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

Kei Yuen Chan

Published 2022-10-31Version 1

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