## Elliptic centralizers in Weyl groups and their coinvariant representations

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- by Mark Reeder PDF
- Represent. Theory
**15**(2011), 63-111 Request permission

## Abstract:

The centralizer $C(w)$ of an elliptic element $w$ in a Weyl group has a natural symplectic representation on the group of $w$-coinvariants in the root lattice. We give the basic properties of this representation, along with applications to $p$-adic groups—classifying maximal tori and computing inducing data in $L$-packets—as well as to elucidating the structure of the centralizer $C(w)$ itself. We give the structure of each elliptic centralizer in $W(E_8)$ in terms of its coinvariant representation, and we refine Springer’s theory for elliptic regular elements to give explicit complex reflections generating $C(w)$. The case where $w$ has order three is examined in detail, with connections to mathematics of the nineteenth century. A variation of the methods recovers the subgroup $W(H_4)\subset W(E_8)$.## References

- Jeffrey D. Adler,
*Refined anisotropic $K$-types and supercuspidal representations*, Pacific J. Math.**185**(1998), no. 1, 1–32. MR**1653184**, DOI 10.2140/pjm.1998.185.1 - Eva Bayer-Fluckiger and Ivan Suarez,
*Ideal lattices over totally real number fields and Euclidean minima*, Arch. Math. (Basel)**86**(2006), no. 3, 217–225. MR**2215310**, DOI 10.1007/s00013-005-1469-9 - Nicolas Bourbaki,
*Lie groups and Lie algebras. Chapters 4–6*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR**1890629**, DOI 10.1007/978-3-540-89394-3 - Egbert Brieskorn and Horst Knörrer,
*Ebene algebraische Kurven*, Birkhäuser Verlag, Basel-Boston, Mass., 1981 (German). MR**646612** - Roger W. Carter,
*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307** - R. W. Carter,
*Conjugacy classes in the Weyl group*, Compositio Math.**25**(1972), 1–59. MR**318337** - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - J. H. Conway and N. J. A. Sloane,
*Sphere packings, lattices and groups*, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR**1662447**, DOI 10.1007/978-1-4757-6568-7 - John H. Conway and Derek A. Smith,
*On quaternions and octonions: their geometry, arithmetic, and symmetry*, A K Peters, Ltd., Natick, MA, 2003. MR**1957212**, DOI 10.1201/9781439864180 - H. S. M. Coxeter,
*The polytope $2_{21}$, whose twenty-seven vertices correspond to the lines on the general cubic surface*, Amer. J. Math.**62**(1940), 457–486. MR**2180**, DOI 10.2307/2371466 - H. S. M. Coxeter,
*Regular polytopes*, 3rd ed., Dover Publications, Inc., New York, 1973. MR**0370327** - Stephen DeBacker,
*Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory*, Michigan Math. J.**54**(2006), no. 1, 157–178. MR**2214792**, DOI 10.1307/mmj/1144437442 - Stephen DeBacker and Mark Reeder,
*Depth-zero supercuspidal $L$-packets and their stability*, Ann. of Math. (2)**169**(2009), no. 3, 795–901. MR**2480618**, DOI 10.4007/annals.2009.169.795 - P. Deligne and G. Lusztig,
*Representations of reductive groups over finite fields*, Ann. of Math. (2)**103**(1976), no. 1, 103–161. MR**393266**, DOI 10.2307/1971021 - Paul Gérardin,
*Weil representations associated to finite fields*, J. Algebra**46**(1977), no. 1, 54–101. MR**460477**, DOI 10.1016/0021-8693(77)90394-5 - Benedict H. Gross and Mark Reeder,
*From Laplace to Langlands via representations of orthogonal groups*, Bull. Amer. Math. Soc. (N.S.)**43**(2006), no. 2, 163–205. MR**2216109**, DOI 10.1090/S0273-0979-06-01100-1 - Benedict H. Gross and Mark Reeder,
*Arithmetic invariants of discrete Langlands parameters*, Duke Math. J.**154**(2010), no. 3, 431–508. MR**2730575**, DOI 10.1215/00127094-2010-043 - B. H. Gross, M. Reeder, J.-K. Yu, work in progress
- Ryoshi Hotta and Kiyoshi Matsui,
*On a lemma of Tate-Thompson*, Hiroshima Math. J.**8**(1978), no. 2, 255–268. MR**486178** - V. Kac
*Infinite dimensional Lie algebras*, third ed., Cambridge, 1995. - David Kazhdan and Yakov Varshavsky,
*Endoscopic decomposition of characters of certain cuspidal representations*, Electron. Res. Announc. Amer. Math. Soc.**10**(2004), 11–20. MR**2048427**, DOI 10.1090/S1079-6762-04-00125-8 - Robert E. Kottwitz,
*Stable trace formula: cuspidal tempered terms*, Duke Math. J.**51**(1984), no. 3, 611–650. MR**757954**, DOI 10.1215/S0012-7094-84-05129-9 - Robert E. Kottwitz,
*Stable trace formula: elliptic singular terms*, Math. Ann.**275**(1986), no. 3, 365–399. MR**858284**, DOI 10.1007/BF01458611 - Robert E. Kottwitz,
*Isocrystals with additional structure. II*, Compositio Math.**109**(1997), no. 3, 255–339. MR**1485921**, DOI 10.1023/A:1000102604688 - Heinrich Maschke,
*Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen*, Math. Ann.**33**(1889), no. 3, 317–344 (German). MR**1510546**, DOI 10.1007/BF01443964 - R. V. Moody and J. Patera,
*Quasicrystals and icosians*, J. Phys. A**26**(1993), no. 12, 2829–2853. MR**1236147**, DOI 10.1088/0305-4470/26/12/022 - M. S. Raghunathan,
*Tori in quasi-split-groups*, J. Ramanujan Math. Soc.**19**(2004), no. 4, 281–287. MR**2125504** - Michael Rapoport,
*A guide to the reduction modulo $p$ of Shimura varieties*, Astérisque**298**(2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR**2141705** - Mark Reeder,
*Level-two structure of simply-laced Coxeter groups*, J. Algebra**285**(2005), no. 1, 29–57. MR**2119103**, DOI 10.1016/j.jalgebra.2004.11.010 - Mark Reeder,
*Supercuspidal $L$-packets of positive depth and twisted Coxeter elements*, J. Reine Angew. Math.**620**(2008), 1–33. MR**2427973**, DOI 10.1515/CRELLE.2008.046 - —,
*Torsion automorphisms of simple Lie algebras*, L’Ens. Math., to appear. - Jean-Pierre Serre,
*Galois cohomology*, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR**1867431** - G. C. Shephard and J. A. Todd,
*Finite unitary reflection groups*, Canad. J. Math.**6**(1954), 274–304. MR**59914**, DOI 10.4153/cjm-1954-028-3 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**, DOI 10.1007/978-1-4757-1920-8 - T. A. Springer,
*Regular elements of finite reflection groups*, Invent. Math.**25**(1974), 159–198. MR**354894**, DOI 10.1007/BF01390173 - T. A. Springer and R. Steinberg,
*Conjugacy classes*, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR**0268192** - Bhama Srinivasan,
*The characters of the finite symplectic group $\textrm {Sp}(4,\,q)$*, Trans. Amer. Math. Soc.**131**(1968), 488–525. MR**220845**, DOI 10.1090/S0002-9947-1968-0220845-7 - J. Tits,
*Sur les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples*, Inst. Hautes Études Sci. Publ. Math.**31**(1966), 21–58 (French). MR**214638**, DOI 10.1007/BF02684801 - È. B. Vinberg,
*The Weyl group of a graded Lie algebra*, Izv. Akad. Nauk SSSR Ser. Mat.**40**(1976), no. 3, 488–526, 709 (Russian). MR**0430168**

## Additional Information

**Mark Reeder**- Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
- Email: reederma@bc.edu
- Received by editor(s): June 9, 2009
- Received by editor(s) in revised form: February 3, 2010
- Published electronically: January 24, 2011
- © Copyright 2011 American Mathematical Society
- Journal: Represent. Theory
**15**(2011), 63-111 - MSC (2010): Primary 11E72, 20G05, 20G25
- DOI: https://doi.org/10.1090/S1088-4165-2011-00377-0
- MathSciNet review: 2765477