On matrix Kloosterman sums and Hall–Littlewood polynomials

Zelingher, Elad

doi:10.1090/tran/9363

On matrix Kloosterman sums and Hall–Littlewood polynomials
HTML articles powered by AMS MathViewer

by Elad Zelingher;

Trans. Amer. Math. Soc. 378 (2025), 3597-3623

DOI: https://doi.org/10.1090/tran/9363

Published electronically: December 27, 2024

HTML | PDF | Request permission

Abstract:

We prove an identity relating twisted matrix Kloosterman sums to modified Hall–Littlewood polynomials evaluated at the roots of the characteristic polynomial associated to a twisted Kloosterman sheaf. This solves a conjecture of Erdélyi and Tóth [Algebra Number Theory 18 (2024), pp. 2247–2308, Conjecture 5.9].

References

Yuanqing Cai, Solomon Friedberg, and Eyal Kaplan, Doubling constructions: global functoriality for non-generic cuspidal representations, Ann. of Math. (2) 200 (2024), no. 3, 893–966. MR 4817565, DOI 10.4007/annals.2024.200.3.2
O. Carmon, Shalika vectors for irreducible representations of $\mathrm {GL}_n$ over a finite field, Master’s thesis, Tel Aviv University, 2023. DaTA:9933623517704146.
O. Carmon and E. Zelingher, On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups, arXiv:2406.14262, 2024.
Hi-joon Chae and Dae San Kim, $L$ functions of some exponential sums of finite classical groups, Math. Ann. 326 (2003), no. 3, 479–487. MR 1992273, DOI 10.1007/s00208-003-0431-z
Charles W. Curtis and Ken-ichi Shinoda, Unitary Kloosterman sums and the Gelfand-Graev representation of $\textrm {GL}_2$, J. Algebra 216 (1999), no. 2, 431–447. MR 1693005, DOI 10.1006/jabr.1998.7807
Charles W. Curtis and Ken-ichi Shinoda, Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 121–139. MR 2074592, DOI 10.2969/aspm/04010121
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
J. Désarménien, B. Leclerc, and J.-Y. Thibon, Hall-Littlewood functions and Kostka-Foulkes polynomials in representation theory, Sém. Lothar. Combin. 32 (1994), Art. B32c, approx. 38 (English, with English and French summaries). MR 1399504
J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. MR 684172, DOI 10.1007/BF01390728
Márton Erdélyi, Will Sawin, and Árpád Tóth, The purity locus of matrix Kloosterman sums, Trans. Amer. Math. Soc. 377 (2024), no. 6, 4117–4132. MR 4748616, DOI 10.1090/tran/9149
Márton Erdélyi and Árpád Tóth, Matrix Kloosterman sums, Algebra Number Theory 18 (2024), no. 12, 2247–2308. MR 4817460, DOI 10.2140/ant.2024.18.2247
M. Erdélyi, Á. Tóth, and G. Zábrádi, Matrix Kloosterman sums modulo prime powers, Math. Z. 306 (2024), no. 4, Paper No. 68, 21. MR 4717900, DOI 10.1007/s00209-024-03467-y
Lei Fu and Daqing Wan, $L$-functions for symmetric products of Kloosterman sums, J. Reine Angew. Math. 589 (2005), 79–103. MR 2194679, DOI 10.1515/crll.2005.2005.589.79
J. Fulman, A new bound for Kloosterman sums, arXiv:0105172, 2001.
S. I. Gel’fand, Representations of the full linear group over a finite field, Math. USSR Sb. 12 (1970), no. 13, 13–39.
Graham Gordon, Cycle type factorizations in $\textrm {GL}_n\Bbb F_q$, Algebr. Comb. 5 (2022), no. 6, 1427–1459. MR 4535800, DOI 10.5802/alco.259
Ofir Gorodetsky and Brad Rodgers, Traces of powers of matrices over finite fields, Trans. Amer. Math. Soc. 374 (2021), no. 7, 4579–4638. MR 4273172, DOI 10.1090/tran/8337
J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447. MR 72878, DOI 10.1090/S0002-9947-1955-0072878-2
Shamgar Gurevich and Roger Howe, Harmonic analysis on $GL_n$ over finite fields, Pure Appl. Math. Q. 17 (2021), no. 4, 1387–1463. MR 4359264, DOI 10.4310/PAMQ.2021.v17.n4.a7
Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39–111. MR 2051783
Jochen Heinloth, Bao-Châu Ngô, and Zhiwei Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), no. 1, 241–310. MR 2999041, DOI 10.4007/annals.2013.177.1.5
John H. Hodges, Weighted partitions for general matrices over a finite field, Duke Math. J. 23 (1956), 545–552. MR 79607
Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
Dae San Kim, Gauss sums for symplectic groups over a finite field, Monatsh. Math. 126 (1998), no. 1, 55–71. MR 1633263, DOI 10.1007/BF01312455
Anatol N. Kirillov, New combinatorial formula for modified Hall-Littlewood polynomials, $q$-series from a contemporary perspective (South Hadley, MA, 1998) Contemp. Math., vol. 254, Amer. Math. Soc., Providence, RI, 2000, pp. 283–333. MR 1768934, DOI 10.1090/conm/254/03959
Takeshi Kondo, On Gaussian sums attached to the general linear groups over finite fields, J. Math. Soc. Japan 15 (1963), 244–255. MR 161914, DOI 10.2969/jmsj/01530244
Emmanuel Kowalski, Philippe Michel, and Will Sawin, Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2) 186 (2017), no. 2, 413–500. MR 3702671, DOI 10.4007/annals.2017.186.2.2
I. G. Macdonald, Zeta functions attached to finite general linear groups, Math. Ann. 249 (1980), no. 1, 1–15. MR 575444, DOI 10.1007/BF01387076
I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
S. Ram, Subspace profiles over finite fields and $q$-Whittaker expansions of symmetric functions, arXiv:2309.16607, 2023.
Allan J. Silberger and Ernst-Wilhelm Zink, The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set, Trans. Amer. Math. Soc. 352 (2000), no. 7, 3339–3356. MR 1650042, DOI 10.1090/S0002-9947-00-02454-5
Zhiwei Yun, Epipelagic representations and rigid local systems, Selecta Math. (N.S.) 22 (2016), no. 3, 1195–1243. MR 3518549, DOI 10.1007/s00029-015-0204-z
Elad Zelingher, On values of the Bessel function for generic representations of finite general linear groups, Adv. Math. 434 (2023), Paper No. 109314, 47. MR 4647744, DOI 10.1016/j.aim.2023.109314

AMS Website Down

Transactions of the American Mathematical Society

On matrix Kloosterman sums and Hall–Littlewood polynomials
HTML articles powered by AMS MathViewer

Abstract: