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Let $G_n$ be an inner form of the general linear group over a non-Archimedeanfield $F$. For a $\square$-irreducible representation $\sigma$ of $G_n$ and anirreducible representation $\pi$ of $G_m$, the parabolically induced modules$\sigma \times \pi$ and $\pi \times \sigma$ have irreducible socles. This wasconjectured by Leclerc (2003) for quantum affine algebras, and is now proved byKang-Kashiwara-Kim-Oh (2015). For $p$-adic general linear groups, it isexplicated by Lapid-M\'inguez. Denote the socles respectively by$I^L_{\sigma}(\pi)$ and $I^R_{\sigma}(\pi)$, called integrals. We denoterespectively the inverse operators by $D^L_{\sigma}$ and $D^R_{\sigma}$, calledderivatives.  We study a sufficient condition arising from geometric lemma such that thecommutation: \[ D^R_{\sigma}\circ I^L_{\sigma'}(\pi) \cong I^L_{\sigma'}\circD^R_{\sigma}(\pi) \] holds. In particular, we develop a dual theory for suchcondition.  When $\sigma$ and $\sigma'$ are essentially square-integrablerepresentations, we give several equivalent sufficient conditions for thecommutation. Such commutation will play an important role in formulating anotion of generalized Gan-Gross-Prasad relevant pairs in the sequel.

On commutations of derivatives and integrals of $\square$-irreducible representations for $p$-adic $\mathrm{GL}$