Restricting representations via restricting $G$-data and Kim-Yu types

Abstract: Let $\mathbb{G}$ be a reductive group defined over a $p$-adic field $F$, and let $G = \mathbb{G}(F)$. We assume that $\mathbb{G}$ splits over a tamely ramified extension of $F$ and that the residual characteristic $p$ of $F$ does not divide the order of the Weyl group of $\mathbb{G}$. Under this assumption, Fintzen showed that all irreducible supercuspidal representations of $G$ are obtained via the J.K.~Yu construction. From a $G$-datum $\Psi$, the J.K.~Yu construction produces an irreducible supercuspidal representation of $G$, which we denote by $\pi_G(\Psi)$.

Given a reductive $F$-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction $\pi_G(\Psi)|_H$ and obtain a description of its decomposition into irreducible components along with their multiplicities. We achieve this by describing a natural restriction process from which we construct $H$-data from the $G$-datum $\Psi$.

To study the restriction to $H$ of irreducible representations of $G$ which are not supercuspidal, one can use the theory of types. More specifically, given the underlying assumption on $p$, Fintzen showed that every irreducible representation of $G$ contains a Kim-Yu type. The construction of Kim-Yu types is very similar to Yu's construction of supercuspidal representations. As such, we can define an analogous restriction process from which we construct Kim-Yu types for $H$ from a Kim-Yu type for $G$, therefore obtaining information on the restriction to $H$ of any irreducible representation of $G$.

number theoryrepresentation theory

Audience: researchers in the topic

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