Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞)

Abstract

Let 𝐺 be a finite group. The trace formula for 𝐺, which is the trivial case of the Arthur trace formula, is well known with many applications. In this note, we further consider a subgroup Γ of 𝐺 and a representation 𝜌 : Γ → GL(𝑉𝜌) of Γ on a finite dimensional C-vector space𝑉𝜌, and compute the trace Tr(Ind𝐺 Γ (𝜌) ( 𝑓 )) of the operator Ind𝐺 Γ 𝜌( 𝑓 ) : Ind𝐺 Γ (𝑉𝜌) → Ind𝐺 Γ (𝑉𝜌) for any function 𝑓 : 𝐺 → C in two different ways. The expressions for Tr(Ind𝐺 Γ (𝜌) ( 𝑓 )) denoted by 𝐽 (𝜌, 𝑓 ) and 𝐼(𝜌, 𝑓 ) are the spectral side and the geometric side of the trace formula for Tr(Ind𝐺 Γ (𝜌) ( 𝑓 )), respectively. The identity 𝐽 (𝜌, 𝑓 ) = Tr(Ind𝐺 Γ (𝜌) ( 𝑓 )) = 𝐼(𝜌, 𝑓 ) is a generalization of the trace formula for the finite group 𝐺. This theory is then applied to the “automorphic side” of the Macdonald correspondence for GL𝑛 (F𝑞); namely, to the “automorphic side” of the local 0-dimensional Langlands correspondence for GL(𝑛), where new identities are obtained for the 𝜖-factors of representations of GL𝑛 (F𝑞).

Keywords

• Trace formula
• finite group
• Macdonald correspondence

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