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Parametrization of supercuspidal representations of depth zero for some simple adjoint groups

Amoru Fujii

TL;DR

This work develops a depth-zero local Langlands framework for simple adjoint groups by constructing a surjective map from depth-zero cuspidal enhanced L-parameters to depth-zero supercuspidal representations, and proving bijectivity in several important types. The approach introduces an intermediate unramified group $H$ attached to a given depth-zero discrete L-parameter, and uses stable apartment embeddings to relate parahoric data between $H$ and $G$, together with Lusztig's Jordan decomposition and the local LLC for unipotent supercuspidals to transport structures and build a concrete LLC for depth-zero cusp representations. It further verifies the bijectivity in a broad family of groups (types $A_n,E_6,E_8,F_4,G_2$, inner form of ${}^3D_4$, split $C_n$, $E_7$, and certain $B_n$/$D$-types with odd residual characteristic) and demonstrates that the Hiraga–Ichino–Ikeda conjecture for formal degrees holds for the resulting parametrization when bijective, by reducing to known unipotent cases. Overall, the paper advances explicit depth-zero L-parameter parametrizations, clarifies the role of parahoric and Jordan-theoretic data, and connects these to precise formal-degree formulas with potential applications to explicit classification and degree computations.

Abstract

We construct a surjective map from the set of conjugacy classes of depth-zero cuspidal enhanced L-parameters to that of isomorphism classes of depth-zero supercuspidal representations for simple adjoint groups, and check the bijectivity in various cases. We also prove that the Hiraga--Ichino--Ikeda conjecture on the formal degree of essentially square-integrable irreducible representations holds for this parametrization if it is bijective.

Parametrization of supercuspidal representations of depth zero for some simple adjoint groups

TL;DR

This work develops a depth-zero local Langlands framework for simple adjoint groups by constructing a surjective map from depth-zero cuspidal enhanced L-parameters to depth-zero supercuspidal representations, and proving bijectivity in several important types. The approach introduces an intermediate unramified group attached to a given depth-zero discrete L-parameter, and uses stable apartment embeddings to relate parahoric data between and , together with Lusztig's Jordan decomposition and the local LLC for unipotent supercuspidals to transport structures and build a concrete LLC for depth-zero cusp representations. It further verifies the bijectivity in a broad family of groups (types , inner form of , split , , and certain /-types with odd residual characteristic) and demonstrates that the Hiraga–Ichino–Ikeda conjecture for formal degrees holds for the resulting parametrization when bijective, by reducing to known unipotent cases. Overall, the paper advances explicit depth-zero L-parameter parametrizations, clarifies the role of parahoric and Jordan-theoretic data, and connects these to precise formal-degree formulas with potential applications to explicit classification and degree computations.

Abstract

We construct a surjective map from the set of conjugacy classes of depth-zero cuspidal enhanced L-parameters to that of isomorphism classes of depth-zero supercuspidal representations for simple adjoint groups, and check the bijectivity in various cases. We also prove that the Hiraga--Ichino--Ikeda conjecture on the formal degree of essentially square-integrable irreducible representations holds for this parametrization if it is bijective.

Paper Structure

This paper contains 9 sections, 30 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lusztig's Jordan decomposition
  4. The local Langlands correspondence for unipotent supercuspidal representations
  5. The reductive group H and its apartments
  6. Construction of the correspondence
  7. The action of the fundamental group on parahoric subgroups
  8. A bijective correspondence for some simple adjoint groups
  9. The Hiraga--Ichino--Ikeda conjecture

Key Result

Theorem 1.1

Let $G$ be a simple adjoint group over $F$ which splits over an unramified extension, and $\Phi_{\mathrm{e}}(G)_{0,\mathrm{cusp}}$ the set of depth-zero discrete L-parameters for $G$ with a cuspidal enhancement. Also we define $\mathop{\mathrm{Irr}}\nolimits(G(F))_{0,\mathrm{cusp}}$ as the set of is where $\sim$ is the conjugacy equivalence under $\widehat{G}$. Moreover, it is bijective in the fol

Theorems & Definitions (58)

  • Theorem 1.1: Theorems \ref{['thm:general_LLC']} and \ref{['thm:LLC_simple_adjoint']}
  • Theorem 1.2: Theorem \ref{['thm:HIIconj']}
  • Proposition 2.1: DL
  • Theorem 2.2: Lusztig_exclusion
  • Theorem 2.3: Lusztig_discon
  • Remark 2.4
  • Theorem 2.5: Geck_Malle_2020
  • Corollary 2.6
  • Conjecture 2.7
  • Theorem 2.8: FOS
  • ...and 48 more