On the Restriction of Supercuspidal Representations: An In-Depth Exploration of the Data

Abstract

Let $\mathbb{G}$ be a connected reductive group defined over a p-adic field F which splits over a tamely ramified extension of F, and let G = $\mathbb{G}(F)$. We also assume that the residual characteristic of F does not divide the order of the Weyl group of $\mathbb{G}$.

Following J.K. Yu's construction, the irreducible supercuspidal representation constructed from the G-datum $\Psi$ is denoted $\pi_G(\Psi)$. The datum $\Psi$ contains an irreducible depth-zero supercuspidal representation, which we refer to as the depth-zero part of the datum. Under our hypotheses, the J.K. Yu Construction is exhaustive.

Given a connected 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 first describing a natural restriction process from which we construct H-data from the G-datum $\Psi$. We then show that the obtained H-data, and conjugates thereof, construct the components of $\pi_G(\Psi)|_H$, thus providing a very precise description of the restriction. Analogously, we also describe an extension process that allows to construct G-data from an H-datum $\Psi_H$. Using Frobenius Reciprocity, we obtain a description for the components of $\Ind_H^G\pi_H(\Psi_H)$.

From the obtained description of $\pi_G(\Psi)|_H$, we prove that the multiplicity in $\pi_G(\Psi)|_H$ is entirely determined by the multiplicity in the restriction of the depth-zero piece of the datum. Furthermore, we use Clifford theory to obtain a formula for the multiplicity of each component in $\pi_G(\Psi)|_H$. As a particular case, we take a look at the regular depth-zero supercuspidal representations and obtain a condition for a multiplicity free restriction.

Finally, we show that our methods can also be used to define a restriction of Kim-Yu types, allowing to study the restriction of irreducible representations which are not supercuspidal.