Reference Request: Possible generalizations of the stability of $\gamma$-factors

Question

$\DeclareMathOperator\GL{GL}$ Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. Here we consider the local automorphic $L$-factors $L(s, \pi \times \sigma)$ defined by Jacquet-Piateskii-Shapiro-Shalika and $L(s, \pi)$ defined by Godement-Jacquet.

The following result is stated and proved as (2.2) Proposition of the article "A lemma on highly ramified $\epsilon$-factors" by Jacquet and Shalika.

Proposition: Notations as above. Then there exists an integer $A>0$, such that for any $a \geq A$ and any character $\chi: F^{\times} \rightarrow \mathbb{C}^{\times}$ of conductor $\varpi^{a}$, $L(s, \pi \otimes \chi) = 1$.

My question is:

1. Has this proposition already been generalized to the case where $\chi$ is replaced by an irreducible admissible representation $\sigma$ of $\GL_{m}(F)$ (with $m \leq n$), i.e. more precisely, for sufficiently ramified such $\sigma$, $L(s, \pi \times \sigma) = 1$? [Here we measure the ramification of $\sigma$ by its conductor $c(\sigma)$, as in the $\GL_1$-case. The conductor is defined by either the power that appears in the functional equation of $\epsilon$-factor, or by the existence of new vectors of "level" $c(\sigma)$.]
2. Can the bound $A$ in the proposition (and its generalization) be made explicit? How does it depend on $\pi$?

I have been trying to find a reference for such results but failed. Maybe it is hidden somewhere, or is there still no result on such problems?

Thank you all for paying attention. :)

Answer 1

Let us first take a look at your first question:

I do not think that the generalization you describe is true for m>1. Indeed we can take a characters $\chi_1,\ldots,\chi_n$ of $F^{\times}$ with conductors $$c(\chi_1)>c(\chi_2)=\ldots = c(\chi_n)=0$$ and consider the representation $$\pi=Ind_B^G(\chi_1\otimes \ldots \otimes \chi_n).$$ Let us also assume that $\pi$ is irreducible. We can choose the representation $\sigma$ of $GL_m(F)$ similarly by putting $$\sigma = Ind_B^G(\xi_1\otimes \ldots \otimes \xi_m)$$ for characters $\xi_1,\ldots,\xi_m$ such that $$c(\xi_1)>c(\xi_2)=\ldots=c(\xi_m)=0.$$ Then $c(\sigma)=c(\xi_1)$, which we can assume to be as large as we want. However, as soon as $m>1$ we see that $L(s,\pi\times\sigma)$ is never trivial. Thus, if one measures the complexity of $\sigma$ only using the conductor one can not expect your suggested generalization to hold.

Concerning your second question:

One can take $A=c(\pi)$. By considering $\pi$ as above one sees that this is optimal in general.