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@WhatsUp Yes. the non-degeneracy condition means that $1\cdot n=n$. Then it is clear that $A\otimes_A N\cong N$ and by "The Homology of Banach and Topological Algebras" Page 122 Equation 1, $A\hat{\otimes}_A N\cong N$ too. The same result works for any finite direct sums of $A$ as well.
@DenisT. That's right but it doesn't solve my problem: is there any difference on $\text{Aut}^{\otimes}(F)$ if we only consider $\text{rep}_{\mathbb{C}}(G)$ as a monoidal category?
@user66288 This is not trivially true. Actually let $\Phi^+$ and $\Psi^+$ be too positive root systems. My question is equivalent to the question whether one of $\Phi^+\cap \Psi^+$ and $\Phi^+\cap \Psi^-$ generates the whole Weyl group.
@GeordieWilliamson Oh, I see. How about the general case. We know that $\text{ch}(L(\lambda))=\sum_{\mu}m_{\lambda,\mu}\text{ch}(M(\mu))$ and the coefficient $m_{\lambda,\mu}\neq 0$ only if $ \mu\leq \lambda$ and $\mu=w\cdot \lambda$ for some $w\in W$. The strongly linkage gives more restrictions. Do we have some "iff" condition that $m_{\lambda,\mu}\neq 0$?
