Abstract
For local non-archimedean fields k of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups GLn(D), SLn(D) (where D is a central division algebra over k) or classical groups, give a new characterisation of the “building set” (defined by DeBacker) of an \(\mathfrak {sl}_{2}(k)\)-triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker’s parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlén, O., Gustafsson, H.P.A., Kleinschmidt, A., Baiying, L., Persson, D.: Fourier coefficients attached to small automorphic representations of \(\text {SL}_{n}(\mathbb {A})\). J. Number Theory 192, 80–142 (2018)
Bernstein, T.: A classification of p-adic quadratic forms, Preprint available at https://bit.ly/2ABZwf9 (2015)
Bourbaki, N.: Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann (1975)
Broussous, P, Lemaire, B: Building of GL(m,D) and centralizers. Transform. Groups 7(1), 15–50 (2002)
Broussous, P., Stevens, S.: Buildings of classical groups and centralizers of Lie algebra elements. J. Lie Theory 19(1), 55–78 (2009)
Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math. 60, 197–376 (1984)
Carter, R.W.: Finite Groups of Lie Type, Wiley Classics Library. Wiley, Chichester (1993). Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication
Christie, A: Fourier Eigenspaces of Waldspurger’s Basis. Preprint. arXiv:1411.1037v2 (2014)
Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)
DeBacker, S: Parametrizing Nilpotent Orbits via Bruhat-Tits Theory. Ann. Math. (2) 156(1), 295–332 (2002)
Diwadkar, J.M.: Nilpotent conjugacy classes in p-adic Lie algebras: The odd orthogonal case. Canad. J. Math. 60(1), 88–108 (2008)
Fintzen, J: On the Moy-Prasad filtration. Preprint. arXiv:1511.00726v3 [math.RT] (2017)
Frechette, S.M., Gordon, J., Robson, L.: Shalika Germs for \(\mathfrak {sl}_{n}\) and \(\mathfrak {sp}_{2n}\) are Motivic, Women in Numbers Europe, Association for Women in Mathematics Series. In: Bertin, M., Bucur, A., Feigon, B., Schneps, L. (eds.) , vol. 2, pp 51–85. Springer, Cham (2015)
Jiang, D., Liu, B.: On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups. J. Number Theory 146, 343–389 (2015)
Lam, T -Y: Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence (2005)
McNinch, G.J.: Optimal SL(2)-homomorphisms. Comment. Math. Helv. 80, 391–426 (2005)
McNinch, G.J.: On the nilpotent orbits of a reductive group over a local field, Preprint, Author’s webpage, https://gmcninch.math.tufts.edu/assets/manuscripts/2019-On-the-nilpotent-orbits-of-a-reductive-group-over-a-local-field.pdf (2018)
Nevins, M.: Admissible nilpotent orbits of real and p-adic split exceptional groups. Represent Theory 6, 160–189 (2002)
Nevins, M.: On nilpotent orbits of SLn and Sp2n over a local non-Archimedean field. Algebr. Represent. Theory 14(1), 161–190 (2011)
Tits, J: Reductive Groups over Local Fields, Automorphic Forms, Representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, pp. 29–69 (1979)
Waldspurger, J.-L.: Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, vol. 269 (2001)
Yap, J.W.: On DeBacker’s parametrization of rational nilpotent orbits of \(\mathbb {O}_{2n}\). Preprint, Overseas UROPS report under the supervision of Jia-Jun Ma. https://www.majiajun.org/pdfs/UROPSR-JitWu.pdf (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Michela Varagnolo
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Monica Nevins’s research is supported by a Discovery Grant from NSERC Canada.
Rights and permissions
About this article
Cite this article
Bernstein, T., Ma, JJ., Nevins, M. et al. Nilpotent Orbits of Orthogonal Groups over p-adic Fields, and the DeBacker Parametrization. Algebr Represent Theor 23, 2033–2058 (2020). https://doi.org/10.1007/s10468-019-09928-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-019-09928-x