Thesis advisor: Mark Reeder / This thesis contributes to the construction of supercuspidal representations in small residual characteristics. Let G be a connected, quasi-split, semisimple reductive algebraic group defined and quasi-split over a non-archimedean local field k and splitting over a tamely, totally ramified extension of k. To each parahoric subgroup of G(k), Moy and Prasad have attached a natural filtration by compact open subgroups, the first of which is called the pro-unipotent radical of the parahoric subgroup. The first main result of this thesis is to characterize shallow characters of a pro-unipotent radical, those being complex characters that vanish on the smallest Moy-Prasad subgroup containing all commutators of linearly-dependent affine k-root groups. Through low-rank examples, we illustrate how this characterization can be used to explicitly construct all shallow characters. Next, we provide a natural sufficient condition under which a shallow character compactly induces as a direct sum of supercuspidal representations of G(k). Through examples, however, we show that this sufficient condition need not be necessary, all while constructing new supercuspidal representations of Sp_4(k) when p = 2 and the split form of G_2 over k when p = 3. This work extends the construction of the simple supercuspidal representations given by Gross and Reeder and the epipelagic supercuspidal representations given by Reeder and Yu. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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.

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Fixons un nombre premier p. Soit k un corps algébriquement clos de caractéristique l différent que p. Nous construisons les k-types maximaux simples cuspidaux des sous-groupes de Levi M' de SL_n(F), où F est un corps local non archimédien de caractéristique résiduelle p. Nous montrons que le support supercuspidal des k-représentations lisses irréductibles de M' est unique à M'-conjugaison près, quand F est soit un corps fini de caractéristique p soit un corps local non-archimédien de caractéristique résiduelle p. / Fix a prime number p. Let k be an algebraically closed field of characteristic l different than p. We construct maximal simple cuspidal k-types of Levi subgroups M' of SL_n(F), where F is a non-archimedean locally compact field of residual characteristic p. And we show that the supercuspidal support of irreducible smooth k-representations of Levi subgroups M' of SL_n(F) is unique up to M'-conjugation, when F is either a finite field of characteristic p or a non-archimedean locally compact field of residual characteristic p.