[Paper Review] Existence of cuspidal representations of p-adic reductive groups

Arno Kret|arXiv (Cornell University)|May 12, 2012

Advanced Algebra and Geometry1 references3 citations

TL;DR

This paper proves the existence of cuspidal complex representations for any reductive group over a non-Archimedean local field, using Deligne-Lusztig theory to reduce the problem to finding characters in general position on elliptic maximal tori. The key result confirms a long-standing folklore conjecture by constructing such representations via geometric and group-theoretic methods in finite groups of Lie type.

ABSTRACT

We prove that any reductive group G over a non-Archimedean local field has a cuspidal complex representation.

Motivation & Objective

• To establish the existence of cuspidal complex representations for any reductive group over a non-Archimedean local field, a result long considered folklore but lacking a published proof.
• To reduce the problem to finite groups of Lie type by analyzing the reductive quotient of a maximal parahoric subgroup.
• To verify the existence of characters in general position on elliptic maximal tori for all classical and exceptional root systems over finite fields.
• To complete the proof by confirming such characters exist for split and non-split classical groups, including types $^2A_n$, $^2D_n$, and $^3D_4$.
• To resolve potential obstructions in small residue fields with large Weyl groups by explicitly constructing suitable characters in general position.

Proposed method

• Apply Deligne-Lusztig theory to associate virtual characters $R_T^ heta$ to rational characters $\theta$ on tori $T$ in finite groups of Lie type.
• Use the criterion that $(-1)^{\sigma(G)-\sigma(T)}R_T^\theta$ lifts to an irreducible representation $\pi_T^\theta$ if $\theta$ is in general position, i.e., the Weyl group acts freely on $\theta$.
• Prove that if $T$ is elliptic and $\theta$ is in general position, then $\pi_T^\theta$ is cuspidal.
• Reduce the problem to checking the existence of such characters $\theta$ on elliptic maximal tori in finite groups of Lie type.
• Analyze the rational Weyl group $W_T(k)$ via the action of Frobenius and the Weyl group $W_0$, using the map $\varphi: W_0 \to \mathfrak{S}_n$ to determine fixed points.
• Verify that the element $2e_n$ modulo $(w\Phi - 1)\Lambda$ is in general position by showing that $2e_n \pm 2e_r$ is not in the image of $w\Phi - 1$, ensuring the character is in general position.

Experimental results

Research questions

• RQ1 Does every reductive group over a non-Archimedean local field admit a cuspidal complex representation?
• RQ2 Can the existence of such representations be reduced to the existence of characters in general position on elliptic maximal tori in finite groups of Lie type?
• RQ3 Are there always characters in general position on elliptic tori for classical groups over finite fields, even in small residue fields?
• RQ4 How does the rational Weyl group action affect the existence of characters in general position for non-split groups like $^2D_n$?
• RQ5 Can the obstruction arising from small residue fields and large Weyl groups be overcome by explicit construction of such characters?

Key findings

• The paper establishes that every reductive group $G$ over a non-Archimedean local field $F$ admits a cuspidal complex representation, confirming a folklore result.
• For finite groups of Lie type, the existence of cuspidal representations is equivalent to the existence of characters in general position on elliptic maximal tori.
• In all exceptional root systems ($E_6$, $E_7$, $E_8$, $F_4$, $G_2$, etc.), Carter’s character tables show a positive number of cuspidal unipotent characters, confirming existence.
• For split classical groups ($A_n$, $B_n$, $C_n$, $D_n$), the existence of characters in general position is established via analysis of the rational Weyl group and the action of Frobenius.
• For non-split groups ($^2A_n$, $^2D_n$), the existence of such characters is confirmed by showing that $2e_n$ is not in the image of $w\Phi - 1$, ensuring general position.
• The proof is completed by verifying that $2e_n \pm 2e_r \notin (w\Phi - 1)\Lambda$ for all $r$, which implies that the character is in general position even in small fields with large Weyl groups.