[Paper Review] Simple wild L-packets

Tasho Kaletha|arXiv (Cornell University)|Aug 11, 2011

Advanced Algebra and Geometry13 references3 citations

TL;DR

This paper constructs simple wild L-packets for p-adic groups by associating a finite set of simple supercuspidal representations to each simple wild Langlands parameter, using a tamely ramified anisotropic torus and affine generic characters. The construction realizes the conjectured correspondence under mild conditions on the residual characteristic and proves stability of these L-packets for an open set of regular semi-simple elements.

ABSTRACT

In a recent paper, Gross and Reeder study arithmetic properties of discrete Langlands parameters for semi-simple p-adic groups and conjecture that a special class of these -- the simple wild parameters -- should correspond to L-packets consisting of simple supercuspidal representations. We provide a construction of this correspondence and show that the simple wild L-packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

Motivation & Objective

• To establish the conjectured correspondence between simple wild Langlands parameters and simple supercuspidal representations in the local Langlands correspondence.
• To construct explicit L-packets for simple supercuspidal representations using a tamely ramified anisotropic torus and stable conjugacy classes of embeddings.
• To verify that the resulting L-packets are stable under the action of the adjoint group and satisfy the expected character-theoretic properties.
• To show compatibility of the construction with finite unramified extensions of the base field.
• To provide evidence for full stability of the L-packets by proving atomic stability on an open subset of regular semi-simple elements.

Proposed method

• Construct a tamely ramified anisotropic torus $S$ over a $p$-adic field $F$ with a stable conjugacy class of embeddings $j: S \to G$ of type (C), characterized by the barycenter of an alcove in the building.
• Use the $L$-group $^L S$ to lift the Langlands parameter $\phi: W_F \to \widehat{G}$ to a character $\chi: S(F) \to \mathbb{C}^\times$ via $\phi_S$.
• Define an affine generic character $\tilde{\chi}$ on the quotient $Z(F)G(F)_{x,1/h}/G(F)_{x,2/h}$ using $\chi$, where $x$ is the point in the building associated to $j$ and $h$ is the Coxeter number.
• Construct the simple supercuspidal representation $\pi$ as a compact induction $\mathrm{c-Ind} \tilde{\chi}$, ensuring it is generic with respect to a Borel subgroup.
• Show that the $G_{\rm ad}(F)$-conjugacy class of the pair $(j, \chi)$ depends only on $\phi$, ensuring well-definedness of the L-packet.
• Prove stability of the Harish-Chandra character sum $S\Theta_\Pi$ on an open set of regular semi-simple elements using the Tate-Nakayama isomorphism and norm maps.

Experimental results

Research questions

• RQ1 Does the conjectured correspondence between simple wild parameters and simple supercuspidal representations hold for $p$-adic groups?
• RQ2 Can the $L$-packet structure be explicitly realized via affine generic characters and torus embeddings of type (C)?
• RQ3 Is the $L$-packet stable on an open set of regular semi-simple elements without restrictions on the residue field?
• RQ4 How does the construction behave under finite unramified extensions of the base field?
• RQ5 Is the $L$-packet uniquely determined by the Langlands parameter $\phi$, up to $G_{\rm ad}(F)$-conjugacy?

Key findings

• The construction yields a well-defined $G_{\rm ad}(F)$-conjugacy class of simple supercuspidal representations from a simple wild parameter $\phi$, realizing the expected $L$-packet.
• The $L$-packet is stable on an open subset of regular semi-simple elements, with no proper subset of the packet having a stable character sum, indicating atomic stability.
• The construction is compatible with finite unramified extensions: if $\tilde{F}/F$ is unramified, the $L$-packet over $\tilde{F}$ is obtained by base change of the character and embedding.
• The $G_{\rm ad}(F)$-orbit of the affine generic character $\tilde{\chi}$ is determined solely by $\phi$, ensuring independence from auxiliary choices in the embedding $j$.
• The $L$-packet structure matches the conjectural form: the set of representations is in bijection with the group of characters of the finite centralizer of $\phi$ in $\widehat{G}$.
• The stability of the character sum $S\Theta_\Pi$ is established via integration over $G_{\rm ad}(F)$, using the norm map and Tate-Nakayama duality.