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DOI :10.26650/ijmath.2024.00016   IUP :10.26650/ijmath.2024.00016    Full Text (PDF)

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

Kazım İlhan İkeda

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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References

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APA

İkeda, K.İ. (2024). Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞). Istanbul Journal of Mathematics, 2(2), 55-69. https://doi.org/10.26650/ijmath.2024.00016


AMA

İkeda K İ. Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞). Istanbul Journal of Mathematics. 2024;2(2):55-69. https://doi.org/10.26650/ijmath.2024.00016


ABNT

İkeda, K.İ. Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞). Istanbul Journal of Mathematics, [Publisher Location], v. 2, n. 2, p. 55-69, 2024.


Chicago: Author-Date Style

İkeda, Kazım İlhan,. 2024. “Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞).” Istanbul Journal of Mathematics 2, no. 2: 55-69. https://doi.org/10.26650/ijmath.2024.00016


Chicago: Humanities Style

İkeda, Kazım İlhan,. “Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞).” Istanbul Journal of Mathematics 2, no. 2 (Feb. 2025): 55-69. https://doi.org/10.26650/ijmath.2024.00016


Harvard: Australian Style

İkeda, Kİ 2024, 'Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞)', Istanbul Journal of Mathematics, vol. 2, no. 2, pp. 55-69, viewed 5 Feb. 2025, https://doi.org/10.26650/ijmath.2024.00016


Harvard: Author-Date Style

İkeda, K.İ. (2024) ‘Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞)’, Istanbul Journal of Mathematics, 2(2), pp. 55-69. https://doi.org/10.26650/ijmath.2024.00016 (5 Feb. 2025).


MLA

İkeda, Kazım İlhan,. “Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞).” Istanbul Journal of Mathematics, vol. 2, no. 2, 2024, pp. 55-69. [Database Container], https://doi.org/10.26650/ijmath.2024.00016


Vancouver

İkeda Kİ. Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞). Istanbul Journal of Mathematics [Internet]. 5 Feb. 2025 [cited 5 Feb. 2025];2(2):55-69. Available from: https://doi.org/10.26650/ijmath.2024.00016 doi: 10.26650/ijmath.2024.00016


ISNAD

İkeda, Kazımİlhan. “Trace formula for finite groups and the Macdonald correspondence for GL𝑛(F𝑞)”. Istanbul Journal of Mathematics 2/2 (Feb. 2025): 55-69. https://doi.org/10.26650/ijmath.2024.00016



TIMELINE


Submitted25.06.2024
Accepted17.12.2024
Published Online31.12.2024

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