2754:
1606:
2212:
1953:
1261:
2749:{\displaystyle {\begin{aligned}152q^{22}&+3,472q^{21}+38,791q^{20}+293,021q^{19}+1,370,892q^{18}+4,067,059q^{17}+7,964,012q^{16}\\&+11,159,003q^{15}+11,808,808q^{14}+9,859,915q^{13}+6,778,956q^{12}+3,964,369q^{11}+2,015,441q^{10}\\&+906,567q^{9}+363,611q^{8}+129,820q^{7}+41,239q^{6}+11,426q^{5}+2,677q^{4}+492q^{3}+61q^{2}+3q\end{aligned}}}
616:
155:). The representation had two natural bases, and the transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actual Kazhdan–Lusztig construction of their polynomials is more elementary. Kazhdan and Lusztig used this to construct a canonical basis in the
3527:
was already known from the interpretation of coefficients of the
Kazhdan–Lusztig polynomials as the dimensions of intersection cohomology groups, irrespective of the conjectures. Conversely, the relation between Kazhdan–Lusztig polynomials and the Ext groups theoretically can be used to prove the
1601:{\displaystyle R_{x,y}={\begin{cases}0,&{\mbox{if }}x\not \leq y\\1,&{\mbox{if }}x=y\\R_{sx,sy},&{\mbox{if }}sx<x{\mbox{ and }}sy<y\\R_{xs,ys},&{\mbox{if }}xs<x{\mbox{ and }}ys<y\\(q-1)R_{sx,y}+qR_{sx,sy},&{\mbox{if }}sx>x{\mbox{ and }}sy<y\end{cases}}}
4141:
Combinatorial properties of
Kazhdan–Lusztig polynomials and their generalizations are a topic of active current research. Given their significance in representation theory and algebraic geometry, attempts have been undertaken to develop the theory of Kazhdan–Lusztig polynomials in purely
1617:
1970:. These formulas are tiresome to use by hand for rank greater than about 3, but are well adapted for computers, and the only limit on computing Kazhdan–Lusztig polynomials with them is that for large rank the number of such polynomials exceeds the storage capacity of computers.
4112:
are all tightly controlled by appropriate analogues of
Kazhdan–Lusztig polynomials. They admit an elementary description, but the deeper properties of these polynomials necessary for representation theory follow from sophisticated techniques of modern algebraic geometry and
3044:
3522:
are determined in terms of coefficients of
Kazhdan–Lusztig polynomials. This result demonstrates that all coefficients of the Kazhdan–Lusztig polynomials of a finite Weyl group are non-negative integers. However, positivity for the case of a finite Weyl group
4132:
The coefficients of the
Kazhdan–Lusztig polynomials are conjectured to be the dimensions of some homomorphism spaces in Soergel's bimodule category. This is the only known positive interpretation of these coefficients for arbitrary Coxeter groups.
3170:
3468:
415:
3821:
993:
1948:{\displaystyle q^{{\frac {1}{2}}(\ell (w)-\ell (x))}D(P_{x,w})-q^{{\frac {1}{2}}(\ell (x)-\ell (w))}P_{x,w}=\sum _{x<y\leq w}(-1)^{\ell (x)+\ell (y)}q^{{\frac {1}{2}}(-\ell (x)+2\ell (y)-\ell (w))}D(R_{x,y})P_{y,w}}
4126:
1250:
4092:. Much of the later work of Lusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic varieties arising in representation theory, in particular, closures of
3224:
2217:
420:
4430:
2901:
794:
4142:
combinatorial fashion, relying to some extent on geometry, but without reference to intersection cohomology and other advanced techniques. This has led to exciting developments in
210:
3678:, and Kazhdan and Lusztig, following a suggestion of Deligne, showed how to express Kazhdan–Lusztig polynomials in terms of intersection cohomology groups of Schubert varieties.
611:{\displaystyle {\begin{aligned}T_{y}T_{w}&=T_{yw},&&{\mbox{if }}\ell (yw)=\ell (y)+\ell (w)\\(T_{s}+1)(T_{s}-q)&=0,&&{\mbox{if }}s\in S.\end{aligned}}}
4059:
3958:
3050:
2778:
paper also put forth two equivalent conjectures, known now as
Kazhdan–Lusztig conjectures, which related the values of their polynomials at 1 with representations of complex
4334:
3321:
1101:
407:
4263:
3613:
2765:
showed that any polynomial with constant term 1 and non-negative integer coefficients is the
Kazhdan–Lusztig polynomial for some pair of elements of some symmetric group.
1031:
63:
264:
4289:
337:
4222:
311:
139:
4190:
4354:
3263:. Thus, the Kazhdan–Lusztig conjectures describe the Jordan–Hölder multiplicities of Verma modules in any regular integral block of Bernstein–Gelfand–Gelfand
3704:
4097:
3211:). The methods introduced in the course of the proof have guided development of representation theory throughout the 1980s and 1990s, under the name
892:
2195:
The simple values of
Kazhdan–Lusztig polynomials for low rank groups are not typical of higher rank groups. For example, for the split form of E
5055:
2774:
The
Kazhdan–Lusztig polynomials arise as transition coefficients between their canonical basis and the natural basis of the Hecke algebra. The
1107:
To establish existence of the
Kazhdan–Lusztig polynomials, Kazhdan and Lusztig gave a simple recursive procedure for computing the polynomials
1137:
5050:
3893:. Their definition is more complicated, reflecting relative complexity of representations of real groups compared to complex groups.
4591:
4084:
The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig polynomials, namely, the
3866:
This gave the first proof that all coefficients of Kazhdan–Lusztig polynomials for finite Weyl groups are non-negative integers.
4575:
2058:
3039:{\displaystyle \operatorname {ch} (L_{w})=\sum _{y\leq w}(-1)^{\ell (w)-\ell (y)}P_{y,w}(1)\operatorname {ch} (M_{y})}
4801:
4545:
183:
5065:
4567:
4359:
279:
156:
121:
162:
In their first paper Kazhdan and Lusztig mentioned that their polynomials were related to the failure of local
732:
4785:
4906:
Soergel, Wolfgang (2006), "Kazhdan–Lusztig polynomials and indecomposable bimodules over polynomial rings",
2839:
218:
4065:
In March 2007, a collaborative project, the "Atlas of Lie groups and representations", announced that the
3896:
The distinction, in the cases directly connection to representation theory, is explained on the level of
3165:{\displaystyle \operatorname {ch} (M_{w})=\sum _{y\leq w}P_{w_{0}w,w_{0}y}(1)\operatorname {ch} (L_{y})}
3463:{\displaystyle P_{y,w}(q)=\sum _{i}q^{i}\dim(\operatorname {Ext} ^{\ell (w)-\ell (y)-2i}(M_{y},L_{w}))}
5060:
4030:
3929:
4294:
1289:
193:
The two bases for the Springer representation reminded Kazhdan and Lusztig of the two bases for the
4869:
Polo, Patrick (1999), "Construction of arbitrary Kazhdan–Lusztig polynomials in symmetric groups",
4820:
4692:
4614:
3991:
4066:
3528:
conjectures, although this approach to proving them turned out to be more difficult to carry out.
1036:
342:
4750:
4227:
4143:
4118:
3578:
3187:
These conjectures were proved over characteristic 0 algebraically closed fields independently by
187:
175:
109:
4450:
3618:
5. Kashiwara (1990) proved a generalization of the Kazhdan–Lusztig conjectures to symmetrizable
26:
4945:
Kobayashi, Masato (2013), "Inequalities on Bruhat graphs, R- and Kazhdan-Lusztig polynomials",
3568:
2783:
3826:
where each term on the right means: take the complex IC of sheaves whose hyperhomology is the
240:
3827:
3619:
17:
4992:
4268:
3885:. They are analogous to Kazhdan–Lusztig polynomials, but are tailored to representations of
3863:. The odd-dimensional cohomology groups do not appear in the sum because they are all zero.
1006:
316:
5045:
4898:
4833:
4705:
4675:
4627:
4478:
4195:
4109:
3965:
3631:
2779:
289:
206:
124:
3531:
4. Some special cases of the Kazhdan–Lusztig conjectures are easy to verify. For example,
8:
4586:
4512:
4495:
4169:
4114:
4105:
3572:
3556:= 1, since the sum reduces to a single term. On the other hand, the first conjecture for
3297:
3188:
2880:
4837:
4709:
4631:
163:
4974:
4954:
4933:
4915:
4857:
4729:
4651:
4605:
4339:
4085:
3675:
3293:
3200:
194:
4978:
4937:
4886:
4861:
4849:
4797:
4764:
4733:
4721:
4643:
4571:
4541:
3292:
in certain subquotient of the Verma module determined by a canonical filtration, the
197:
of certain infinite dimensional representations of semisimple Lie algebras, given by
5022:
4655:
4555:
1611:
The Kazhdan–Lusztig polynomials can then be computed recursively using the relation
186:, and gave another definition of such a basis in terms of the dimensions of certain
4964:
4925:
4878:
4841:
4789:
4759:
4713:
4663:
4635:
4609:
4516:
4499:
4122:
4011:
4001:
3889:
semisimple Lie groups, and play major role in the conjectural description of their
3831:
3301:
3204:
3192:
167:
151:. They found a new construction of these representations over the complex numbers (
4882:
3852:, and then take the dimension of the stalk of this sheaf at any point of the cell
3816:{\displaystyle P_{y,w}(q)=\sum _{i}q^{i}\dim IH_{X_{y}}^{2i}({\overline {X_{w}}})}
4894:
4829:
4793:
4701:
4671:
4623:
4474:
4093:
3296:. The Jantzen conjecture in regular integral case was proved in a later paper of
214:
144:
117:
5007:
4969:
4811:
4777:
4745:
4687:
4670:, Progress in Mathematics, vol. 87, Boston: Birkhauser, pp. 407–433,
4101:
3920:
3538:
is the antidominant Verma module, which is known to be simple. This means that
700:
70:
4929:
4666:(1990), "The Kazhdan–Lusztig conjecture for symmetrizable KacMoody algebras",
4587:"Kazhdan–Lusztig Polynomials: History, Problems, and Combinatorial Invariance"
5039:
4890:
4853:
4773:
4741:
4725:
4683:
4647:
4533:
3969:
3901:
1967:
202:
86:
66:
3625:
988:{\displaystyle C'_{w}=q^{-{\frac {\ell (w)}{2}}}\sum _{y\leq w}P_{y,w}T_{y}}
221:
to representations of Weyl groups, led to the Kazhdan–Lusztig conjectures.
5003:
4192:
for crystallographic Coxeter groups satisfy certain strict inequality: Let
4089:
4079:
3973:
3897:
3890:
2809:
841:
198:
179:
2871:
are locally-finite weight modules over the complex semisimple Lie algebra
4815:
4529:
3312:
2828:
5025:
software for computing Kazhdan–Lusztig polynomials for any Coxeter group
5015:
4996:
4845:
4717:
4639:
3515:
3264:
2794:
113:
93:
5029:
4506:, Sér. I Math., vol. 292, Paris: C. R. Acad. Sci., pp. 15–18
4166:
Kobayashi (2013) proved that values of Kazhdan–Lusztig polynomials at
1245:{\displaystyle T_{y^{-1}}^{-1}=\sum _{x}D(R_{x,y})q^{-\ell (x)}T_{x}.}
4920:
4690:(June 1979), "Representations of Coxeter groups and Hecke algebras",
3977:
2200:
148:
97:
4818:(1983), "Singularities of closures of K-orbits on flag manifolds.",
4612:(October 1981), "Kazhdan–Lusztig conjecture and holonomic systems",
3923:. The original (K-L) case is then about the details of decomposing
4959:
4540:, Progress in Mathematics, vol. 182, Boston, MA: Birkhäuser,
703:. From this it follows that the Hecke algebra has an automorphism
4748:(1980a), "A topological approach to Springer's representations",
1958:
using the fact that the two terms on the left are polynomials in
4784:, Proceedings of Symposia in Pure Mathematics, vol. XXXVI,
2786:, addressing a long-standing problem in representation theory.
3575:, together with the fact that all Kazhdan–Lusztig polynomials
2099:(or more generally any Coxeter group of rank at most 2) then
65:
is a member of a family of integral polynomials introduced by
4523:, Advances in Soviet Mathematics, vol. 16, pp. 1–50
3274:
coefficients of Kazhdan–Lusztig polynomials follows from the
3223:
1. The two conjectures are known to be equivalent. Moreover,
1594:
2203:(a variation of Kazhdan–Lusztig polynomials: see below) is
621:
The quadratic second relation implies that each generator
3626:
Relation to intersection cohomology of Schubert varieties
4558:; Brenti, Francesco (2005), "Ch. 5: Kazhdan–Lusztig and
4080:
Generalization to other objects in representation theory
828:, and uniquely determined by the following properties.
108:
In the spring of 1978 Kazhdan and Lusztig were studying
4451:"Computing Kazhdan-Lusztig-Vogan polynomials for split
3900:; or in other terms of actions on analogues of complex
4037:
3936:
1573:
1554:
1464:
1445:
1395:
1376:
1329:
1301:
586:
470:
4995:
from Spring 2005 course on Kazhdan–Lusztig Theory at
4362:
4342:
4297:
4271:
4230:
4198:
4172:
4033:
3932:
3707:
3581:
3324:
3278:, which roughly says that individual coefficients of
3053:
2904:
2215:
1620:
1264:
1140:
1039:
1009:
895:
735:
418:
345:
319:
292:
243:
127:
29:
4599:, Ellwangen: Haus Schönenberg: Research article B49b
3184:
is the element of maximal length of the Weyl group.
682:(obtained by multiplying the quadratic relation for
1255:They can be computed using the recursion relations
4908:Journal of the Institute of Mathematics of Jussieu
4604:
4511:
4494:
4424:
4348:
4328:
4283:
4257:
4216:
4184:
4100:. It turned out that the representation theory of
4053:
3952:
3815:
3607:
3462:
3305:
3208:
3196:
3164:
3038:
2748:
1947:
1600:
1244:
1118:) in terms of more elementary polynomials denoted
1095:
1025:
987:
788:
610:
401:
331:
305:
258:
133:
57:
628:is invertible in the Hecke algebra, with inverse
5037:
5032:for computing Kazhdan–Lusztig-Vogan polynomials.
4566:, Graduate Texts in Mathematics, vol. 231,
4528:
4155:
3869:
3315:showed as a consequence of the conjectures that
4772:
4740:
4150:. Some references are given in the textbook of
3681:More precisely, the Kazhdan–Lusztig polynomial
171:
152:
4682:
4088:of singularities of Schubert varieties in the
2827:where ρ is the half-sum of positive roots (or
2769:
2192:, giving examples of non-constant polynomials.
74:
4554:
4425:{\displaystyle P_{uw}(1)>P_{tu,w}(1)>0}
4151:
4810:
3882:
3877:(also called Kazhdan–Lusztig polynomials or
4448:
5018:for computing Kazhdan–Lusztig polynomials.
4224:be a crystallographic Coxeter system and
1103:-module, called the Kazhdan–Lusztig basis.
270:(the smallest length of an expression for
4968:
4958:
4947:Journal of Combinatorial Theory, Series A
4944:
4919:
4763:
4662:
4154:. A research monograph on the subject is
4127:Beilinson–Bernstein–Deligne decomposition
3859:whose closure is the Schubert variety of
3552:, establishing the second conjecture for
2895:. The Kazhdan–Lusztig conjectures state:
2201:most complicated Lusztig–Vogan polynomial
2059:longest element of a finite Coxeter group
1041:
347:
103:
77:). They are indexed by pairs of elements
789:{\displaystyle D(T_{w})=T_{w^{-1}}^{-1}}
174:they reinterpreted this in terms of the
5008:"Tables of Kazhdan–Lusztig polynomials"
4905:
4782:Schubert varieties and Poincaré duality
4136:
4024:. Then the relevant object of study is
1033:form a basis of the Hecke algebra as a
5038:
4592:Séminaire Lotharingien de Combinatoire
4584:
3514:is odd, so the dimensions of all such
656:. These inverses satisfy the relation
5056:Representation theory of Lie algebras
3650:is a disjoint union of affine spaces
3225:Borho–Jantzen's translation principle
4868:
4265:its Kazhdan–Lusztig polynomials. If
2762:
816:) are indexed by a pair of elements
5051:Representation theory of Lie groups
5002:
4538:Singular loci of Schubert varieties
4067:L–V polynomials had been calculated
1003:of the Hecke algebra. The elements
999:are invariant under the involution
13:
4521:A proof of the Jantzen conjectures
157:Hecke algebra of the Coxeter group
14:
5077:
4986:
4336:, then there exists a reflection
3879:Kazhdan–Lusztig–Vogan polynomials
3845:), take its cohomology of degree
3259:for any dominant integral weight
2838:be its irreducible quotient, the
800:can be seen to be an involution.
409:, with multiplication defined by
92:, which can in particular be the
4668:The Grothendieck Festschrift, II
803:The Kazhdan–Lusztig polynomials
205:. This analogy, and the work of
4564:Combinatorics of Coxeter Groups
4054:{\displaystyle K\backslash G/B}
3953:{\displaystyle B\backslash G/B}
3667:. The closures of these spaces
3270:2. A similar interpretation of
3213:geometric representation theory
4442:
4413:
4407:
4382:
4376:
4329:{\displaystyle P_{uw}(1)>1}
4317:
4311:
4251:
4245:
4211:
4199:
4156:Billey & Lakshmibai (2000)
3810:
3790:
3730:
3724:
3457:
3454:
3428:
3411:
3405:
3396:
3390:
3379:
3347:
3341:
3159:
3146:
3137:
3131:
3073:
3060:
3033:
3020:
3011:
3005:
2984:
2978:
2969:
2963:
2956:
2946:
2924:
2911:
1926:
1907:
1899:
1896:
1890:
1881:
1875:
1863:
1857:
1848:
1828:
1822:
1813:
1807:
1800:
1790:
1744:
1741:
1735:
1726:
1720:
1714:
1693:
1674:
1666:
1663:
1657:
1648:
1642:
1636:
1498:
1486:
1224:
1218:
1204:
1185:
1090:
1045:
932:
926:
752:
739:
566:
547:
544:
525:
518:
512:
503:
497:
488:
479:
396:
351:
253:
247:
52:
46:
1:
4883:10.1090/S1088-4165-99-00074-6
4786:American Mathematical Society
4488:
4161:
3870:Generalization to real groups
266:for the length of an element
224:
172:Kazhdan & Lusztig (1980b)
16:In the mathematical field of
4765:10.1016/0001-8708(80)90005-5
4148:pattern-avoidance phenomenon
3805:
2840:simple highest weight module
1096:{\displaystyle \mathbb {Z} }
402:{\displaystyle \mathbb {Z} }
274:as a product of elements of
7:
4467:Nieuw Archief voor Wiskunde
4258:{\displaystyle {P_{uw}(q)}}
4152:Björner & Brenti (2005)
3915:is a complex Lie group and
3608:{\displaystyle P_{y,w_{0}}}
3573:character of a Verma module
2770:Kazhdan–Lusztig conjectures
1973:
153:Kazhdan & Lusztig 1980a
10:
5082:
4970:10.1016/j.jcta.2012.10.001
4585:Brenti, Francesco (2003),
4449:van Leeuwen, Marc (2008),
3883:Lusztig & Vogan (1983)
3659:parameterized by elements
3218:
58:{\displaystyle P_{y,w}(q)}
22:Kazhdan–Lusztig polynomial
4930:10.1017/S1474748007000023
4504:Localisation de g-modules
3964:a classical theme of the
3875:Lusztig–Vogan polynomials
3838:(the closure of the cell
2887:) for the character of a
2879:, and therefore admit an
864:their degree is at most (
729:. More generally one has
159:and its representations.
4821:Inventiones Mathematicae
4794:10.1090/pspum/036/573434
4693:Inventiones Mathematicae
4615:Inventiones Mathematicae
4435:
3992:maximal compact subgroup
3571:and the formula for the
286:has a basis of elements
259:{\displaystyle \ell (w)}
110:Springer representations
5066:Algebraic combinatorics
4751:Advances in Mathematics
4144:algebraic combinatorics
4119:intersection cohomology
3976:. The L-V case takes a
3642:of the algebraic group
3189:Alexander Beilinson
188:intersection cohomology
176:intersection cohomology
141:-adic cohomology groups
4426:
4350:
4330:
4285:
4284:{\displaystyle u<w}
4259:
4218:
4186:
4069:for the split form of
4055:
3954:
3817:
3609:
3569:Weyl character formula
3464:
3285:are multiplicities of
3201:Jean-Luc Brylinski
3166:
3040:
2750:
1949:
1602:
1246:
1097:
1027:
1026:{\displaystyle C'_{w}}
989:
790:
612:
403:
333:
332:{\displaystyle w\in W}
307:
260:
135:
104:Motivation and history
59:
4871:Representation Theory
4427:
4351:
4331:
4286:
4260:
4219:
4217:{\displaystyle (W,S)}
4187:
4117:, such as the use of
4110:affine Hecke algebras
4056:
3968:, and before that of
3955:
3881:) were introduced in
3828:intersection homology
3818:
3610:
3465:
3167:
3041:
2780:semisimple Lie groups
2751:
2127:is the Coxeter group
2085:is the Coxeter group
1950:
1603:
1247:
1098:
1028:
990:
791:
613:
404:
334:
308:
306:{\displaystyle T_{w}}
261:
136:
134:{\displaystyle \ell }
60:
18:representation theory
4788:, pp. 185–203,
4513:Beilinson, Alexandre
4496:Beilinson, Alexandre
4360:
4340:
4295:
4269:
4228:
4196:
4170:
4137:Combinatorial theory
4106:modular Lie algebras
4031:
3966:Bruhat decomposition
3930:
3705:
3632:Bruhat decomposition
3579:
3322:
3051:
2902:
2875:with the Weyl group
2213:
2134:with generating set
2000:has constant term 1.
1618:
1262:
1138:
1037:
1007:
893:
733:
416:
343:
317:
290:
241:
233:with generating set
229:Fix a Coxeter group
207:Jens Carsten Jantzen
125:
27:
4838:1983InMat..71..365L
4710:1979InMat..53..165K
4632:1981InMat..64..387B
4606:Brylinski, Jean-Luc
4185:{\displaystyle q=1}
4115:homological algebra
3789:
3241:can be replaced by
2883:. Let us write ch(
2881:algebraic character
1168:
1022:
908:
785:
219:enveloping algebras
4999:by Monica Vazirani
4846:10.1007/BF01389103
4718:10.1007/BF01390031
4640:10.1007/BF01389272
4422:
4346:
4326:
4281:
4255:
4214:
4182:
4051:
3950:
3813:
3765:
3745:
3676:Schubert varieties
3620:Kac–Moody algebras
3605:
3460:
3362:
3294:Jantzen filtration
3276:Jantzen conjecture
3162:
3094:
3036:
2945:
2842:of highest weight
2812:of highest weight
2746:
2744:
1945:
1789:
1598:
1593:
1577:
1558:
1468:
1449:
1399:
1380:
1333:
1305:
1242:
1181:
1141:
1093:
1023:
1010:
985:
958:
896:
832:They are 0 unless
786:
758:
608:
606:
590:
474:
399:
329:
303:
256:
195:Grothendieck group
168:Schubert varieties
131:
55:
5021:Fokko du Cloux's
4664:Kashiwara, Masaki
4610:Kashiwara, Masaki
4577:978-3-540-44238-7
4517:Bernstein, Joseph
4500:Bernstein, Joseph
4349:{\displaystyle t}
3808:
3736:
3567:follows from the
3353:
3079:
2930:
1846:
1768:
1712:
1634:
1576:
1557:
1467:
1448:
1398:
1379:
1332:
1304:
1172:
943:
939:
589:
473:
184:Robert MacPherson
145:conjugacy classes
67:David Kazhdan
5073:
5061:Algebraic groups
5011:
4981:
4972:
4962:
4940:
4923:
4901:
4864:
4806:
4768:
4767:
4736:
4678:
4658:
4600:
4580:
4550:
4524:
4507:
4482:
4481:
4464:
4459:
4446:
4431:
4429:
4428:
4423:
4406:
4405:
4375:
4374:
4355:
4353:
4352:
4347:
4335:
4333:
4332:
4327:
4310:
4309:
4290:
4288:
4287:
4282:
4264:
4262:
4261:
4256:
4254:
4244:
4243:
4223:
4221:
4220:
4215:
4191:
4189:
4188:
4183:
4123:perverse sheaves
4098:quiver varieties
4094:nilpotent orbits
4060:
4058:
4057:
4052:
4047:
4023:
4012:complexification
4010:, and makes the
4009:
4002:semisimple group
3999:
3985:
3959:
3957:
3956:
3951:
3946:
3858:
3851:
3844:
3832:Schubert variety
3822:
3820:
3819:
3814:
3809:
3804:
3803:
3794:
3788:
3780:
3779:
3778:
3755:
3754:
3744:
3723:
3722:
3673:
3646:with Weyl group
3615:are equal to 1.
3614:
3612:
3611:
3606:
3604:
3603:
3602:
3601:
3513:
3490:
3469:
3467:
3466:
3461:
3453:
3452:
3440:
3439:
3424:
3423:
3372:
3371:
3361:
3340:
3339:
3291:
3284:
3262:
3258:
3240:
3205:Masaki Kashiwara
3193:Joseph Bernstein
3183:
3171:
3169:
3168:
3163:
3158:
3157:
3130:
3129:
3125:
3124:
3109:
3108:
3093:
3072:
3071:
3045:
3043:
3042:
3037:
3032:
3031:
3004:
3003:
2988:
2987:
2944:
2923:
2922:
2870:
2863:
2856:
2837:
2826:
2807:
2755:
2753:
2752:
2747:
2745:
2732:
2731:
2716:
2715:
2700:
2699:
2678:
2677:
2656:
2655:
2634:
2633:
2612:
2611:
2590:
2589:
2565:
2561:
2560:
2533:
2532:
2505:
2504:
2477:
2476:
2449:
2448:
2421:
2420:
2390:
2386:
2385:
2358:
2357:
2330:
2329:
2302:
2301:
2280:
2279:
2258:
2257:
2232:
2231:
2120:and 0 otherwise.
2029:
1954:
1952:
1951:
1946:
1944:
1943:
1925:
1924:
1903:
1902:
1847:
1839:
1832:
1831:
1788:
1764:
1763:
1748:
1747:
1713:
1705:
1692:
1691:
1670:
1669:
1635:
1627:
1607:
1605:
1604:
1599:
1597:
1596:
1578:
1574:
1559:
1555:
1547:
1546:
1519:
1518:
1469:
1465:
1450:
1446:
1438:
1437:
1400:
1396:
1381:
1377:
1369:
1368:
1334:
1330:
1306:
1302:
1280:
1279:
1251:
1249:
1248:
1243:
1238:
1237:
1228:
1227:
1203:
1202:
1180:
1167:
1159:
1158:
1157:
1102:
1100:
1099:
1094:
1089:
1088:
1084:
1065:
1064:
1060:
1044:
1032:
1030:
1029:
1024:
1018:
994:
992:
991:
986:
984:
983:
974:
973:
957:
942:
941:
940:
935:
921:
904:
795:
793:
792:
787:
784:
776:
775:
774:
751:
750:
728:
721:
699:), and also the
695:
688:
681:
655:
627:
617:
615:
614:
609:
607:
591:
587:
583:
559:
558:
537:
536:
475:
471:
467:
462:
461:
442:
441:
432:
431:
408:
406:
405:
400:
395:
394:
390:
371:
370:
366:
350:
338:
336:
335:
330:
312:
310:
309:
304:
302:
301:
265:
263:
262:
257:
215:primitive ideals
164:Poincaré duality
140:
138:
137:
132:
64:
62:
61:
56:
45:
44:
5081:
5080:
5076:
5075:
5074:
5072:
5071:
5070:
5036:
5035:
4989:
4830:Springer-Verlag
4812:Lusztig, George
4804:
4778:Lusztig, George
4746:Lusztig, George
4702:Springer-Verlag
4688:Lusztig, George
4624:Springer-Verlag
4578:
4556:Björner, Anders
4548:
4491:
4486:
4485:
4462:
4458:
4452:
4447:
4443:
4438:
4392:
4388:
4367:
4363:
4361:
4358:
4357:
4341:
4338:
4337:
4302:
4298:
4296:
4293:
4292:
4270:
4267:
4266:
4236:
4232:
4231:
4229:
4226:
4225:
4197:
4194:
4193:
4171:
4168:
4167:
4164:
4139:
4082:
4075:
4043:
4032:
4029:
4028:
4022:
4018:
4008:
4004:
3998:
3994:
3984:
3980:
3942:
3931:
3928:
3927:
3872:
3857:
3853:
3846:
3843:
3839:
3799:
3795:
3793:
3781:
3774:
3770:
3769:
3750:
3746:
3740:
3712:
3708:
3706:
3703:
3702:
3698:) is equal to
3693:
3672:
3668:
3658:
3628:
3597:
3593:
3586:
3582:
3580:
3577:
3576:
3566:
3551:
3544:
3537:
3492:
3487:
3480:
3474:
3448:
3444:
3435:
3431:
3386:
3382:
3367:
3363:
3357:
3329:
3325:
3323:
3320:
3319:
3290:
3286:
3283:
3279:
3260:
3242:
3228:
3221:
3182:
3176:
3153:
3149:
3120:
3116:
3104:
3100:
3099:
3095:
3083:
3067:
3063:
3052:
3049:
3048:
3027:
3023:
2993:
2989:
2959:
2955:
2934:
2918:
2914:
2903:
2900:
2899:
2869:
2865:
2862:
2858:
2843:
2836:
2832:
2813:
2806:
2802:
2797:. For each w ∈
2772:
2743:
2742:
2727:
2723:
2711:
2707:
2695:
2691:
2673:
2669:
2651:
2647:
2629:
2625:
2607:
2603:
2585:
2581:
2563:
2562:
2556:
2552:
2528:
2524:
2500:
2496:
2472:
2468:
2444:
2440:
2416:
2412:
2388:
2387:
2381:
2377:
2353:
2349:
2325:
2321:
2297:
2293:
2275:
2271:
2253:
2249:
2233:
2227:
2223:
2216:
2214:
2211:
2210:
2198:
2187:
2170:
2158:commuting then
2133:
2111:
2098:
2091:
2073:
2056:
2042:
2012:
1999:
1976:
1933:
1929:
1914:
1910:
1838:
1837:
1833:
1803:
1799:
1772:
1753:
1749:
1704:
1703:
1699:
1681:
1677:
1626:
1625:
1621:
1619:
1616:
1615:
1592:
1591:
1575: and
1572:
1553:
1551:
1530:
1526:
1505:
1501:
1483:
1482:
1466: and
1463:
1444:
1442:
1421:
1417:
1414:
1413:
1397: and
1394:
1375:
1373:
1352:
1348:
1345:
1344:
1328:
1326:
1317:
1316:
1300:
1298:
1285:
1284:
1269:
1265:
1263:
1260:
1259:
1233:
1229:
1211:
1207:
1192:
1188:
1176:
1160:
1150:
1146:
1145:
1139:
1136:
1135:
1126:
1112:
1080:
1073:
1069:
1056:
1052:
1048:
1040:
1038:
1035:
1034:
1014:
1008:
1005:
1004:
979:
975:
963:
959:
947:
922:
920:
916:
912:
900:
894:
891:
890:
811:
777:
767:
763:
762:
746:
742:
734:
731:
730:
727:
723:
720:
716:
701:braid relations
694:
690:
687:
683:
675:
666:
657:
649:
637:
629:
626:
622:
605:
604:
585:
582:
569:
554:
550:
532:
528:
522:
521:
469:
466:
454:
450:
443:
437:
433:
427:
423:
419:
417:
414:
413:
386:
379:
375:
362:
358:
354:
346:
344:
341:
340:
318:
315:
314:
297:
293:
291:
288:
287:
242:
239:
238:
227:
126:
123:
122:
118:algebraic group
106:
34:
30:
28:
25:
24:
12:
11:
5:
5079:
5069:
5068:
5063:
5058:
5053:
5048:
5034:
5033:
5026:
5019:
5012:
5000:
4988:
4987:External links
4985:
4984:
4983:
4953:(2): 470–482,
4942:
4914:(3): 501–525,
4903:
4866:
4808:
4802:
4774:Kazhdan, David
4770:
4758:(2): 222–228,
4742:Kazhdan, David
4738:
4684:Kazhdan, David
4680:
4660:
4602:
4582:
4576:
4562:polynomials",
4552:
4546:
4534:Lakshmibai, V.
4526:
4509:
4490:
4487:
4484:
4483:
4473:(2): 113–116,
4456:
4440:
4439:
4437:
4434:
4421:
4418:
4415:
4412:
4409:
4404:
4401:
4398:
4395:
4391:
4387:
4384:
4381:
4378:
4373:
4370:
4366:
4345:
4325:
4322:
4319:
4316:
4313:
4308:
4305:
4301:
4280:
4277:
4274:
4253:
4250:
4247:
4242:
4239:
4235:
4213:
4210:
4207:
4204:
4201:
4181:
4178:
4175:
4163:
4160:
4138:
4135:
4102:quantum groups
4081:
4078:
4073:
4063:
4062:
4050:
4046:
4042:
4039:
4036:
4020:
4006:
3996:
3982:
3970:Schubert cells
3962:
3961:
3949:
3945:
3941:
3938:
3935:
3921:Borel subgroup
3902:flag manifolds
3871:
3868:
3855:
3841:
3824:
3823:
3812:
3807:
3802:
3798:
3792:
3787:
3784:
3777:
3773:
3768:
3764:
3761:
3758:
3753:
3749:
3743:
3739:
3735:
3732:
3729:
3726:
3721:
3718:
3715:
3711:
3685:
3670:
3654:
3627:
3624:
3600:
3596:
3592:
3589:
3585:
3564:
3549:
3542:
3535:
3485:
3478:
3471:
3470:
3459:
3456:
3451:
3447:
3443:
3438:
3434:
3430:
3427:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3385:
3381:
3378:
3375:
3370:
3366:
3360:
3356:
3352:
3349:
3346:
3343:
3338:
3335:
3332:
3328:
3288:
3281:
3220:
3217:
3180:
3173:
3172:
3161:
3156:
3152:
3148:
3145:
3142:
3139:
3136:
3133:
3128:
3123:
3119:
3115:
3112:
3107:
3103:
3098:
3092:
3089:
3086:
3082:
3078:
3075:
3070:
3066:
3062:
3059:
3056:
3046:
3035:
3030:
3026:
3022:
3019:
3016:
3013:
3010:
3007:
3002:
2999:
2996:
2992:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2958:
2954:
2951:
2948:
2943:
2940:
2937:
2933:
2929:
2926:
2921:
2917:
2913:
2910:
2907:
2867:
2860:
2834:
2804:
2771:
2768:
2767:
2766:
2759:
2758:
2757:
2756:
2741:
2738:
2735:
2730:
2726:
2722:
2719:
2714:
2710:
2706:
2703:
2698:
2694:
2690:
2687:
2684:
2681:
2676:
2672:
2668:
2665:
2662:
2659:
2654:
2650:
2646:
2643:
2640:
2637:
2632:
2628:
2624:
2621:
2618:
2615:
2610:
2606:
2602:
2599:
2596:
2593:
2588:
2584:
2580:
2577:
2574:
2571:
2568:
2566:
2564:
2559:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2531:
2527:
2523:
2520:
2517:
2514:
2511:
2508:
2503:
2499:
2495:
2492:
2489:
2486:
2483:
2480:
2475:
2471:
2467:
2464:
2461:
2458:
2455:
2452:
2447:
2443:
2439:
2436:
2433:
2430:
2427:
2424:
2419:
2415:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2391:
2389:
2384:
2380:
2376:
2373:
2370:
2367:
2364:
2361:
2356:
2352:
2348:
2345:
2342:
2339:
2336:
2333:
2328:
2324:
2320:
2317:
2314:
2311:
2308:
2305:
2300:
2296:
2292:
2289:
2286:
2283:
2278:
2274:
2270:
2267:
2264:
2261:
2256:
2252:
2248:
2245:
2242:
2239:
2236:
2234:
2230:
2226:
2222:
2219:
2218:
2205:
2204:
2196:
2193:
2179:
2162:
2131:
2121:
2103:
2096:
2089:
2079:
2065:
2054:
2044:
2034:
2028:) ∈ {0, 1, 2}
2001:
1991:
1975:
1972:
1968:constant terms
1956:
1955:
1942:
1939:
1936:
1932:
1928:
1923:
1920:
1917:
1913:
1909:
1906:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1845:
1842:
1836:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1802:
1798:
1795:
1792:
1787:
1784:
1781:
1778:
1775:
1771:
1767:
1762:
1759:
1756:
1752:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1711:
1708:
1702:
1698:
1695:
1690:
1687:
1684:
1680:
1676:
1673:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1633:
1630:
1624:
1609:
1608:
1595:
1590:
1587:
1584:
1581:
1571:
1568:
1565:
1562:
1552:
1550:
1545:
1542:
1539:
1536:
1533:
1529:
1525:
1522:
1517:
1514:
1511:
1508:
1504:
1500:
1497:
1494:
1491:
1488:
1485:
1484:
1481:
1478:
1475:
1472:
1462:
1459:
1456:
1453:
1443:
1441:
1436:
1433:
1430:
1427:
1424:
1420:
1416:
1415:
1412:
1409:
1406:
1403:
1393:
1390:
1387:
1384:
1374:
1372:
1367:
1364:
1361:
1358:
1355:
1351:
1347:
1346:
1343:
1340:
1337:
1327:
1325:
1322:
1319:
1318:
1315:
1312:
1309:
1299:
1297:
1294:
1291:
1290:
1288:
1283:
1278:
1275:
1272:
1268:
1253:
1252:
1241:
1236:
1232:
1226:
1223:
1220:
1217:
1214:
1210:
1206:
1201:
1198:
1195:
1191:
1187:
1184:
1179:
1175:
1171:
1166:
1163:
1156:
1153:
1149:
1144:
1131:). defined by
1122:
1110:
1105:
1104:
1092:
1087:
1083:
1079:
1076:
1072:
1068:
1063:
1059:
1055:
1051:
1047:
1043:
1021:
1017:
1013:
997:
996:
995:
982:
978:
972:
969:
966:
962:
956:
953:
950:
946:
938:
934:
931:
928:
925:
919:
915:
911:
907:
903:
899:
885:
884:
881:
807:
783:
780:
773:
770:
766:
761:
757:
754:
749:
745:
741:
738:
725:
718:
692:
685:
671:
662:
645:
633:
624:
619:
618:
603:
600:
597:
594:
584:
581:
578:
575:
572:
570:
568:
565:
562:
557:
553:
549:
546:
543:
540:
535:
531:
527:
524:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
468:
465:
460:
457:
453:
449:
446:
444:
440:
436:
430:
426:
422:
421:
398:
393:
389:
385:
382:
378:
374:
369:
365:
361:
357:
353:
349:
339:over the ring
328:
325:
322:
300:
296:
255:
252:
249:
246:
226:
223:
211:Anthony Joseph
203:simple modules
130:
105:
102:
71:George Lusztig
54:
51:
48:
43:
40:
37:
33:
9:
6:
4:
3:
2:
5078:
5067:
5064:
5062:
5059:
5057:
5054:
5052:
5049:
5047:
5044:
5043:
5041:
5031:
5027:
5024:
5020:
5017:
5013:
5009:
5005:
5004:Goresky, Mark
5001:
4998:
4994:
4991:
4990:
4980:
4976:
4971:
4966:
4961:
4956:
4952:
4948:
4943:
4939:
4935:
4931:
4927:
4922:
4917:
4913:
4909:
4904:
4900:
4896:
4892:
4888:
4884:
4880:
4877:(4): 90–104,
4876:
4872:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4839:
4835:
4831:
4827:
4823:
4822:
4817:
4813:
4809:
4805:
4803:9780821814390
4799:
4795:
4791:
4787:
4783:
4779:
4775:
4771:
4766:
4761:
4757:
4753:
4752:
4747:
4743:
4739:
4735:
4731:
4727:
4723:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4694:
4689:
4685:
4681:
4677:
4673:
4669:
4665:
4661:
4657:
4653:
4649:
4645:
4641:
4637:
4633:
4629:
4625:
4621:
4617:
4616:
4611:
4607:
4603:
4598:
4594:
4593:
4588:
4583:
4579:
4573:
4569:
4565:
4561:
4557:
4553:
4549:
4547:0-8176-4092-4
4543:
4539:
4535:
4531:
4527:
4522:
4518:
4514:
4510:
4505:
4501:
4497:
4493:
4492:
4480:
4476:
4472:
4468:
4461:
4455:
4445:
4441:
4433:
4419:
4416:
4410:
4402:
4399:
4396:
4393:
4389:
4385:
4379:
4371:
4368:
4364:
4343:
4323:
4320:
4314:
4306:
4303:
4299:
4278:
4275:
4272:
4248:
4240:
4237:
4233:
4208:
4205:
4202:
4179:
4176:
4173:
4159:
4157:
4153:
4149:
4145:
4134:
4130:
4128:
4124:
4120:
4116:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4077:
4072:
4068:
4048:
4044:
4040:
4034:
4027:
4026:
4025:
4016:
4013:
4003:
3993:
3989:
3979:
3975:
3971:
3967:
3947:
3943:
3939:
3933:
3926:
3925:
3924:
3922:
3918:
3914:
3910:
3906:
3903:
3899:
3898:double cosets
3894:
3892:
3891:unitary duals
3888:
3884:
3880:
3876:
3867:
3864:
3862:
3850:
3837:
3833:
3829:
3800:
3796:
3785:
3782:
3775:
3771:
3766:
3762:
3759:
3756:
3751:
3747:
3741:
3737:
3733:
3727:
3719:
3716:
3713:
3709:
3701:
3700:
3699:
3697:
3692:
3688:
3684:
3679:
3677:
3666:
3662:
3657:
3653:
3649:
3645:
3641:
3637:
3633:
3623:
3621:
3616:
3598:
3594:
3590:
3587:
3583:
3574:
3570:
3563:
3559:
3555:
3548:
3541:
3534:
3529:
3526:
3521:
3517:
3511:
3507:
3503:
3499:
3495:
3488:
3481:
3449:
3445:
3441:
3436:
3432:
3425:
3420:
3417:
3414:
3408:
3402:
3399:
3393:
3387:
3383:
3376:
3373:
3368:
3364:
3358:
3354:
3350:
3344:
3336:
3333:
3330:
3326:
3318:
3317:
3316:
3314:
3309:
3307:
3303:
3300: and
3299:
3295:
3277:
3273:
3268:
3266:
3257:
3253:
3249:
3245:
3239:
3235:
3231:
3227:implies that
3226:
3216:
3214:
3210:
3206:
3203: and
3202:
3198:
3194:
3191: and
3190:
3185:
3179:
3154:
3150:
3143:
3140:
3134:
3126:
3121:
3117:
3113:
3110:
3105:
3101:
3096:
3090:
3087:
3084:
3080:
3076:
3068:
3064:
3057:
3054:
3047:
3028:
3024:
3017:
3014:
3008:
3000:
2997:
2994:
2990:
2981:
2975:
2972:
2966:
2960:
2952:
2949:
2941:
2938:
2935:
2931:
2927:
2919:
2915:
2908:
2905:
2898:
2897:
2896:
2894:
2890:
2886:
2882:
2878:
2874:
2855:
2851:
2847:
2841:
2830:
2825:
2821:
2817:
2811:
2800:
2796:
2792:
2787:
2785:
2781:
2777:
2764:
2761:
2760:
2739:
2736:
2733:
2728:
2724:
2720:
2717:
2712:
2708:
2704:
2701:
2696:
2692:
2688:
2685:
2682:
2679:
2674:
2670:
2666:
2663:
2660:
2657:
2652:
2648:
2644:
2641:
2638:
2635:
2630:
2626:
2622:
2619:
2616:
2613:
2608:
2604:
2600:
2597:
2594:
2591:
2586:
2582:
2578:
2575:
2572:
2569:
2567:
2557:
2553:
2549:
2546:
2543:
2540:
2537:
2534:
2529:
2525:
2521:
2518:
2515:
2512:
2509:
2506:
2501:
2497:
2493:
2490:
2487:
2484:
2481:
2478:
2473:
2469:
2465:
2462:
2459:
2456:
2453:
2450:
2445:
2441:
2437:
2434:
2431:
2428:
2425:
2422:
2417:
2413:
2409:
2406:
2403:
2400:
2397:
2394:
2392:
2382:
2378:
2374:
2371:
2368:
2365:
2362:
2359:
2354:
2350:
2346:
2343:
2340:
2337:
2334:
2331:
2326:
2322:
2318:
2315:
2312:
2309:
2306:
2303:
2298:
2294:
2290:
2287:
2284:
2281:
2276:
2272:
2268:
2265:
2262:
2259:
2254:
2250:
2246:
2243:
2240:
2237:
2235:
2228:
2224:
2220:
2209:
2208:
2207:
2206:
2202:
2194:
2191:
2186:
2182:
2178:
2174:
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2130:
2126:
2122:
2119:
2115:
2110:
2106:
2102:
2095:
2088:
2084:
2080:
2077:
2072:
2068:
2064:
2060:
2053:
2049:
2045:
2041:
2037:
2033:
2027:
2023:
2019:
2015:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1977:
1971:
1969:
1965:
1961:
1940:
1937:
1934:
1930:
1921:
1918:
1915:
1911:
1904:
1893:
1887:
1884:
1878:
1872:
1869:
1866:
1860:
1854:
1851:
1843:
1840:
1834:
1825:
1819:
1816:
1810:
1804:
1796:
1793:
1785:
1782:
1779:
1776:
1773:
1769:
1765:
1760:
1757:
1754:
1750:
1738:
1732:
1729:
1723:
1717:
1709:
1706:
1700:
1696:
1688:
1685:
1682:
1678:
1671:
1660:
1654:
1651:
1645:
1639:
1631:
1628:
1622:
1614:
1613:
1612:
1588:
1585:
1582:
1579:
1569:
1566:
1563:
1560:
1548:
1543:
1540:
1537:
1534:
1531:
1527:
1523:
1520:
1515:
1512:
1509:
1506:
1502:
1495:
1492:
1489:
1479:
1476:
1473:
1470:
1460:
1457:
1454:
1451:
1439:
1434:
1431:
1428:
1425:
1422:
1418:
1410:
1407:
1404:
1401:
1391:
1388:
1385:
1382:
1370:
1365:
1362:
1359:
1356:
1353:
1349:
1341:
1338:
1335:
1323:
1320:
1313:
1310:
1307:
1295:
1292:
1286:
1281:
1276:
1273:
1270:
1266:
1258:
1257:
1256:
1239:
1234:
1230:
1221:
1215:
1212:
1208:
1199:
1196:
1193:
1189:
1182:
1177:
1173:
1169:
1164:
1161:
1154:
1151:
1147:
1142:
1134:
1133:
1132:
1130:
1125:
1121:
1117:
1113:
1085:
1081:
1077:
1074:
1070:
1066:
1061:
1057:
1053:
1049:
1019:
1015:
1011:
1002:
998:
980:
976:
970:
967:
964:
960:
954:
951:
948:
944:
936:
929:
923:
917:
913:
909:
905:
901:
897:
889:
888:
887:
886:
882:
879:
875:
871:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
830:
829:
827:
823:
819:
815:
810:
806:
801:
799:
781:
778:
771:
768:
764:
759:
755:
747:
743:
736:
714:
710:
706:
702:
698:
679:
674:
670:
665:
661:
653:
648:
644:
641:
636:
632:
601:
598:
595:
592:
579:
576:
573:
571:
563:
560:
555:
551:
541:
538:
533:
529:
515:
509:
506:
500:
494:
491:
485:
482:
476:
463:
458:
455:
451:
447:
445:
438:
434:
428:
424:
412:
411:
410:
391:
387:
383:
380:
376:
372:
367:
363:
359:
355:
326:
323:
320:
298:
294:
285:
281:
280:Hecke algebra
277:
273:
269:
250:
244:
236:
232:
222:
220:
216:
212:
208:
204:
200:
199:Verma modules
196:
191:
189:
185:
181:
177:
173:
169:
165:
160:
158:
154:
150:
146:
142:
128:
119:
115:
111:
101:
99:
95:
91:
88:
87:Coxeter group
84:
80:
76:
72:
69: and
68:
49:
41:
38:
35:
31:
23:
19:
4950:
4946:
4921:math/0403496
4911:
4907:
4874:
4870:
4825:
4819:
4816:Vogan, David
4781:
4755:
4749:
4697:
4691:
4667:
4619:
4613:
4596:
4590:
4563:
4559:
4537:
4530:Billey, Sara
4520:
4503:
4470:
4466:
4453:
4444:
4165:
4147:
4140:
4131:
4090:flag variety
4083:
4070:
4064:
4014:
3987:
3974:Grassmannian
3963:
3916:
3912:
3908:
3904:
3895:
3886:
3878:
3874:
3873:
3865:
3860:
3848:
3835:
3825:
3695:
3690:
3686:
3682:
3680:
3664:
3660:
3655:
3651:
3647:
3643:
3639:
3635:
3629:
3617:
3561:
3557:
3553:
3546:
3539:
3532:
3530:
3524:
3519:
3518:in category
3509:
3505:
3501:
3497:
3493:
3491:vanishes if
3483:
3476:
3472:
3310:
3275:
3271:
3269:
3255:
3251:
3247:
3243:
3237:
3233:
3229:
3222:
3212:
3186:
3177:
3174:
2892:
2888:
2884:
2876:
2872:
2853:
2849:
2845:
2823:
2819:
2815:
2810:Verma module
2798:
2793:be a finite
2790:
2788:
2784:Lie algebras
2775:
2773:
2189:
2184:
2180:
2176:
2172:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2128:
2124:
2117:
2113:
2108:
2104:
2100:
2093:
2086:
2082:
2075:
2074:= 1 for all
2070:
2066:
2062:
2051:
2047:
2039:
2035:
2031:
2025:
2021:
2017:
2013:
2008:
2004:
1996:
1992:
1988:
1984:
1980:
1963:
1959:
1957:
1610:
1254:
1128:
1123:
1119:
1115:
1108:
1106:
1000:
883:The elements
877:
873:
869:
865:
861:
857:
853:
849:
845:
842:Bruhat order
837:
833:
825:
821:
817:
813:
808:
804:
802:
797:
712:
708:
704:
696:
677:
672:
668:
663:
659:
651:
646:
642:
639:
634:
630:
620:
283:
275:
271:
267:
237:, and write
234:
230:
228:
192:
180:Mark Goresky
161:
107:
89:
82:
78:
21:
15:
5046:Polynomials
4832:: 365–379,
4704:: 165–184,
4626:: 387–410,
3674:are called
3313:David Vogan
2831:), and let
2829:Weyl vector
2776:Inventiones
2763:Polo (1999)
707:that sends
143:related to
5040:Categories
4997:U.C. Davis
4489:References
4356:such that
4162:Inequality
4146:, such as
3634:the space
3516:Ext groups
3265:category O
2801:denote by
2795:Weyl group
856:, and for
225:Definition
147:which are
114:Weyl group
94:Weyl group
4979:205929043
4960:1211.4305
4938:120459494
4891:1088-4165
4862:120917588
4854:0020-9910
4780:(1980b),
4734:120098142
4726:0020-9910
4648:0020-9910
4038:∖
3978:real form
3937:∖
3806:¯
3760:
3738:∑
3473:and that
3426:
3415:−
3403:ℓ
3400:−
3388:ℓ
3377:
3355:∑
3302:Bernstein
3298:Beilinson
3199:) and by
3144:
3088:≤
3081:∑
3058:
3018:
2976:ℓ
2973:−
2961:ℓ
2950:−
2939:≤
2932:∑
2909:
1888:ℓ
1885:−
1873:ℓ
1855:ℓ
1852:−
1820:ℓ
1805:ℓ
1794:−
1783:≤
1770:∑
1733:ℓ
1730:−
1718:ℓ
1697:−
1655:ℓ
1652:−
1640:ℓ
1493:−
1216:ℓ
1213:−
1174:∑
1162:−
1152:−
1075:−
952:≤
945:∑
924:ℓ
918:−
880:) − 1)/2.
779:−
769:−
715:and each
596:∈
561:−
510:ℓ
495:ℓ
477:ℓ
381:−
324:∈
245:ℓ
213:relating
149:unipotent
129:ℓ
98:Lie group
5030:software
5016:programs
5014:The GAP
4993:Readings
4656:18403883
4568:Springer
4536:(2000),
4519:(1993),
4502:(1981),
4086:geometry
4000:in that
2891:-module
2112:is 1 if
1974:Examples
1966:without
1556:if
1447:if
1378:if
1331:if
1311:≰
1303:if
1020:′
906:′
848:), 1 if
840:(in the
588:if
472:if
278:). The
190:groups.
5023:Coxeter
4899:1698201
4834:Bibcode
4706:Bibcode
4676:1106905
4628:Bibcode
4479:2454587
3830:of the
3630:By the
3304: (
3219:Remarks
3207: (
3195: (
2857:. Both
2808:be the
2150:} with
2057:is the
796:; also
112:of the
73: (
5028:Atlas
4977:
4936:
4897:
4889:
4860:
4852:
4800:
4732:
4724:
4674:
4654:
4646:
4574:
4544:
4477:
3911:where
3175:where
2188:= 1 +
2171:= 1 +
116:of an
85:of a
4975:S2CID
4955:arXiv
4934:S2CID
4916:arXiv
4858:S2CID
4828:(2),
4730:S2CID
4700:(2),
4652:S2CID
4622:(3),
4463:(PDF)
4436:Notes
3972:in a
2185:acbca
2175:and
2061:then
2030:then
1987:then
860:<
680:) = 0
667:+ 1)(
170:. In
96:of a
4887:ISSN
4850:ISSN
4798:ISBN
4722:ISSN
4644:ISSN
4572:ISBN
4542:ISBN
4417:>
4386:>
4321:>
4291:and
4276:<
4125:and
4108:and
4096:and
3990:, a
3887:real
3504:) +
3475:Ext(
3306:1993
3254:) −
3236:) −
3209:1981
3197:1981
2864:and
2852:) −
2822:) −
2789:Let
2782:and
2199:the
2168:bacb
2154:and
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2011:and
1962:and
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313:for
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4965:doi
4951:120
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2138:= {
2123:If
2092:or
2081:If
2046:If
2003:If
1979:If
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722:to
711:to
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2007:≤
1983:≤
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2022:ℓ
2018:w
2016:(
2014:ℓ
2009:w
2005:y
1997:w
1995:,
1993:y
1989:P
1985:w
1981:y
1964:q
1960:q
1941:w
1938:,
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