Ž. JOURNAL OF ALGEBRA 196, 490498 1997 ARTICLE NO. JA977119 Nlpotent Orbts and Commutatve Elements C. Kenneth Fan* Department of Mathematcs, Harard Unersty, Cambrdge, Massachusetts 02138 and John R. Stembrdge Department of Mathematcs, Unersty of Mchgan, Ann Arbor, Mchgan 48109 Communcated by Georga Benkart Receved Aprl 1, 1996 Let W be a smply laced Coxeter group th generatng set S, and let Wc denote the subset consstng of those elements hose reduced expressons have no substrngs of the form sts for any non-commutng s, t S. We gve a root system characterzaton of W c, and n the case here W corresponds to a fnte Weyl group, sho that Wc s a unon of SpaltenstenSprngerStenberg cells. The latter s vald also for affne Weyl groups of type A, but not for type D or E. 1997 Academc Press 1. INTRODUCTION Let W be a Coxeter group th Ž fnte. generatng set S s 4 I. In the Weyl group case, the commutatve element of W ere defned n F1 to be those elements havng no reduced expresson contanng a substrng of the form sss, here s and s are Ž non-commutng. j j generators such that the smple root correspondng to sj s at least as long as the smple root correspondng to s. The fully commutatve elements of a general Research of the frst author partally supported by a NSF Postdoctoral Felloshp. The frst author thanks George Lusztg thout hom ths paper ould not exst. E-mal address: ckfan@math.harvard.edu. Research of the second author partally supported by NSF Grant DMS-9401575. E-mal address: jrs@math.lsa.umch.edu. 0021-869397 $25.00 Copyrght 1997 by Academc Press All rghts of reproducton n any form reserved. 490
COMMUTATIVE ELEMENTS 491 Coxeter group ere defned n S1 to be those elements havng no reduced expresson contanng a substrng ssss j j of length m 3, here m s the order of ss j n W. In the smply laced case these to defntons agree, snce the product of any par of generators has order 2 or 3, and all roots have the same length. There are numerous characterzatons and propertes of Ž fully. commutatve elements n F1, F3, S1, S2. In ths paper, e extend some prevous characterzatons n F1 for fnte, smply laced Coxeter groups to arbtrary smply laced Coxeter groups. In partcular, n Secton 2, e provde a root system characterzaton of commutatvty. ŽThe specal case correspondng to fnte Weyl groups as frst proved n F1, by a dfferent argument.. Ths can be veed as a generalzaton of the fact that n the symmetrc group, the commutatve elements are the permutatons th no decreasng subsequence of length 3. In Secton 3, e study the relatonshp beteen commutatve elements and certan nlpotent orbts n the assocated Le algebra hen W s a Ž smply laced. fnte or affne Weyl group. In partcular, e obtan that W c s a unon of SpaltenstenSprngerStenberg cells f and only f W s affne of type A, or fnte. 2. ROOT SYSTEM CHARACTERIZATIONS OF W c We assume henceforth that the Coxeter group W s smply laced; thus ssss j j or ssssss j j j for all, j I. Let denote the Coxeter graph correspondng to W,.e., the smple graph th vertex set I and adjacent to j f and only f s and s do not commute. We let W denote the subset j c of W consstng of those elements th no reduced expresson contanng a substrng sss j for any adjacent par, j of. Let V be a vector space over th bass 4, and let ², : denote the symmetrc blnear form on V defned by I 2 f j, ², : j 1 f and j are adjacent n, 0 otherse. The space V carres the reflecton representaton of W; namely, ² : s, for all V, I. Furthermore, ², : s W-nvarant relatve to ths acton.
492 FAN AND STEMBRIDGE Let denote the Ž generalzed. root system generated by the acton of W on ;.e., W, I 4. Every s an nteger lnear combnaton of the smple roots. Let denote the set of postve roots,.e., the set of hose coeffcents relatve to are nonnega- tve. For every root, e have ether or Že.g., H, Sect. 5.4.. We rte 0 and 0 n these cases, respectvely. For W, let Ž. denote the set of roots 0 such that 0. The cardnalty of Ž. s the length l of any reduced expresson s s, also denoted l Ž.. In fact Ž.,..., 4, here 1 l 1 l, s,..., s s. 1 2 l l l l1 l 2 1 We refer to Ž,...,. 1 l as the root sequence of the reduced expresson s s. 1 l We remark that Ž. s bconvex Žcf. Bj, Sect. 3. n the sense that for all, and all ntegers c 1, c2 0 such that c1 c2, e have, Ž. c c Ž. 1 2, Ž. c1c2 Ž.. Ž 2.1. In fact, these convexty propertes characterze the fnte subsets of of Ž. the form for some W. ² : Ž. LEMMA 2.1. We hae, 1 for all,. Proof. If, Ž. are roots such that ², : c2, then the reflecton correspondng to maps to c, a root n the postve lnear span of and. Hence c Ž., by Ž 2.1.. Ho- ² : 2 ever,, c 2 c 2, so teratons of the map Ž,. Ž, c. generate an nfnte sequence n the fnte set Ž.. Gven a root sequence Ž,...,. for, let us partally order Ž. 1 l by takng the transtve closure of the relatons j for all j such that ², : 0. j Ž. PROPOSITION 2.2. The partal orderng of s ndependent of the choce of root sequence f and only f W. c Proof. Any reduced expresson for W can be obtaned from any other by a sequence of brad moves Ž.e., ssssss j j j or ssss, j j accordng to hether and j are adjacent n. B, Sect. IV.1.5. Therefore, f there are no opportuntes to apply brad moves of length three Ž.e., W. c, all reduced expressons for can be generated merely by nterchangng consecutve pars of commutng generators. In the root sequence, these moves correspond to nterchangng consecutve pars of orthogonal roots and clearly have no effect on the partal order.
COMMUTATIVE ELEMENTS 493 On the other hand, f and j are adjacent n, then the root sequences correspondng to the to reduced expressons for x ssssss j j j are Ž,,. and Ž,,. j j j j, and the partal orders are total. It follos that f sss s a substrng of some reduced expresson for Ž j.e., W. c, then there exst root sequences for contanng W-conjugates of these to subsequences, and hence the correspondng partal orders dffer. Remark 2.3. The partal orderng of a root sequence s somorphc to the dual of the heap Žsee S1, Sect. 1. of the correspondng reduced expresson. In partcular, t follos that the extensons of the partal order to a total order are the root sequences that can be generated from the gven root sequence by nterchangng consecutve pars of orthogonal roots. In the follong, let denote the customary partal orderng of n hch henever has nonnegatve coordnates relatve to the smple roots. THEOREM 2.4. For W, the follong are equalent. Ž. a W. c Ž b. ², : 0 for all, Ž.. Ž. c There does not exst a trple,, Ž.. Ž d. The partal orderng of Ž. relate to some Ž equalently, eery. root sequence s consstent th Ž.e., n Ž. mples.. Proof. We demonstrate that the negatons of these propertes are equvalent. ab Ž. Ž.. If Ž. a fals, then has a reduced expresson of the form xssjsy for some adjacent par, j. It follos that the correspondng root sequence ncludes y 1 and y 1 ssy j 1, j for hch ², : ², : j 1. Ž b. Ž c.. If, Ž. are roots such that ², : 0, then ², : 1 by Lemma 2.1. Therefore s a root Žbeng the reflecton of through., and hence by Ž 2.1. must belong to Ž.. cd Ž. Ž.. Every ntal segment of a root sequence s also a root sequence, and hence the subset of Ž. formed by such an ntal segment must satsfy Ž 2.1.. It follos that a set of roots of the form,, Ž. must occur n the order Ž,,. or Ž,,. n every root sequence, and hence also n the correspondng partal order. Hoever, nether of these orderngs s consstent th. Ž d. Ž a.. If Ž d. fals, then there s a root sequence for hose partal order ncludes a coverng relaton that s not consstent th
494 FAN AND STEMBRIDGE ; n partcular,. By choosng a sutable lnear extenson of the partal order, e may obtan a root sequence for n hch and appear consecutvely, and hence y 1, y 1 s j, gven that the correspondng reduced expresson for s of the form xsjsy. Snce ², : 0 Ž otherse could not be a coverng relaton., t follos that ², : ² y 1, y 1 s : ², : j j 1. Hence y 1 j s a root, necessarly postve, snce. Hoever, y 1 j 0 mples that there s a reduced expresson for y that begns th s Že.g., H, Sect. 5.4. j. Hence there s a reduced expresson for contanng the substrng sss, j j and W c. Remark 2.5. The prevous result can be veed as a generalzaton of the fact that the commutatve elements of the symmetrc group Sn are the permutatons Ž,...,. of 1,..., n4 1 n that do not contan a decreasng subsequence of length 3. Indeed, usng 1 j n4 j as the set of postve roots for A n1, one sees that the trples of postve roots of the form,, are j, k j, k, here 1 j kn. Havng such a trple occur n Ž. s equvalent to havng j. A smlar descrpton can be provded n type D; see F1, Sect. 7 k or S2, Sect. 10. 3. CELLS No suppose that W s the Weyl group of a semsmple, smply laced, smply connected algebrac group G over th Le algebra. We may assume that s the root system of relatve to some choce of a Cartan subalgebra, and that s the Borel subalgebra correspondng to the chosen smple roots. Let be the nlpotent radcal of, and defne, here 0 0 0 0 denotes the longest element of W. For W, set 0 0 1. Let N be the subvarety of nlpotent elements n, and let NG denote the G-orbts of such elements. Follong Spaltensten, Sprnger, and Stenberg Ž et al.., one may defne a map : W NG by takng Ž. to Ž. be the unque nlpotent orbt O such that O 0 s dense n. Ž Ths dffers from the standard defnton by a factor of.. 0 0 The fbers of are cells. We no pass to analogous structures for the affne Weyl group W. ˆ It should be noted that Wˆ s also smply laced Ž n the sense of Secton 2. unless W s of type A 1. In ths exceptonal case, e can mantan the valdty of Theorem 2.4 by defnng Wˆ W. ˆ c
COMMUTATIVE ELEMENTS 495 Let Gˆ GF, Ž. here F ŽŽ t... The abstract root system ˆ generated by Wˆ Ž n the sense of Secton 2. can be dentfed th the real roots of the Le algebra F. Let ˆ be the Iahor subalgebra hch sts n t as the nverse mage of relatve to the canoncal projecton t 0, and let ˆ ˆ be the nverse mage of relatve to the same projecton. Let ˆ 0 be the nverse mage of relatve to the canoncal projecton t 1 0 ˆ 1 defned by t. For W, e set ˆ0 ˆ0. For further detals on ths setup, see KL, Sect. 10. Each nlpotent orbt O n NG also ndexes a GF-orbt Ž. O, ˆ here F denotes the algebrac closure of F. Follong Lusztg L, e may defne a map : ˆ Wˆ NG by takng ˆŽ. to be the Ž unque. nlpotent orbt O ˆ such that O ˆ0 ˆ s dense n ˆ0. ˆ Note that ˆ0 ˆ s fnte-dmensonal over ; n fact, t s spanned by the root spaces ndexed by ˆ Ž 1.. Let N n N adž n. 4 04 4, and let N4G denote the nlpotent orbts n N 4. THEOREM 3.1. We hae Ž. 1 a W Ž N G. c 4. Ž b. Wˆ ˆ1 Ž N G. c 4, th equalty f and only f W s of type A. Let E denote a generator for the root space correspondng to. ˆ ˆ Ž. 4 c ˆ0 ˆ LEMMA 3.2. For W, there exsts n such that ad n 0. ˆ 1 Proof. Gven that W Ž and hence W ˆ. c c, Theorem 2.4 mples ˆŽ 1 that there s a trple,,.. Ths gven, e take n E E ˆ 0. ˆ Snce n s the regular element of an 3 subalgebra, t follos that adž n. 4 0 Že.g., see K.. Alternatvely, one can drectly compute adž n. 4 E and verfy that t s a non-zero multple of E. LEMMA 3.3. In an affne root system of type D or E, there exsts a quadruple of orthogonal smple roots 1,...,4 and a root such that ², : 1 for all. Proof. For type D, e may take 1,...,4 to be the smple roots correspondng to the four end nodes of, and to be the sum of the remanng smple roots. For E, use I 0, 1,..., m 4 m, th the ndexng arranged so that 4 labels the node of degree three, 0 labels the node correspondng to the hghest root, and 1, 3, 4,..., m 4 labels a path n. In E t suffces to take,..., 4,,, 4 6 1 4 0 1 4 6 and 2 ; n E,,..., 4,,, 4 3 4 5 7 1 4 0 3 5 7 and 1 2 2 ;n E,,..., 4,,, 4 3 4 5 6 8 1 4 0 2 5 7 and 1 2233425267 8.
496 FAN AND STEMBRIDGE Proof of Theorem 3.1. Lemma 3.2 mples that ˆ1 Ž N G. W ˆ 4 c, and 1 essentally the same argument proves Ž N G. 4 W. To prove the ˆ reverse nclusons, t ould suffce to sho that for Wc and n ˆ 0 Ž ˆ resp., W and n. c 0, e have n N 4. Snce any n ˆ 0 ˆ s a lnear combnaton of those E such that ˆŽ 1. Ž. 4, t follos that ad n s a lnear combnaton of monomals of the form Ž. Ž. Ž. Ž. M ad E ad E ad E ad E, Ž 3.1. 1 2 3 4 ˆŽ 1 here,...,. 1 4. If adž n. 4 0, at least one such monomal must be non-zero. Let us therefore suppose MŽ E. 0 for some ˆ 0 4, follong the conventon that E0 represents an arbtrary member of F. Settng Ý, t s clearly necessary that ˆ 0 4. Furthermore, ²,: ², : 82 ², : 2 ², : Ý Ý Ý j j ², : 82 ², :, Ž 3.2. snce ², : j 0 by Theorem 2.4. If 0, ths mples ², : 8, hch s mpossble. Hence ˆ and ², : 2. Snce ², : s postve semdefnte, t follos that ², : 2 for all.if², : 2, then ould belong to the radcal of ², : 1 1, and therefore ² : ² :,, 6, 2 3 4 2 3 4 a contradcton. Thus ², : 1 and the bound mpled by Ž 3.2. yelds ²,: 2. Ths bound s tght, so equalty occurs n Ž 3.2.; n partcular, the must be parse orthogonal and ², : 1 for all. Conversely, n any such confguraton of roots, e have ², : 1 4 1, so ˆ for all, and hence MŽ E. 4 0. Furthermore, f e set n E E, then the above analyss shos that every term 1 4 n the expanson of adž n. 4 s 0 except for the 24 monomals that correspond to selectng a permutaton of Ž 3.1.. Hoever, adž E. and adž E. j commute parse for j, soadn Ž. 4 24M 0. If W s of type A, e clam that there can be no confguraton,,..., ˆ 1 4 as above. Indeed, snce the nner products among these roots concde th those formed by the smple roots of an affne system of type D, they generate ether a fnte or affne subsystem of type D n, ˆ 4 4 accordng to hether s n the lnear span of 1,..., 4. In ether case,
COMMUTATIVE ELEMENTS 497 modulo the radcal of ², :, e ould have an embeddng of a fnte root system of type D4 n a fnte root system of type A, hch s mpossbleevery rreducble subsystem n type A s also of type A. If W s of type D or E, Lemma 3.3 mples that there s a sutable confguraton of roots, 1,...,4 n hch the are smple. If e take to be the product of the smple reflectons correspondng to the,ts 1 clear that Wˆ and ˆŽ.,..., 4 c 1 4. Hence there exsts nˆ such that adž n. 4 0, and the ncluson n Ž b. 0 ˆ s proper. Turnng no to Ž. a, the above reasonng also proves that for W c and n Ž. 0, e have ad n 4 0 unless there exst parse orthogonal roots,..., Ž. and satsfyng ², : 1 4 1 for all. Hoever, n ths case ², : s postve defnte, so, 1,...,4 must generate a fnte root system of type D 4. Settng, e can choose an orthogonal bass 1,...,4 for the span of so that 1 j 44 j. There are three quadruples of parse orthogonal roots n ; namely,, 4 j k l, here, j 4, k, l44 ranges over the three parttons of 1,..., 44 nto doubletons. We clam that f any of these confguratons occurs n Ž., then there ould exst a root Ž. such that ², : 1 for some, contradctng the fact that W Ž cf. Theorem 2.4. c. If, Ž. 1 2 3 4, then the decomposton 1 2 Ž. Ž. together th the convexty propertes of Ž 2.1. 1 4 2 4 mply or Ž.. Hoever, ², : 1 4 2 4 1 4 3 4 1 and ², : 2 4 3 4 1, so both cases lead to a contradcton. Smlarly, f, Ž. 1 3 2 4, then the decomposton 1 Ž. Ž. 3 1 2 2 3 and convexty together mply 1 2 or 2 Ž.. Hoever, ², : 1 and ², : 3 1 2 2 4 2 3 1 3 1, so agan both cases yeld contradctons. Fnally, f, Ž. 1 4 2 3, then the decomposton 2 3 Ž. Ž. and convexty mply or Ž. 2 4 3 4 2 4 3 4. Hoever, ², : 1 and ², : 2 4 1 4 3 4 1 4 1, so both cases yeld contradctons. Remark 3.4. Ž. a Part Ž. a of Theorem 3.1 as frst proved n F1, Sect. 7 and used there to determne the longest elements n Wc for fnte smply laced Weyl groups W. Ž b. If W s of type D or E, Theorem 3.1Ž b. mples that there exsts Wˆ such that ˆŽ. c O N4G. On the other hand, t s knon that 1 every fber of the map s non-empty, so Ž O. must contan elements not n W, by Theorem 3.1Ž. a. Snce and ˆ c commute th the natural embeddng of W n W, ˆ t follos that n these cases, Wˆc s not a unon of cells.
498 FAN AND STEMBRIDGE Ž. c From the tables n C, t can be shon that for E8 there are exactly fve nlpotent orbts n N 4. On the other hand, t s knon that there are only fntely many commutatve elements n the affne Weyl group Eˆ F1, Sect. 3. Ž In fact, there are exactly 44,199 such elements.. 8 Thus Eˆ 8 has at least fve fnte cells. Ž d. The members of N4 are precsely the sphercal nlpotents as classfed by Panyushev P. ŽA nlpotent element s sphercal f ts orbt under the acton of some Borel subgroup s dense n ts G-orbt.. Panyushev s classfcaton s acheved on a case-by-case bass; t s possble that further analyss of the fbers of ll lead to a unform proof, at least n the smply laced cases. REFERENCES Bj A. Bjorner, Orderngs of Coxeter groups, Contemp. Math. 34 Ž 1984., 175195. B N. Bourbak, Groupes et Algebres ` de Le, Chaps. IVVI, Masson, Pars, 1981. C R. W. Carter, Fnte Groups of Le Type, Wley, Ne York, 1985. F1 C. K. Fan, A Hecke Algebra Quotent and Propertes of Commutatve Elements of a Weyl Group, Ph.D. thess, MIT, 1995. F2 C. K. Fan, A Hecke algebra quotent and some combnatoral applcatons, J. Algebrac Combn. 5, Ž 1996., 175189. F3 C. K. Fan, Schubert varetes and short bradedness, preprnt. H J. E. Humphreys, Reflecton Groups and Coxeter Groups, Cambrdge Unv. Press, Cambrdge, 1990. KL D. Kazhdan and G. Lusztg, Fxed pont varetes on affne flag manfolds, Israel J. Math. 62 Ž 1988., 129168. K B. Kostant, The prncpal three-dmensonal subgroup and the Bett numbers of a complex Le group, Amer. J. Math. 81 Ž 1959., 9731032. L G. Lusztg, Affne Weyl groups and conjugacy classes n Weyl groups, Transform. Groups 1 Ž 1996., 8397. P D. I. Panyushev, Complexty and nlpotent orbts, Manuscrpta Math. 83 Ž 1994., 223237. S1 J. R. Stembrdge, On the fully commutatve elements of Coxeter groups, J. Algebrac Combn. 5 Ž 1996., 353385. S2 J. R. Stembrdge, Some combnatoral aspects of reduced ords n fnte Coxeter groups, Trans. Amer. Math. Soc. 349 Ž 1997., 12851332.