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

Year 2024, Volume: 2 Issue: 2, 55 - 69, 31.12.2024

Kazım İlhan İkeda

https://doi.org/10.26650/ijmath.2024.00016

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

References

• Arthur, J., 2005, An introduction to the trace formula. Clay Math. Proc. 4, 1-263. google scholar
• Bump, D., 2013, Lie Groups, 2nd edition. Graduate Texts in Mathematics, Springer-Verlag, New York. google scholar
• Carter, R. W., 1993, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester. google scholar
• Chasek, M. D., 2023, A Vector-valued Trace Formula for Finite Groups. M.A. Thesis, The University of Maine, US. google scholar
• Green, J. A., 1955, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80, 402-447. google scholar
• Lee, S., 2022, Trace formula for finite groups, https://seewoo5.github.io/jekyll/update/2022/11/18/finite-group-trace-formula.html. google scholar
• Macdonald, I. G., 1980, Zeta functions attached to finite general linear group. Math. Ann. 249, 1-15. google scholar
• Piatetski-Shapiro, I., 1983, Complex Representations of GL(2, k) for Finite Fields K. Contemp. Math. 16. google scholar
• Serre, J. P., 1972, Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer-Verlag, New York and Heidelberg. google scholar
• Silberger A. andZink, E. W., 2008, Explicit Shintani base change and the Macdonald correspondence for characters of GL„ (k). Journal of Algebra 319, 4147-4176. google scholar
• Terras, A., 1999, Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts 43, Cambridge University Press, Cambridge. google scholar
• Vogan, D., 2020, Local Langlands conjecture for finite groups of Lie type. Joint Mathematics Meetings AMS-MAA, Denver, 15-18. google scholar
• Yang J. H, 2006, Trace formulas on finite groups. Commun. Korean Math. Soc. 21, 17-25. google scholar
• Ye, R. and Zelingher, E., 2021, Epsilon factors of representations of finite general linear groups. Journal of Number Theory 221, 122-142. google scholar