We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,\mathscr{R})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.
Y. Hironaka introduced the spherical functions on the p-adic space of Hermitian matrices. For the space of 2\times2 Hermitian matrices, we complete Hironaka's work by also considering the case of a wildly ramified quadratic extension. We compute the spherical functions explicitly and obtain the functional equations.
We construct infinite families of irreducible supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any finite extension with residue field $\mathbb{F}_{p^2}$ and ramification degree $e \leq (p-1)/2$. These are the first such examples for ramified $F / \mathbb{Q}_p$.
For certain characters of the compact torus of a reductive $p$-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
On restriction to the maximal compact subgroup $\mathrm{GL}(3,\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents.
Building on prior work, we analyze the decomposition of the restriction of an irreducible representation of SL_2(k), for k a p-adic field of odd residual characteristic, to a maximal compact subgroup K. The pattern of the decomposition varies between principal series and different supercuspidal representations, whereas the K-representations which occur in the "tail end" of these decompositions are precisely those occurring in the decomposition of depth-zero supercuspidal representations. Various applications are considered.
Let G be a reductive group over a non-archimedean local field k. We provide necessary conditions and sufficient conditions for all tori of G to split over a tamely ramified extension of k. We then show the existence of good semisimple elements in every Moy-Prasad filtration coset of the group G(k) and its Lie algebra, assuming the above sufficient conditions are met.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $Π$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $\mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method s…
Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contr…
We study the geometry and the cohomology of the tamely ramified cover of Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $\ell$-adic cohomologies, as well as the torsion freeness of the integral cohomology. During the study, we obtain some results of independent interest on the coefficient systems over the Bruhat-Tits building associated to $GL_d(K).$
We prove a conjecture of Dipendra Prasad on the Ext-branching problem from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$, where $F$ is a $p$-adic field.
Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,π)$, for any irreducible representation $π$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde π} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of …
Using Bruhat-Tits theory, we analyse the restriction of depth-zero representations of a semisimple simply connected $p$-adic group $G$ to a maximal compact subgroup $K$. We prove the coincidence of branching rules within classes of Deligne-Lusztig supercuspidal representations. Furthermore, we show that under obvious compatibility conditions, the restriction to $K$ of a Deligne-Lusztig supercuspidal representation of $G$ intertwines with the restriction of a depth-zero principal series representation in infinitely many distinct components of arbitrarily large depth. Several qualitative and qua…
Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
Institut de Mathématiques de Jussieu, Sorbonne Université – Université de Paris – CNRS（巴黎数学研究所，索邦大学-巴黎大学-法国国家科学研究中心）；CNRS, Laboratoire de mathématiques de Versailles, Université Paris-Saclay, UVSQ（法国国家科学研究中心，凡尔赛数学实验室，巴黎萨克雷大学，凡尔赛大学）
Let $\{ρ_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Let $Γ_\ell$ and $\mathbf{G}_\ell$ be respectively the image and the algebraic monodromy group of $ρ_\ell$. We prove that the reductive quotient of $\mathbf{G}_\ell^\circ$ is unramified over every degree 12 totally ramified extension of $\mathbb{Q}_\ell$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(\ast)$ on $\{ρ_\ell\}_\ell$ such that for all sufficiently large $\ell$, the subgrou…
The group $\GL_2$ over a local field with (residue) characteristic $2$ possesses much more smooth supercuspidal $\ell$-adic representations, than over a local field of residue characteristic $> 2$. One way to construct these representations is via the theory of types of Bushnell-Kutzko. We construct many of them in the cohomology of certain extended affine Deligne-Lusztig varieties attached to $\GL_2$ and wildly ramified maximal tori in it. Then we compare our construction with the type-theoretic one. The corresponding extended affine Deligne-Lusztig varieties were introduced in a preceding ar…
In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(π,n+l)$ of $\mathrm{GL}_{2n+2l}$ with respect to the block diagonal subgroup $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$ for any irreducible generic representation $π$ of $\mathrm{GL}_2$ over any $p$-adic field. We use the Shalika model of $\mathrm{Sp}(π,n)$ to construct certain zeta integrals, which were defined by Ginzburg and Kaplan independently, and study them. Finally, using these zeta integrals, we obtain a nonzero $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$-map from $\mathrm{Sp}(π,n+l)$ to $τ\boxtimesτ^\veeχ_π…
