2765:
1617:
2223:
1964:
1272:
2760:{\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}}}
627:
166:). 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
3538:
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
1612:{\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}}}
4152:
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
1628:
1981:. 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.
4123:
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
3055:
3533:
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
4143:
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.
3181:
3479:
426:
3832:
1004:
1959:{\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}}
1261:
4103:. 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
2228:
431:
4441:
2912:
805:
4153:
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
3689:, 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.
622:{\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}}}
4070:
3969:
3061:
2789:
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
4345:
3332:
1112:
418:
4274:
3624:
2776:
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.
1042:
74:
275:
4300:
348:
4233:
322:
150:
5066:
4201:
4365:
3274:. Thus, the Kazhdan–Lusztig conjectures describe the Jordan–Hölder multiplicities of Verma modules in any regular integral block of Bernstein–Gelfand–Gelfand
5061:
3715:
4602:
3222:). The methods introduced in the course of the proof have guided development of representation theory throughout the 1980s and 1990s, under the name
903:
2206:
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
2785:
The
Kazhdan–Lusztig polynomials arise as transition coefficients between their canonical basis and the natural basis of the Hecke algebra. The
1118:
To establish existence of the
Kazhdan–Lusztig polynomials, Kazhdan and Lusztig gave a simple recursive procedure for computing the polynomials
4586:
1148:
2069:
3904:. Their definition is more complicated, reflecting relative complexity of representations of real groups compared to complex groups.
4812:
4556:
194:
4095:
The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig polynomials, namely, the
4578:
5076:
3877:
This gave the first proof that all coefficients of Kazhdan–Lusztig polynomials for finite Weyl groups are non-negative integers.
290:
167:
4137:
3235:
4796:
2850:
229:
3050:{\displaystyle \operatorname {ch} (L_{w})=\sum _{y\leq w}(-1)^{\ell (w)-\ell (y)}P_{y,w}(1)\operatorname {ch} (M_{y})}
221:
4370:
5071:
4831:
4703:
4625:
4002:
173:
In their first paper Kazhdan and Lusztig mentioned that their polynomials were related to the failure of local
743:
4917:
Soergel, Wolfgang (2006), "Kazhdan–Lusztig polynomials and indecomposable bimodules over polynomial rings",
4761:
4154:
4129:
198:
186:
120:
4076:
In March 2007, a collaborative project, the "Atlas of Lie groups and representations", announced that the
3579:
2794:
3907:
The distinction, in the cases directly connection to representation theory, is explained on the level of
3630:
3838:
3176:{\displaystyle \operatorname {ch} (M_{w})=\sum _{y\leq w}P_{w_{0}w,w_{0}y}(1)\operatorname {ch} (L_{y})}
28:
3474:{\displaystyle P_{y,w}(q)=\sum _{i}q^{i}\dim(\operatorname {Ext} ^{\ell (w)-\ell (y)-2i}(M_{y},L_{w}))}
4108:
5056:
4844:
4716:
4638:
4120:
4041:
3976:
3940:
3642:
2790:
217:
4305:
1300:
204:
The two bases for the Springer representation reminded Kazhdan and Lusztig of the two bases for the
4880:
Polo, Patrick (1999), "Construction of arbitrary Kazhdan–Lusztig polynomials in symmetric groups",
4523:
4506:
4125:
4116:
3583:
3308:
3199:
2891:
174:
132:
3539:
conjectures, although this approach to proving them turned out to be more difficult to carry out.
1047:
353:
4985:
4965:
4944:
4926:
4868:
4740:
4662:
4616:
4238:
4096:
3686:
3589:
3304:
3211:
3198:
These conjectures were proved over characteristic 0 algebraically closed fields independently by
205:
4461:
3629:
5. Kashiwara (1990) proved a generalization of the Kazhdan–Lusztig conjectures to symmetrizable
37:
4956:
Kobayashi, Masato (2013), "Inequalities on Bruhat graphs, R- and Kazhdan-Lusztig polynomials",
4897:
4860:
4808:
4732:
4654:
4582:
4566:
4552:
3837:
where each term on the right means: take the complex IC of sheaves whose hyperhomology is the
251:
5003:
4975:
4936:
4889:
4852:
4800:
4770:
4724:
4674:
4646:
4620:
4527:
4510:
4279:
4133:
4077:
4022:
4012:
3896:. They are analogous to Kazhdan–Lusztig polynomials, but are tailored to representations of
3874:. The odd-dimensional cohomology groups do not appear in the sum because they are all zero.
3842:
3312:
3215:
3203:
1017:
327:
178:
4909:
4686:
4489:
4206:
300:
135:
4905:
4840:
4712:
4682:
4634:
4485:
4104:
3542:
4. Some special cases of the Kazhdan–Lusztig conjectures are easy to verify. For example,
225:
155:
128:
4597:
4180:
3567:= 1, since the sum reduces to a single term. On the other hand, the first conjecture for
4848:
4720:
4642:
17:
4822:
4788:
4756:
4698:
4350:
4112:
3931:
711:
81:
5050:
4989:
4948:
4872:
4784:
4775:
4752:
4744:
4694:
4544:
3980:
3912:
3303:
in certain subquotient of the Verma module determined by a canonical filtration, the
1978:
213:
208:
of certain infinite dimensional representations of semisimple Lie algebras, given by
97:
77:
5033:
4666:
1622:
The Kazhdan–Lusztig polynomials can then be computed recursively using the relation
197:, and gave another definition of such a basis in terms of the dimensions of certain
5014:
4100:
3984:
3908:
3901:
3900:
semisimple Lie groups, and play major role in the conjectural description of their
2820:
852:
209:
190:
162:. They found a new construction of these representations over the complex numbers (
4893:
3863:, and then take the dimension of the stalk of this sheaf at any point of the cell
3827:{\displaystyle P_{y,w}(q)=\sum _{i}q^{i}\dim IH_{X_{y}}^{2i}({\overline {X_{w}}})}
4804:
3307:. The Jantzen conjecture in regular integral case was proved in a later paper of
4826:
4540:
3323:
2839:
5018:
4980:
4681:, Progress in Mathematics, vol. 87, Boston: Birkhauser, pp. 407–433,
3549:
is the antidominant Verma module, which is known to be simple. This means that
5007:
4940:
4677:(1990), "The Kazhdan–Lusztig conjecture for symmetrizable KacMoody algebras",
4598:"Kazhdan–Lusztig Polynomials: History, Problems, and Combinatorial Invariance"
3526:
3275:
2805:
124:
104:
4901:
4864:
4736:
4658:
3988:
999:{\displaystyle C'_{w}=q^{-{\frac {\ell (w)}{2}}}\sum _{y\leq w}P_{y,w}T_{y}}
232:
to representations of Weyl groups, led to the Kazhdan–Lusztig conjectures.
159:
108:
4203:
for crystallographic Coxeter groups satisfy certain strict inequality: Let
2882:
are locally-finite weight modules over the complex semisimple Lie algebra
5036:
software for computing Kazhdan–Lusztig polynomials for any Coxeter group
5026:
4856:
4728:
4650:
5040:
4517:, Sér. I Math., vol. 292, Paris: C. R. Acad. Sci., pp. 15–18
4177:
Kobayashi (2013) proved that values of Kazhdan–Lusztig polynomials at
1256:{\displaystyle T_{y^{-1}}^{-1}=\sum _{x}D(R_{x,y})q^{-\ell (x)}T_{x}.}
4931:
4701:(June 1979), "Representations of Coxeter groups and Hecke algebras",
2211:
4829:(1983), "Singularities of closures of K-orbits on flag manifolds.",
4623:(October 1981), "Kazhdan–Lusztig conjecture and holonomic systems",
3934:. The original (K-L) case is then about the details of decomposing
4970:
4551:, Progress in Mathematics, vol. 182, Boston, MA: Birkhäuser,
714:. From this it follows that the Hecke algebra has an automorphism
4759:(1980a), "A topological approach to Springer's representations",
1969:
using the fact that the two terms on the left are polynomials in
4795:, Proceedings of Symposia in Pure Mathematics, vol. XXXVI,
2797:, addressing a long-standing problem in representation theory.
3586:, together with the fact that all Kazhdan–Lusztig polynomials
2110:(or more generally any Coxeter group of rank at most 2) then
76:
is a member of a family of integral polynomials introduced by
4534:, Advances in Soviet Mathematics, vol. 16, pp. 1–50
3285:
coefficients of Kazhdan–Lusztig polynomials follows from the
3234:
1. The two conjectures are known to be equivalent. Moreover,
4048:
3947:
1605:
2214:(a variation of Kazhdan–Lusztig polynomials: see below) is
632:
The quadratic second relation implies that each generator
3637:
Relation to intersection cohomology of Schubert varieties
4569:; Brenti, Francesco (2005), "Ch. 5: Kazhdan–Lusztig and
4091:
Generalization to other objects in representation theory
839:, and uniquely determined by the following properties.
119:
In the spring of 1978 Kazhdan and Lusztig were studying
4462:"Computing Kazhdan-Lusztig-Vogan polynomials for split
3911:; or in other terms of actions on analogues of complex
1584:
1565:
1475:
1456:
1406:
1387:
1340:
1312:
597:
481:
5006:
from Spring 2005 course on Kazhdan–Lusztig Theory at
4373:
4353:
4308:
4282:
4241:
4209:
4183:
4044:
3943:
3718:
3592:
3335:
3289:, which roughly says that individual coefficients of
3064:
2915:
2226:
1631:
1275:
1151:
1050:
1020:
906:
746:
429:
356:
330:
303:
254:
138:
40:
4610:, Ellwangen: Haus Schönenberg: Research article B49b
3195:
is the element of maximal length of the Weyl group.
693:(obtained by multiplying the quadratic relation for
1266:They can be computed using the recursion relations
4919:Journal of the Institute of Mathematics of Jussieu
4435:
4359:
4339:
4294:
4268:
4227:
4195:
4111:. It turned out that the representation theory of
4064:
3963:
3826:
3618:
3473:
3316:
3219:
3207:
3175:
3049:
2759:
1958:
1611:
1255:
1129:) in terms of more elementary polynomials denoted
1106:
1036:
998:
799:
621:
412:
342:
316:
269:
144:
68:
639:is invertible in the Hecke algebra, with inverse
5043:for computing Kazhdan–Lusztig-Vogan polynomials.
4577:, Graduate Texts in Mathematics, vol. 231,
4166:
3326:showed as a consequence of the conjectures that
4161:. Some references are given in the textbook of
3692:More precisely, the Kazhdan–Lusztig polynomial
182:
163:
4099:of singularities of Schubert varieties in the
2838:where ρ is the half-sum of positive roots (or
2203:, giving examples of non-constant polynomials.
85:
4436:{\displaystyle P_{uw}(1)>P_{tu,w}(1)>0}
4162:
8:
3893:
3888:(also called Kazhdan–Lusztig polynomials or
5029:for computing Kazhdan–Lusztig polynomials.
4235:be a crystallographic Coxeter system and
1114:-module, called the Kazhdan–Lusztig basis.
281:(the smallest length of an expression for
4979:
4969:
4958:Journal of Combinatorial Theory, Series A
4930:
4774:
4403:
4378:
4372:
4352:
4313:
4307:
4281:
4247:
4242:
4240:
4208:
4182:
4165:. A research monograph on the subject is
4138:Beilinson–Bernstein–Deligne decomposition
4054:
4043:
3953:
3942:
3870:whose closure is the Schubert variety of
3810:
3804:
3792:
3785:
3780:
3761:
3751:
3723:
3717:
3608:
3597:
3591:
3563:, establishing the second conjecture for
3459:
3446:
3397:
3378:
3368:
3340:
3334:
3164:
3131:
3115:
3110:
3094:
3078:
3063:
3038:
3004:
2970:
2945:
2929:
2914:
2906:. The Kazhdan–Lusztig conjectures state:
2738:
2722:
2706:
2684:
2662:
2640:
2618:
2596:
2567:
2539:
2511:
2483:
2455:
2427:
2392:
2364:
2336:
2308:
2286:
2264:
2238:
2227:
2225:
2212:most complicated Lusztig–Vogan polynomial
2070:longest element of a finite Coxeter group
1944:
1925:
1849:
1848:
1814:
1783:
1764:
1715:
1714:
1692:
1637:
1636:
1630:
1583:
1564:
1541:
1516:
1474:
1455:
1432:
1405:
1386:
1363:
1339:
1311:
1295:
1280:
1274:
1244:
1222:
1203:
1187:
1171:
1161:
1156:
1150:
1091:
1084:
1067:
1063:
1052:
1051:
1049:
1025:
1019:
990:
974:
958:
931:
927:
911:
905:
788:
778:
773:
757:
745:
596:
565:
543:
480:
465:
448:
438:
430:
428:
397:
390:
373:
369:
358:
357:
355:
329:
308:
302:
253:
137:
88:). They are indexed by pairs of elements
45:
39:
800:{\displaystyle D(T_{w})=T_{w^{-1}}^{-1}}
185:they reinterpreted this in terms of the
5019:"Tables of Kazhdan–Lusztig polynomials"
4793:Schubert varieties and Poincaré duality
4452:
4035:. Then the relevant object of study is
1044:form a basis of the Hecke algebra as a
4603:Séminaire Lotharingien de Combinatoire
3525:is odd, so the dimensions of all such
667:. These inverses satisfy the relation
5067:Representation theory of Lie algebras
3661:is a disjoint union of affine spaces
3236:Borho–Jantzen's translation principle
7:
4276:its Kazhdan–Lusztig polynomials. If
2773:
827:) are indexed by a pair of elements
5062:Representation theory of Lie groups
4549:Singular loci of Schubert varieties
4078:L–V polynomials had been calculated
1014:of the Hecke algebra. The elements
1010:are invariant under the involution
4532:A proof of the Jantzen conjectures
168:Hecke algebra of the Coxeter group
25:
4347:, then there exists a reflection
3890:Kazhdan–Lusztig–Vogan polynomials
3856:), take its cohomology of degree
3270:for any dominant integral weight
2849:be its irreducible quotient, the
811:can be seen to be an involution.
420:, with multiplication defined by
103:, which can in particular be the
4679:The Grothendieck Festschrift, II
814:The Kazhdan–Lusztig polynomials
216:. This analogy, and the work of
4575:Combinatorics of Coxeter Groups
4065:{\displaystyle K\backslash G/B}
3964:{\displaystyle B\backslash G/B}
3678:. The closures of these spaces
3281:2. A similar interpretation of
3224:geometric representation theory
4424:
4418:
4393:
4387:
4340:{\displaystyle P_{uw}(1)>1}
4328:
4322:
4262:
4256:
4222:
4210:
4167:Billey & Lakshmibai (2000)
3821:
3801:
3741:
3735:
3468:
3465:
3439:
3422:
3416:
3407:
3401:
3390:
3358:
3352:
3170:
3157:
3148:
3142:
3084:
3071:
3044:
3031:
3022:
3016:
2995:
2989:
2980:
2974:
2967:
2957:
2935:
2922:
1937:
1918:
1910:
1907:
1901:
1892:
1886:
1874:
1868:
1859:
1839:
1833:
1824:
1818:
1811:
1801:
1755:
1752:
1746:
1737:
1731:
1725:
1704:
1685:
1677:
1674:
1668:
1659:
1653:
1647:
1509:
1497:
1235:
1229:
1215:
1196:
1101:
1056:
943:
937:
763:
750:
577:
558:
555:
536:
529:
523:
514:
508:
499:
490:
407:
362:
264:
258:
63:
57:
1:
4894:10.1090/S1088-4165-99-00074-6
4797:American Mathematical Society
3881:Generalization to real groups
277:for the length of an element
183:Kazhdan & Lusztig (1980b)
27:In the mathematical field of
4776:10.1016/0001-8708(80)90005-5
4159:pattern-avoidance phenomenon
3816:
2851:simple highest weight module
1107:{\displaystyle \mathbb {Z} }
413:{\displaystyle \mathbb {Z} }
285:as a product of elements of
4478:Nieuw Archief voor Wiskunde
4269:{\displaystyle {P_{uw}(q)}}
4163:Björner & Brenti (2005)
3926:is a complex Lie group and
3619:{\displaystyle P_{y,w_{0}}}
3584:character of a Verma module
2781:Kazhdan–Lusztig conjectures
164:Kazhdan & Lusztig 1980a
5093:
4981:10.1016/j.jcta.2012.10.001
4596:Brenti, Francesco (2003),
4460:van Leeuwen, Marc (2008),
3894:Lusztig & Vogan (1983)
3670:parameterized by elements
69:{\displaystyle P_{y,w}(q)}
33:Kazhdan–Lusztig polynomial
4941:10.1017/S1474748007000023
4515:Localisation de g-modules
3975:a classical theme of the
3886:Lusztig–Vogan polynomials
3849:(the closure of the cell
2898:) for the character of a
2890:, and therefore admit an
875:their degree is at most (
740:. More generally one has
170:and its representations.
4832:Inventiones Mathematicae
4805:10.1090/pspum/036/573434
4704:Inventiones Mathematicae
4626:Inventiones Mathematicae
4003:maximal compact subgroup
3582:and the formula for the
297:has a basis of elements
270:{\displaystyle \ell (w)}
121:Springer representations
18:Lusztig–Vogan polynomial
5077:Algebraic combinatorics
4762:Advances in Mathematics
4155:algebraic combinatorics
4130:intersection cohomology
3987:. The L-V case takes a
3653:of the algebraic group
3200:Alexander Beilinson
199:intersection cohomology
187:intersection cohomology
152:-adic cohomology groups
4437:
4361:
4341:
4296:
4295:{\displaystyle u<w}
4270:
4229:
4197:
4080:for the split form of
4066:
3965:
3828:
3620:
3580:Weyl character formula
3475:
3296:are multiplicities of
3212:Jean-Luc Brylinski
3177:
3051:
2761:
1960:
1613:
1257:
1108:
1038:
1037:{\displaystyle C'_{w}}
1000:
801:
623:
414:
344:
343:{\displaystyle w\in W}
318:
271:
146:
115:Motivation and history
70:
4882:Representation Theory
4438:
4362:
4342:
4297:
4271:
4230:
4228:{\displaystyle (W,S)}
4198:
4128:, such as the use of
4121:affine Hecke algebras
4067:
3979:, and before that of
3966:
3892:) were introduced in
3839:intersection homology
3829:
3621:
3476:
3178:
3052:
2791:semisimple Lie groups
2762:
2138:is the Coxeter group
2096:is the Coxeter group
1961:
1614:
1258:
1109:
1039:
1001:
802:
624:
415:
345:
319:
317:{\displaystyle T_{w}}
272:
147:
145:{\displaystyle \ell }
71:
29:representation theory
4799:, pp. 185–203,
4524:Beilinson, Alexandre
4507:Beilinson, Alexandre
4371:
4351:
4306:
4280:
4239:
4207:
4181:
4148:Combinatorial theory
4117:modular Lie algebras
4042:
3977:Bruhat decomposition
3941:
3716:
3643:Bruhat decomposition
3590:
3333:
3062:
2913:
2886:with the Weyl group
2224:
2145:with generating set
2011:has constant term 1.
1629:
1273:
1149:
1048:
1018:
904:
744:
427:
354:
328:
301:
252:
244:with generating set
240:Fix a Coxeter group
218:Jens Carsten Jantzen
136:
38:
4849:1983InMat..71..365L
4721:1979InMat..53..165K
4643:1981InMat..64..387B
4617:Brylinski, Jean-Luc
4196:{\displaystyle q=1}
4126:homological algebra
3800:
3252:can be replaced by
2894:. Let us write ch(
2892:algebraic character
1179:
1033:
919:
796:
230:enveloping algebras
5010:by Monica Vazirani
4857:10.1007/BF01389103
4729:10.1007/BF01390031
4651:10.1007/BF01389272
4433:
4357:
4337:
4292:
4266:
4225:
4193:
4062:
3961:
3824:
3776:
3756:
3687:Schubert varieties
3631:Kac–Moody algebras
3616:
3471:
3373:
3305:Jantzen filtration
3287:Jantzen conjecture
3173:
3105:
3047:
2956:
2853:of highest weight
2823:of highest weight
2757:
2755:
1956:
1800:
1609:
1604:
1588:
1569:
1479:
1460:
1410:
1391:
1344:
1316:
1253:
1192:
1152:
1104:
1034:
1021:
996:
969:
907:
843:They are 0 unless
797:
769:
619:
617:
601:
485:
410:
340:
314:
267:
206:Grothendieck group
179:Schubert varieties
142:
66:
5032:Fokko du Cloux's
4675:Kashiwara, Masaki
4621:Kashiwara, Masaki
4588:978-3-540-44238-7
4528:Bernstein, Joseph
4511:Bernstein, Joseph
4360:{\displaystyle t}
3819:
3747:
3578:follows from the
3364:
3090:
2941:
1857:
1779:
1723:
1645:
1587:
1568:
1478:
1459:
1409:
1390:
1343:
1315:
1183:
954:
950:
600:
484:
195:Robert MacPherson
156:conjugacy classes
78:David Kazhdan
16:(Redirected from
5084:
5072:Algebraic groups
5022:
4992:
4983:
4973:
4951:
4934:
4912:
4875:
4817:
4779:
4778:
4747:
4689:
4669:
4611:
4591:
4561:
4535:
4518:
4493:
4492:
4475:
4470:
4457:
4442:
4440:
4439:
4434:
4417:
4416:
4386:
4385:
4366:
4364:
4363:
4358:
4346:
4344:
4343:
4338:
4321:
4320:
4301:
4299:
4298:
4293:
4275:
4273:
4272:
4267:
4265:
4255:
4254:
4234:
4232:
4231:
4226:
4202:
4200:
4199:
4194:
4134:perverse sheaves
4109:quiver varieties
4105:nilpotent orbits
4071:
4069:
4068:
4063:
4058:
4034:
4023:complexification
4021:, and makes the
4020:
4013:semisimple group
4010:
3996:
3970:
3968:
3967:
3962:
3957:
3869:
3862:
3855:
3843:Schubert variety
3833:
3831:
3830:
3825:
3820:
3815:
3814:
3805:
3799:
3791:
3790:
3789:
3766:
3765:
3755:
3734:
3733:
3684:
3657:with Weyl group
3626:are equal to 1.
3625:
3623:
3622:
3617:
3615:
3614:
3613:
3612:
3524:
3501:
3480:
3478:
3477:
3472:
3464:
3463:
3451:
3450:
3435:
3434:
3383:
3382:
3372:
3351:
3350:
3302:
3295:
3273:
3269:
3251:
3216:Masaki Kashiwara
3204:Joseph Bernstein
3194:
3182:
3180:
3179:
3174:
3169:
3168:
3141:
3140:
3136:
3135:
3120:
3119:
3104:
3083:
3082:
3056:
3054:
3053:
3048:
3043:
3042:
3015:
3014:
2999:
2998:
2955:
2934:
2933:
2881:
2874:
2867:
2848:
2837:
2818:
2766:
2764:
2763:
2758:
2756:
2743:
2742:
2727:
2726:
2711:
2710:
2689:
2688:
2667:
2666:
2645:
2644:
2623:
2622:
2601:
2600:
2576:
2572:
2571:
2544:
2543:
2516:
2515:
2488:
2487:
2460:
2459:
2432:
2431:
2401:
2397:
2396:
2369:
2368:
2341:
2340:
2313:
2312:
2291:
2290:
2269:
2268:
2243:
2242:
2131:and 0 otherwise.
2040:
1965:
1963:
1962:
1957:
1955:
1954:
1936:
1935:
1914:
1913:
1858:
1850:
1843:
1842:
1799:
1775:
1774:
1759:
1758:
1724:
1716:
1703:
1702:
1681:
1680:
1646:
1638:
1618:
1616:
1615:
1610:
1608:
1607:
1589:
1585:
1570:
1566:
1558:
1557:
1530:
1529:
1480:
1476:
1461:
1457:
1449:
1448:
1411:
1407:
1392:
1388:
1380:
1379:
1345:
1341:
1317:
1313:
1291:
1290:
1262:
1260:
1259:
1254:
1249:
1248:
1239:
1238:
1214:
1213:
1191:
1178:
1170:
1169:
1168:
1113:
1111:
1110:
1105:
1100:
1099:
1095:
1076:
1075:
1071:
1055:
1043:
1041:
1040:
1035:
1029:
1005:
1003:
1002:
997:
995:
994:
985:
984:
968:
953:
952:
951:
946:
932:
915:
806:
804:
803:
798:
795:
787:
786:
785:
762:
761:
739:
732:
710:), and also the
706:
699:
692:
666:
638:
628:
626:
625:
620:
618:
602:
598:
594:
570:
569:
548:
547:
486:
482:
478:
473:
472:
453:
452:
443:
442:
419:
417:
416:
411:
406:
405:
401:
382:
381:
377:
361:
349:
347:
346:
341:
323:
321:
320:
315:
313:
312:
276:
274:
273:
268:
226:primitive ideals
175:Poincaré duality
151:
149:
148:
143:
75:
73:
72:
67:
56:
55:
21:
5092:
5091:
5087:
5086:
5085:
5083:
5082:
5081:
5047:
5046:
5013:
5000:
4955:
4916:
4879:
4841:Springer-Verlag
4823:Lusztig, George
4821:
4815:
4789:Lusztig, George
4783:
4757:Lusztig, George
4751:
4713:Springer-Verlag
4699:Lusztig, George
4693:
4673:
4635:Springer-Verlag
4615:
4595:
4589:
4567:Björner, Anders
4565:
4559:
4539:
4522:
4505:
4502:
4497:
4496:
4473:
4469:
4463:
4459:
4458:
4454:
4449:
4399:
4374:
4369:
4368:
4349:
4348:
4309:
4304:
4303:
4278:
4277:
4243:
4237:
4236:
4205:
4204:
4179:
4178:
4175:
4150:
4093:
4086:
4040:
4039:
4033:
4029:
4019:
4015:
4009:
4005:
3995:
3991:
3939:
3938:
3883:
3868:
3864:
3857:
3854:
3850:
3806:
3781:
3757:
3719:
3714:
3713:
3709:) is equal to
3704:
3683:
3679:
3669:
3639:
3604:
3593:
3588:
3587:
3577:
3562:
3555:
3548:
3503:
3498:
3491:
3485:
3455:
3442:
3393:
3374:
3336:
3331:
3330:
3301:
3297:
3294:
3290:
3271:
3253:
3239:
3232:
3193:
3187:
3160:
3127:
3111:
3106:
3074:
3060:
3059:
3034:
3000:
2966:
2925:
2911:
2910:
2880:
2876:
2873:
2869:
2854:
2847:
2843:
2824:
2817:
2813:
2808:. For each w ∈
2783:
2754:
2753:
2734:
2718:
2702:
2680:
2658:
2636:
2614:
2592:
2574:
2573:
2563:
2535:
2507:
2479:
2451:
2423:
2399:
2398:
2388:
2360:
2332:
2304:
2282:
2260:
2244:
2234:
2222:
2221:
2209:
2198:
2181:
2169:commuting then
2144:
2122:
2109:
2102:
2084:
2067:
2053:
2023:
2010:
1987:
1940:
1921:
1844:
1810:
1760:
1710:
1688:
1632:
1627:
1626:
1603:
1602:
1586: and
1562:
1537:
1512:
1494:
1493:
1477: and
1453:
1428:
1425:
1424:
1408: and
1384:
1359:
1356:
1355:
1337:
1328:
1327:
1309:
1296:
1276:
1271:
1270:
1240:
1218:
1199:
1157:
1147:
1146:
1137:
1123:
1080:
1059:
1046:
1045:
1016:
1015:
986:
970:
933:
923:
902:
901:
822:
774:
753:
742:
741:
738:
734:
731:
727:
712:braid relations
705:
701:
698:
694:
686:
677:
668:
660:
648:
640:
637:
633:
616:
615:
593:
580:
561:
539:
533:
532:
477:
461:
454:
444:
434:
425:
424:
386:
365:
352:
351:
326:
325:
304:
299:
298:
250:
249:
238:
134:
133:
129:algebraic group
117:
41:
36:
35:
23:
22:
15:
12:
11:
5:
5090:
5088:
5080:
5079:
5074:
5069:
5064:
5059:
5049:
5048:
5045:
5044:
5037:
5030:
5023:
5011:
4999:
4998:External links
4996:
4995:
4994:
4964:(2): 470–482,
4953:
4925:(3): 501–525,
4914:
4877:
4819:
4813:
4785:Kazhdan, David
4781:
4769:(2): 222–228,
4753:Kazhdan, David
4749:
4695:Kazhdan, David
4691:
4671:
4613:
4593:
4587:
4573:polynomials",
4563:
4557:
4545:Lakshmibai, V.
4537:
4520:
4501:
4498:
4495:
4494:
4484:(2): 113–116,
4467:
4451:
4450:
4448:
4445:
4432:
4429:
4426:
4423:
4420:
4415:
4412:
4409:
4406:
4402:
4398:
4395:
4392:
4389:
4384:
4381:
4377:
4356:
4336:
4333:
4330:
4327:
4324:
4319:
4316:
4312:
4291:
4288:
4285:
4264:
4261:
4258:
4253:
4250:
4246:
4224:
4221:
4218:
4215:
4212:
4192:
4189:
4186:
4174:
4171:
4149:
4146:
4113:quantum groups
4092:
4089:
4084:
4074:
4073:
4061:
4057:
4053:
4050:
4047:
4031:
4017:
4007:
3993:
3981:Schubert cells
3973:
3972:
3960:
3956:
3952:
3949:
3946:
3932:Borel subgroup
3913:flag manifolds
3882:
3879:
3866:
3852:
3835:
3834:
3823:
3818:
3813:
3809:
3803:
3798:
3795:
3788:
3784:
3779:
3775:
3772:
3769:
3764:
3760:
3754:
3750:
3746:
3743:
3740:
3737:
3732:
3729:
3726:
3722:
3696:
3681:
3665:
3638:
3635:
3611:
3607:
3603:
3600:
3596:
3575:
3560:
3553:
3546:
3496:
3489:
3482:
3481:
3470:
3467:
3462:
3458:
3454:
3449:
3445:
3441:
3438:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3396:
3392:
3389:
3386:
3381:
3377:
3371:
3367:
3363:
3360:
3357:
3354:
3349:
3346:
3343:
3339:
3299:
3292:
3231:
3228:
3191:
3184:
3183:
3172:
3167:
3163:
3159:
3156:
3153:
3150:
3147:
3144:
3139:
3134:
3130:
3126:
3123:
3118:
3114:
3109:
3103:
3100:
3097:
3093:
3089:
3086:
3081:
3077:
3073:
3070:
3067:
3057:
3046:
3041:
3037:
3033:
3030:
3027:
3024:
3021:
3018:
3013:
3010:
3007:
3003:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2969:
2965:
2962:
2959:
2954:
2951:
2948:
2944:
2940:
2937:
2932:
2928:
2924:
2921:
2918:
2878:
2871:
2845:
2815:
2782:
2779:
2778:
2777:
2770:
2769:
2768:
2767:
2752:
2749:
2746:
2741:
2737:
2733:
2730:
2725:
2721:
2717:
2714:
2709:
2705:
2701:
2698:
2695:
2692:
2687:
2683:
2679:
2676:
2673:
2670:
2665:
2661:
2657:
2654:
2651:
2648:
2643:
2639:
2635:
2632:
2629:
2626:
2621:
2617:
2613:
2610:
2607:
2604:
2599:
2595:
2591:
2588:
2585:
2582:
2579:
2577:
2575:
2570:
2566:
2562:
2559:
2556:
2553:
2550:
2547:
2542:
2538:
2534:
2531:
2528:
2525:
2522:
2519:
2514:
2510:
2506:
2503:
2500:
2497:
2494:
2491:
2486:
2482:
2478:
2475:
2472:
2469:
2466:
2463:
2458:
2454:
2450:
2447:
2444:
2441:
2438:
2435:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2402:
2400:
2395:
2391:
2387:
2384:
2381:
2378:
2375:
2372:
2367:
2363:
2359:
2356:
2353:
2350:
2347:
2344:
2339:
2335:
2331:
2328:
2325:
2322:
2319:
2316:
2311:
2307:
2303:
2300:
2297:
2294:
2289:
2285:
2281:
2278:
2275:
2272:
2267:
2263:
2259:
2256:
2253:
2250:
2247:
2245:
2241:
2237:
2233:
2230:
2229:
2216:
2215:
2207:
2204:
2190:
2173:
2142:
2132:
2114:
2107:
2100:
2090:
2076:
2065:
2055:
2045:
2039:) ∈ {0, 1, 2}
2012:
2002:
1986:
1983:
1979:constant terms
1967:
1966:
1953:
1950:
1947:
1943:
1939:
1934:
1931:
1928:
1924:
1920:
1917:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1856:
1853:
1847:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1813:
1809:
1806:
1803:
1798:
1795:
1792:
1789:
1786:
1782:
1778:
1773:
1770:
1767:
1763:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1722:
1719:
1713:
1709:
1706:
1701:
1698:
1695:
1691:
1687:
1684:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1644:
1641:
1635:
1620:
1619:
1606:
1601:
1598:
1595:
1592:
1582:
1579:
1576:
1573:
1563:
1561:
1556:
1553:
1550:
1547:
1544:
1540:
1536:
1533:
1528:
1525:
1522:
1519:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1495:
1492:
1489:
1486:
1483:
1473:
1470:
1467:
1464:
1454:
1452:
1447:
1444:
1441:
1438:
1435:
1431:
1427:
1426:
1423:
1420:
1417:
1414:
1404:
1401:
1398:
1395:
1385:
1383:
1378:
1375:
1372:
1369:
1366:
1362:
1358:
1357:
1354:
1351:
1348:
1338:
1336:
1333:
1330:
1329:
1326:
1323:
1320:
1310:
1308:
1305:
1302:
1301:
1299:
1294:
1289:
1286:
1283:
1279:
1264:
1263:
1252:
1247:
1243:
1237:
1234:
1231:
1228:
1225:
1221:
1217:
1212:
1209:
1206:
1202:
1198:
1195:
1190:
1186:
1182:
1177:
1174:
1167:
1164:
1160:
1155:
1142:). defined by
1133:
1121:
1116:
1115:
1103:
1098:
1094:
1090:
1087:
1083:
1079:
1074:
1070:
1066:
1062:
1058:
1054:
1032:
1028:
1024:
1008:
1007:
1006:
993:
989:
983:
980:
977:
973:
967:
964:
961:
957:
949:
945:
942:
939:
936:
930:
926:
922:
918:
914:
910:
896:
895:
892:
818:
794:
791:
784:
781:
777:
772:
768:
765:
760:
756:
752:
749:
736:
729:
703:
696:
682:
673:
656:
644:
635:
630:
629:
614:
611:
608:
605:
595:
592:
589:
586:
583:
581:
579:
576:
573:
568:
564:
560:
557:
554:
551:
546:
542:
538:
535:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
479:
476:
471:
468:
464:
460:
457:
455:
451:
447:
441:
437:
433:
432:
409:
404:
400:
396:
393:
389:
385:
380:
376:
372:
368:
364:
360:
350:over the ring
339:
336:
333:
311:
307:
266:
263:
260:
257:
237:
234:
222:Anthony Joseph
214:simple modules
141:
116:
113:
82:George Lusztig
65:
62:
59:
54:
51:
48:
44:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5089:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5054:
5052:
5042:
5038:
5035:
5031:
5028:
5024:
5020:
5016:
5015:Goresky, Mark
5012:
5009:
5005:
5002:
5001:
4997:
4991:
4987:
4982:
4977:
4972:
4967:
4963:
4959:
4954:
4950:
4946:
4942:
4938:
4933:
4928:
4924:
4920:
4915:
4911:
4907:
4903:
4899:
4895:
4891:
4888:(4): 90–104,
4887:
4883:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4850:
4846:
4842:
4838:
4834:
4833:
4828:
4824:
4820:
4816:
4814:9780821814390
4810:
4806:
4802:
4798:
4794:
4790:
4786:
4782:
4777:
4772:
4768:
4764:
4763:
4758:
4754:
4750:
4746:
4742:
4738:
4734:
4730:
4726:
4722:
4718:
4714:
4710:
4706:
4705:
4700:
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4627:
4622:
4618:
4614:
4609:
4605:
4604:
4599:
4594:
4590:
4584:
4580:
4576:
4572:
4568:
4564:
4560:
4558:0-8176-4092-4
4554:
4550:
4546:
4542:
4538:
4533:
4529:
4525:
4521:
4516:
4512:
4508:
4504:
4503:
4499:
4491:
4487:
4483:
4479:
4472:
4466:
4456:
4453:
4446:
4444:
4430:
4427:
4421:
4413:
4410:
4407:
4404:
4400:
4396:
4390:
4382:
4379:
4375:
4354:
4334:
4331:
4325:
4317:
4314:
4310:
4289:
4286:
4283:
4259:
4251:
4248:
4244:
4219:
4216:
4213:
4190:
4187:
4184:
4172:
4170:
4168:
4164:
4160:
4156:
4147:
4145:
4141:
4139:
4135:
4131:
4127:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4090:
4088:
4083:
4079:
4059:
4055:
4051:
4045:
4038:
4037:
4036:
4027:
4024:
4014:
4004:
4000:
3990:
3986:
3982:
3978:
3958:
3954:
3950:
3944:
3937:
3936:
3935:
3933:
3929:
3925:
3921:
3917:
3914:
3910:
3909:double cosets
3905:
3903:
3902:unitary duals
3899:
3895:
3891:
3887:
3880:
3878:
3875:
3873:
3861:
3848:
3844:
3840:
3811:
3807:
3796:
3793:
3786:
3782:
3777:
3773:
3770:
3767:
3762:
3758:
3752:
3748:
3744:
3738:
3730:
3727:
3724:
3720:
3712:
3711:
3710:
3708:
3703:
3699:
3695:
3690:
3688:
3677:
3673:
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3636:
3634:
3632:
3627:
3609:
3605:
3601:
3598:
3594:
3585:
3581:
3574:
3570:
3566:
3559:
3552:
3545:
3540:
3537:
3532:
3528:
3522:
3518:
3514:
3510:
3506:
3499:
3492:
3460:
3456:
3452:
3447:
3443:
3436:
3431:
3428:
3425:
3419:
3413:
3410:
3404:
3398:
3394:
3387:
3384:
3379:
3375:
3369:
3365:
3361:
3355:
3347:
3344:
3341:
3337:
3329:
3328:
3327:
3325:
3320:
3318:
3314:
3311: and
3310:
3306:
3288:
3284:
3279:
3277:
3268:
3264:
3260:
3256:
3250:
3246:
3242:
3238:implies that
3237:
3229:
3227:
3225:
3221:
3217:
3214: and
3213:
3209:
3205:
3202: and
3201:
3196:
3190:
3165:
3161:
3154:
3151:
3145:
3137:
3132:
3128:
3124:
3121:
3116:
3112:
3107:
3101:
3098:
3095:
3091:
3087:
3079:
3075:
3068:
3065:
3058:
3039:
3035:
3028:
3025:
3019:
3011:
3008:
3005:
3001:
2992:
2986:
2983:
2977:
2971:
2963:
2960:
2952:
2949:
2946:
2942:
2938:
2930:
2926:
2919:
2916:
2909:
2908:
2907:
2905:
2901:
2897:
2893:
2889:
2885:
2866:
2862:
2858:
2852:
2841:
2836:
2832:
2828:
2822:
2811:
2807:
2803:
2798:
2796:
2792:
2788:
2780:
2775:
2772:
2771:
2750:
2747:
2744:
2739:
2735:
2731:
2728:
2723:
2719:
2715:
2712:
2707:
2703:
2699:
2696:
2693:
2690:
2685:
2681:
2677:
2674:
2671:
2668:
2663:
2659:
2655:
2652:
2649:
2646:
2641:
2637:
2633:
2630:
2627:
2624:
2619:
2615:
2611:
2608:
2605:
2602:
2597:
2593:
2589:
2586:
2583:
2580:
2578:
2568:
2564:
2560:
2557:
2554:
2551:
2548:
2545:
2540:
2536:
2532:
2529:
2526:
2523:
2520:
2517:
2512:
2508:
2504:
2501:
2498:
2495:
2492:
2489:
2484:
2480:
2476:
2473:
2470:
2467:
2464:
2461:
2456:
2452:
2448:
2445:
2442:
2439:
2436:
2433:
2428:
2424:
2420:
2417:
2414:
2411:
2408:
2405:
2403:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2370:
2365:
2361:
2357:
2354:
2351:
2348:
2345:
2342:
2337:
2333:
2329:
2326:
2323:
2320:
2317:
2314:
2309:
2305:
2301:
2298:
2295:
2292:
2287:
2283:
2279:
2276:
2273:
2270:
2265:
2261:
2257:
2254:
2251:
2248:
2246:
2239:
2235:
2231:
2220:
2219:
2218:
2217:
2213:
2205:
2202:
2197:
2193:
2189:
2185:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2141:
2137:
2133:
2130:
2126:
2121:
2117:
2113:
2106:
2099:
2095:
2091:
2088:
2083:
2079:
2075:
2071:
2064:
2060:
2056:
2052:
2048:
2044:
2038:
2034:
2030:
2026:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1988:
1984:
1982:
1980:
1976:
1972:
1951:
1948:
1945:
1941:
1932:
1929:
1926:
1922:
1915:
1904:
1898:
1895:
1889:
1883:
1880:
1877:
1871:
1865:
1862:
1854:
1851:
1845:
1836:
1830:
1827:
1821:
1815:
1807:
1804:
1796:
1793:
1790:
1787:
1784:
1780:
1776:
1771:
1768:
1765:
1761:
1749:
1743:
1740:
1734:
1728:
1720:
1717:
1711:
1707:
1699:
1696:
1693:
1689:
1682:
1671:
1665:
1662:
1656:
1650:
1642:
1639:
1633:
1625:
1624:
1623:
1599:
1596:
1593:
1590:
1580:
1577:
1574:
1571:
1559:
1554:
1551:
1548:
1545:
1542:
1538:
1534:
1531:
1526:
1523:
1520:
1517:
1513:
1506:
1503:
1500:
1490:
1487:
1484:
1481:
1471:
1468:
1465:
1462:
1450:
1445:
1442:
1439:
1436:
1433:
1429:
1421:
1418:
1415:
1412:
1402:
1399:
1396:
1393:
1381:
1376:
1373:
1370:
1367:
1364:
1360:
1352:
1349:
1346:
1334:
1331:
1324:
1321:
1318:
1306:
1303:
1297:
1292:
1287:
1284:
1281:
1277:
1269:
1268:
1267:
1250:
1245:
1241:
1232:
1226:
1223:
1219:
1210:
1207:
1204:
1200:
1193:
1188:
1184:
1180:
1175:
1172:
1165:
1162:
1158:
1153:
1145:
1144:
1143:
1141:
1136:
1132:
1128:
1124:
1096:
1092:
1088:
1085:
1081:
1077:
1072:
1068:
1064:
1060:
1030:
1026:
1022:
1013:
1009:
991:
987:
981:
978:
975:
971:
965:
962:
959:
955:
947:
940:
934:
928:
924:
920:
916:
912:
908:
900:
899:
898:
897:
893:
890:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
846:
842:
841:
840:
838:
834:
830:
826:
821:
817:
812:
810:
792:
789:
782:
779:
775:
770:
766:
758:
754:
747:
725:
721:
717:
713:
709:
690:
685:
681:
676:
672:
664:
659:
655:
652:
647:
643:
612:
609:
606:
603:
590:
587:
584:
582:
574:
571:
566:
562:
552:
549:
544:
540:
526:
520:
517:
511:
505:
502:
496:
493:
487:
474:
469:
466:
462:
458:
456:
449:
445:
439:
435:
423:
422:
421:
402:
398:
394:
391:
387:
383:
378:
374:
370:
366:
337:
334:
331:
309:
305:
296:
292:
291:Hecke algebra
288:
284:
280:
261:
255:
247:
243:
235:
233:
231:
227:
223:
219:
215:
211:
210:Verma modules
207:
202:
200:
196:
192:
188:
184:
180:
176:
171:
169:
165:
161:
157:
153:
139:
130:
126:
122:
114:
112:
110:
106:
102:
99:
98:Coxeter group
95:
91:
87:
83:
80: and
79:
60:
52:
49:
46:
42:
34:
30:
19:
4961:
4957:
4932:math/0403496
4922:
4918:
4885:
4881:
4836:
4830:
4827:Vogan, David
4792:
4766:
4760:
4708:
4702:
4678:
4630:
4624:
4607:
4601:
4574:
4570:
4548:
4541:Billey, Sara
4531:
4514:
4481:
4477:
4464:
4455:
4176:
4158:
4151:
4142:
4101:flag variety
4094:
4081:
4075:
4025:
3998:
3985:Grassmannian
3974:
3927:
3923:
3919:
3915:
3906:
3897:
3889:
3885:
3884:
3876:
3871:
3859:
3846:
3836:
3706:
3701:
3697:
3693:
3691:
3675:
3671:
3666:
3662:
3658:
3654:
3650:
3646:
3640:
3628:
3572:
3568:
3564:
3557:
3550:
3543:
3541:
3535:
3530:
3529:in category
3520:
3516:
3512:
3508:
3504:
3502:vanishes if
3494:
3487:
3483:
3321:
3286:
3282:
3280:
3266:
3262:
3258:
3254:
3248:
3244:
3240:
3233:
3223:
3197:
3188:
3185:
2903:
2899:
2895:
2887:
2883:
2864:
2860:
2856:
2834:
2830:
2826:
2821:Verma module
2809:
2804:be a finite
2801:
2799:
2795:Lie algebras
2786:
2784:
2200:
2195:
2191:
2187:
2183:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2139:
2135:
2128:
2124:
2119:
2115:
2111:
2104:
2097:
2093:
2086:
2085:= 1 for all
2081:
2077:
2073:
2062:
2058:
2050:
2046:
2042:
2036:
2032:
2028:
2024:
2019:
2015:
2007:
2003:
1999:
1995:
1991:
1974:
1970:
1968:
1621:
1265:
1139:
1134:
1130:
1126:
1119:
1117:
1011:
894:The elements
888:
884:
880:
876:
872:
868:
864:
860:
856:
853:Bruhat order
848:
844:
836:
832:
828:
824:
819:
815:
813:
808:
723:
719:
715:
707:
688:
683:
679:
674:
670:
662:
657:
653:
650:
645:
641:
631:
294:
286:
282:
278:
248:, and write
245:
241:
239:
203:
191:Mark Goresky
172:
118:
100:
93:
89:
32:
26:
5057:Polynomials
4843:: 365–379,
4715:: 165–184,
4637:: 387–410,
3685:are called
3324:David Vogan
2842:), and let
2840:Weyl vector
2787:Inventiones
2774:Polo (1999)
718:that sends
154:related to
5051:Categories
5008:U.C. Davis
4500:References
4367:such that
4173:Inequality
4157:, such as
3645:the space
3527:Ext groups
3276:category O
2812:denote by
2806:Weyl group
867:, and for
236:Definition
158:which are
125:Weyl group
105:Weyl group
4990:205929043
4971:1211.4305
4949:120459494
4902:1088-4165
4873:120917588
4865:0020-9910
4791:(1980b),
4745:120098142
4737:0020-9910
4659:0020-9910
4049:∖
3989:real form
3948:∖
3817:¯
3771:
3749:∑
3484:and that
3437:
3426:−
3414:ℓ
3411:−
3399:ℓ
3388:
3366:∑
3313:Bernstein
3309:Beilinson
3210:) and by
3155:
3099:≤
3092:∑
3069:
3029:
2987:ℓ
2984:−
2972:ℓ
2961:−
2950:≤
2943:∑
2920:
1899:ℓ
1896:−
1884:ℓ
1866:ℓ
1863:−
1831:ℓ
1816:ℓ
1805:−
1794:≤
1781:∑
1744:ℓ
1741:−
1729:ℓ
1708:−
1666:ℓ
1663:−
1651:ℓ
1504:−
1227:ℓ
1224:−
1185:∑
1173:−
1163:−
1086:−
963:≤
956:∑
935:ℓ
929:−
891:) − 1)/2.
790:−
780:−
726:and each
607:∈
572:−
521:ℓ
506:ℓ
488:ℓ
392:−
335:∈
256:ℓ
224:relating
160:unipotent
140:ℓ
109:Lie group
5041:software
5027:programs
5025:The GAP
5004:Readings
4667:18403883
4579:Springer
4547:(2000),
4530:(1993),
4513:(1981),
4097:geometry
4011:in that
2902:-module
2123:is 1 if
1985:Examples
1977:without
1567:if
1458:if
1389:if
1342:if
1322:≰
1314:if
1031:′
917:′
859:), 1 if
851:(in the
599:if
483:if
289:). The
201:groups.
5034:Coxeter
4910:1698201
4845:Bibcode
4717:Bibcode
4687:1106905
4639:Bibcode
4490:2454587
3841:of the
3641:By the
3315: (
3230:Remarks
3218: (
3206: (
2868:. Both
2819:be the
2161:} with
2068:is the
807:; also
123:of the
84: (
5039:Atlas
4988:
4947:
4908:
4900:
4871:
4863:
4811:
4743:
4735:
4685:
4665:
4657:
4585:
4555:
4488:
3922:where
3186:where
2199:= 1 +
2182:= 1 +
127:of an
96:of a
4986:S2CID
4966:arXiv
4945:S2CID
4927:arXiv
4869:S2CID
4839:(2),
4741:S2CID
4711:(2),
4663:S2CID
4633:(3),
4474:(PDF)
4447:Notes
3983:in a
2196:acbca
2186:and
2072:then
2041:then
1998:then
871:<
691:) = 0
678:+ 1)(
181:. In
107:of a
4898:ISSN
4861:ISSN
4809:ISBN
4733:ISSN
4655:ISSN
4583:ISBN
4553:ISBN
4428:>
4397:>
4332:>
4302:and
4287:<
4136:and
4119:and
4107:and
4001:, a
3898:real
3515:) +
3486:Ext(
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3220:1981
3208:1981
2875:and
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2800:Let
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2210:the
2179:bacb
2165:and
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2031:) −
2022:and
1973:and
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324:for
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4976:doi
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2324:370
2302:021
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2149:= {
2134:If
2103:or
2092:If
2057:If
2014:If
1990:If
855:of
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722:to
700:by
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2018:≤
1994:≤
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2016:y
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1992:y
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