Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets by biquadratic extensions. We compute the formal degrees of the elements in the supercuspidal packets.
Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of certain L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets of cardinality 4 are parametrized by biquadratic extensions. Each supercuspidal packet of cardinality 4 is accompanied by a singleton packet for SL_1(D). We compute the depths of the irreducible constituents of all these L-packets …
Relying on work of Kashiwara-Schapira and Schmid-Vilonen, we describe the behaviour of characteristic cycles with respect to the operation of geometric induction, the geometric counterpart of taking parabolic or cohomological induction in representation theory. By doing this, we are able to describe to some extent the characteristic cycle associated to an induced representation, in terms of the characteristic cycle of the representation being induced. More precisely, under the hypothesis that the infinitesimal character is regular (and dominant), we show that the characteristic cycle of an ind…
Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$-factors of L-parameters. This can be regar…
We prove the conjectural endoscopic transfer of L-packets for the local Langlands correspondence for pure inner forms of unramified p-adic groups and depth-zero parameters established by DeBacker and Reeder. More precisely, we show that under mild conditions on the residual characteristic, endoscopic induction identifies an unstable character of such an L-packet with the stable character of the corresponding endoscopic L-packet.
Let $G \subseteq \tilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and $G_{der} = \tilde{G}_{der}$. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of that of $\tilde{G}$. Motivated by this, we hope to construct the L-packets of $\tilde{G}$ from that of $G$. The primary example in our mind is when $G = Sp(2n)$, whose L-packets have been determined by Arthur (2013), and $\tilde{G} = GSp(2n)$. As a first step, we need to consider some well-known conjectural prop…
We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of $SL_2 (K)$ is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all $L$-packets for $SL_2 (K)$ and for its non-split inner form, and we provide explicit formulae for the depths of their $L$-parameters.
Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GL_n(F) and its derived subgroup SL_n(F). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL_2(Q_p) and smooth mod-p representations of SL_2(Q_p) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modu…
Let $F$ be a local field of characteristic not $2$. We propose a definition of stable conjugacy for all the covering groups of $\mathrm{Sp}(2n,F)$ constructed by Brylinski and Deligne, whose degree we denote by $m$. To support this notion, we follow Kaletha's approach to construct genuine epipelagic $L$-packets for such covers in the non-archimedean case with $p \nmid 2m$, or some weaker variant when $4 \mid m$; we also prove the stability of packets when $F \supset \mathbb{Q}_p$ with $p$ large. When $m=2$, the stable conjugacy reduces to that defined by J. Adams, and the epipelagic $L$-packet…
For a quasi-split classical group over a p-adic field with sufficiently large residual characteristic, we prove that the maximum of depth of representations in each L-packet equals the depth of the corresponding L-parameter. Furthermore, for quasi-split unitary groups, we show that the depth is constant in each L-packet. The key is an analysis of the endoscopic character relation via harmonic analysis based on the Bruhat--Tits theory. These results are slight generalizations of a result of Ganapathy and Varma.
We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies the expected properties of such a correspondence. More precisely, we show that each L-packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with twists by central and cocentral characters, provides a stable virtual char…
Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its $L$-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second, using the first, is to verify that Kaletha's regular supercuspidal $L$-packets satisfy the conjecture.
Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(τ,ν)$ and $I(τ',ν')$ for $G$ and $G'$ respectively with $τ'$ a generic tempered representation, such that the Harish-Chandra's $μ$-functions of a representation in the supercuspidal support of $τ$ and of a generic essentially square-integral representation in some Jacquet module of $τ'$ agree (after a suitable identification of the unde…
Let F be a non-archimedean local field and let G be a connected reductive group defined over F. We assume that G splits over a tame extension of F and that the residual characteristic p does not divide the order of the Weyl group. To each discrete Langlands parameter of the Weil group of F into the complex L-group of G we associate explicitly a finite set of irreducible supercuspidal representations of G(F), and relate its internal structure to the centralizer of the parameter. We give evidence that this assignment is an explicit realization of the local Langlands correspondence.
