[Paper Review] On L-embeddings and double covers of tori over local fields

Tasho Kaletha|arXiv (Cornell University)|Jul 11, 2019

Advanced Algebra and Geometry11 references4 citations

TL;DR

This paper introduces a canonical double cover $T(F)_{\pm}$ of the $F$-rational points of a torus $T$ over a local field $F$, equipped with a canonical $L$-group ${}^{L}T_{\pm}$, and establishes a natural bijection between $L$-parameters valued in ${}^{L}T_{\pm}$ and genuine characters of $T(F)_{\pm}$. It further constructs a canonical $L$-embedding ${}^{L}T_{\pm} \to {}^{L}G$ for a maximal torus $T$ in a reductive group $G$, enabling a factorization of Langlands parameters and a conjectural characterization of the supercuspidal local Langlands correspondence via a Harish-Chandra character formula.

ABSTRACT

To a torus T over a local field F and a subset of its character module subject to certain properties, we associate a canonical double cover of the topological group T(F). We further associate an L-group to this double cover and establish a natural bijection between L-parameters valued in this L-group and genuine characters of the double cover. When T is a maximal torus of a connected reductive group G, we show that there is a canonical L-embedding from the L-group of the double cover of T to the L-group of G. This leads to a canonical factorization of Langlands parameters. We associate to a genuine character of the double cover subject to certain conditions a Harish-Chandra character formula and use it to give a conjectural characterization of the supercuspidal local Langlands correspondence for G, subject to a certain condition on p. This generalizes previous work of Adams and Vogan F=R, and reinterprets computations of Langlands and Shelstad.

Motivation & Objective

• To construct a canonical double cover $T(F)_{\pm}$ of the $F$-rational points of a torus $T$ over a local field $F$, equipped with additional structure derived from $\chi$-data.
• To define an $L$-group ${}^{L}T_{\pm}$ associated to this double cover, generalizing the Weil-form of the $L$-group and extending the framework of $\rho$-covers.
• To establish a natural bijection between $L$-parameters with values in ${}^{L}T_{\pm}$ and genuine characters of $T(F)_{\pm}$, thereby linking representation-theoretic data to Langlands parameters.
• To construct a canonical $L$-embedding ${}^{L}T_{\pm} \to {}^{L}G$ when $T$ is a maximal torus in a reductive group $G$, enabling a factorization of Langlands parameters.
• To propose a conjectural characterization of the supercuspidal local Langlands correspondence for $G$ using a Harish-Chandra character formula associated to genuine characters of $T(F)_{\pm}$.

Proposed method

• The double cover $T(F)_{\pm}$ is constructed as a pullback via a canonical morphism involving induced one-dimensional anisotropic tori $J_O$ associated to a $\Sigma$-invariant subset $R \subset X^*(T)$.
• The $L$-group ${}^{L}T_{\pm}$ is defined as the pushout of $\widehat{S} \leftarrow \prod \widehat{J}_O \to {}^{L}(\prod J_O)_{\pm}$, where the latter is twisted by the Tits cocycle arising from $\chi$-data.
• The $L$-parameter of a genuine character $\chi_S$ is obtained as the image under the pushout map of the product of $L$-parameters of the $\chi_\alpha$'s on the $J_\alpha(F)_{\pm}$, using the Shapiro map in degree 1.
• The construction uses the Shapiro isomorphism to relate $L$-parameters for ${}^{L}T_{\pm}$ to cochains in $C^1(W_{E_\pm}, \mathbb{C}^\times_{(-1)})$ with differential equal to the canonical 2-cocycle $z$, ensuring compatibility with $\chi$-data.
• The $L$-embedding ${}^{L}T_{\pm} \to {}^{L}G$ is shown to exist canonically when $T$ is a maximal torus in $G$, using the $\chi$-data and the structure of the Weil form of the $L$-group.
• A Harish-Chandra character formula is associated to each genuine character of $T(F)_{\pm}$, which is used to conjecture a characterization of supercuspidal Langlands parameters via trace identities.

Experimental results

Research questions

• RQ1 Can a canonical double cover $T(F)_{\pm}$ of the $F$-rational points of a torus $T$ over a local field $F$ be constructed, equipped with a compatible $L$-group ${}^{L}T_{\pm}$?
• RQ2 Is there a natural bijection between $L$-parameters with values in ${}^{L}T_{\pm}$ and genuine characters of $T(F)_{\pm}$?
• RQ3 Does a canonical $L$-embedding ${}^{L}T_{\pm} \to {}^{L}G$ exist when $T$ is a maximal torus in a reductive group $G$?
• RQ4 Can the local Langlands correspondence for supercuspidal representations be characterized via a Harish-Chandra character formula associated to genuine characters of $T(F)_{\pm}$?
• RQ5 How does the $L$-group ${}^{L}T_{\pm}$ relate to the $\rho_i$-cover and $E$-group constructions of Adams-Vogan in the real case, and how does it generalize them to $p$-adic fields?

Key findings

• A canonical double cover $T(F)_{\pm}$ is constructed for any torus $T$ over a local field $F$, using $\chi$-data and a decomposition into induced one-dimensional anisotropic tori $J_O$.
• The $L$-group ${}^{L}T_{\pm}$ is defined as the pushout of $\widehat{S} \leftarrow \prod \widehat{J}_O \to {}^{L}(\prod J_O)_{\pm}$, with the twisting cocycle given by the Tits cocycle from $\chi$-data.
• There is a natural bijection between $L$-parameters with values in ${}^{L}T_{\pm}$ and genuine characters of $T(F)_{\pm}$, realized via the Shapiro isomorphism in degree 1.
• A canonical $L$-embedding ${}^{L}T_{\pm} \to {}^{L}G$ exists when $T$ is a maximal torus in a reductive group $G$, providing a factorization of Langlands parameters.
• A Harish-Chandra character formula is associated to each genuine character of $T(F)_{\pm}$, and this formula is conjectured to characterize the supercuspidal local Langlands correspondence for $G$ under a condition on the residual characteristic $p$.
• The construction generalizes the Adams-Vogan $\rho_i$-cover to $p$-adic fields and provides a uniform framework for the local Langlands correspondence via $L$-groups and genuine characters.