[Paper Review] Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences

Mitya Boyarchenko, Jared Weinstein|arXiv (Cornell University)|Mar 22, 2013

Advanced Algebra and Geometry8 references8 citations

TL;DR

This paper provides a geometric construction of supercuspidal representations arising from the local Langlands and Jacquet-Langlands correspondences for a specific class of characters on a degree-$n$ unramified extension of a $p$-adic field. By leveraging affinoid subspaces in the Lubin-Tate tower, the authors construct the corresponding $GL_n(F)$ and $D^ imes$-representations without relying on Weil representations over finite fields, thereby simplifying and streamlining the proofs of these correspondences in the minimal admissible case.

ABSTRACT

Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation π of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, π corresponds to an irreducible representation ρ of the multiplicative group of the central division algebra over F with invariant 1/n. In this note we give a new geometric construction of the representations π and ρ, which is simpler than the existing algebraic approaches (in particular, the use of the Weil representation over finite fields is eliminated).

Motivation & Objective

• To provide an explicit geometric construction of irreducible supercuspidal representations of $GL_n(F)$ and the corresponding representations of $D^ imes$, where $D$ is the central division algebra over $F$ with invariant $1/n$.
• To simplify the proofs of the local Langlands and Jacquet-Langlands correspondences in the minimal admissible case by avoiding the use of Weil representations over finite fields.
• To offer a self-contained, detailed exposition that serves as a pedagogical resource for researchers new to the area of $p$-adic representation theory.

Proposed method

• The construction uses a geometric framework based on affinoid subspaces in the Lubin-Tate tower of the local field $F$, introduced by the second author.
• It defines a compact open subgroup $J$ of $GL_n(F)$ and a subgroup $J_+$, whose quotient $J/J_+$ is identified with the unipotent group $U(ar{bF}_{q^n})$.
• The representation $ ho$ of $D^ imes$ is constructed via a pullback from a cohomological representation on $H^{n-1}_c(X imes ar{bF}_q, ar{bQ}_l)$ twisted by a character $ heta$ of level $r_0 \geq 2$.
• The character $ heta$ is extended to a representation of $E^ imes \ltimes J$, which descends to a representation $ heta \otimes \sigma_0$ on $E^ imes \cdot U^{(r_0-1)/2}_G$, and then induced to obtain the supercuspidal representation $\pi$ of $GL_n(F)$.
• The proof of irreducibility relies on intertwining properties and the fact that any intertwiner must preserve the level-$r_0$ structure, leading to a control of the normalizer of the compact open subgroup.
• The construction avoids Weil representations by using geometric data from the Lubin-Tate tower, offering a more direct and streamlined argument than previous algebraic approaches.

Experimental results

Research questions

• RQ1 How can the local Langlands correspondence be realized geometrically for minimal admissible pairs $(E^\times, \theta)$ with $\theta$ of level $r_0 \geq 2$?
• RQ2 Can the Jacquet-Langlands correspondence be constructed explicitly without relying on Weil representations over finite fields?
• RQ3 What is the role of affinoid subspaces in the Lubin-Tate tower in realizing supercuspidal representations of $GL_n(F)$ and $D^\times$?
• RQ4 How does the geometric construction simplify the standard algebraic proofs of the local Langlands and Jacquet-Langlands correspondences?

Key findings

• The representation $\pi = \operatorname{Ind}_{E^\times \cdot U^{(r_0-1)/2}_G}^{GL_n(F)}(\sigma)$ is an irreducible supercuspidal representation of $GL_n(F)$, constructed via geometric induction from a representation $\sigma$ on a compact open subgroup.
• The representation $\rho$ of $D^\times$ is constructed as a pullback from the cohomology of a suitable affinoid subvariety in the Lubin-Tate tower, and it corresponds to $\pi$ under the Jacquet-Langlands correspondence.
• The character $\theta$ of $E^\times$ of level $r_0 \geq 2$ with trivial ${\rm Gal}(E/F)$-stabilizer gives rise to a smooth irreducible $n$-dimensional representation $\sigma_\theta$ of the Weil group $\mathcal{W}_F$, via induction from $\mathcal{W}_E$.
• The trace of the representation $\sigma$ on $E^\times \cdot U^{(r_0-1)/2}_G$ satisfies $\operatorname{tr}\sigma(x) = (-1)^{n-1} \cdot \theta(x)$ for very regular $x \in \mathcal{O}_E^\times$, confirming the expected character relation.
• The construction avoids the use of Weil representations over finite fields, providing a more geometric and streamlined proof of the correspondence in the minimal case.
• The normalizer of the compact open subgroup $E^\times \cdot U^{(r_0-1)/2}_G$ in $GL_n(F)$ is shown to be itself, ensuring the irreducibility of the induced representation $\pi$ via standard intertwining arguments.