A=\sum_{d=0}^r M_d^+\otimes \mathrm{Sym}^d(\mathbb C^2)\oplus\sum_{d=1}^s M_d^-\otimes \mathrm{Sym}^{d-1}(\mathbb C^2),$$

and

$$N_A=\sum_{d=1}^r M_d^+\otimes \mathrm{Sym}^{d-1}(\mathbb C^2)\oplus\sum_{d=0}^s M_d^-\otimes \mathrm{Sym}^d(\mathbb C^2).$$

Here $G_n=\mathrm{GL}_n(F)$. Non-tempered Gan–Gross–Prasad conjecture. Let $\pi_M$ and $\pi_N$ be Arthur type representations of $\mathrm{GL}_{n+1}(F)$ and $\mathrm{GL}_n(F)$, respectively. Then

$$\mathrm{Hom}_{G_n}(\pi_M,\pi_N)\neq 0$$

if and only if their associated Arthur parameters $M_A$ and $N_A$ are relevant. This predicts the branching law for Arthur type representations under the standard embedding $G_n\hookrightarrow G_{n+1}$ and extends the tempered Gan–Gross–Prasad framework to the non-tempered setting; the source does not provide evidence resolving the conjecture.