1589:
1447:
3064:(1954) , "Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)" [Linear representations and Kähler homogeneous spaces of compact Lie groups (after Armand Borel and André Weil)],
2727:
2468:
1452:
1584:{\displaystyle {\begin{aligned}H&=x{\frac {\partial }{\partial x}}-y{\frac {\partial }{\partial y}},\\X&=x{\frac {\partial }{\partial y}},\\Y&=y{\frac {\partial }{\partial x}}.\end{aligned}}}
470:
1439:
1290:
1067:
2876:
979:
2236:
1840:
1701:
873:
798:
532:
1374:
1332:
1940:
335:
2566:
669:
1205:
908:
1962:
366:
122:
1735:
1009:
727:
611:
1866:
1761:
1637:
1109:
701:
637:
585:
404:
2007:
of the group. The Borel–Weil–Bott theorem is its generalization to higher cohomology spaces. The theorem dates back to the early 1950s and can be found in
3127:
2643:
2370:
17:
1384:. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: if
3187:
3192:
66:, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by
1884:
is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules
2139:
2988:
2941:
421:
2305:. (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is
1399:
1250:
1018:
2278:
2819:
917:
3100:
372:-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as
2185:
1781:
1642:
814:
739:
495:
1337:
1295:
2960:
1887:
282:
147:
3043:
3033:
2972:
2913:
2256:
2508:
2610:, with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for
1872:-module is simple in general, although it does contain the unique highest weight module of highest weight
3038:
642:
2964:
1981:
1610:
1177:
95:
75:
46:, showing how a family of representations can be obtained from holomorphic sections of certain complex
878:
3136:
1945:
344:
105:
2752:
2743:
2000:
1768:
235:
2620:, with dominant weights corresponding to nonnegative integers, and the corresponding characters
1714:
988:
706:
590:
2030:
1996:
1213:
1209:
543:
99:
39:
1845:
1740:
1616:
1088:
680:
616:
564:
386:
3157:
2998:
2600:
1992:
480:. It is straightforward to check that this defines a group action, although this action is
8:
272:
3171:
3096:
3061:
3026:
3002:
2984:
2937:
2102:
1985:
194:
3050:
1771:. However, the other statements of the theorem do not remain valid in this setting.
3145:
3016:
2976:
2722:{\displaystyle \chi _{n}{\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}}=a^{n}.}
2089:
1605:
One also has a weaker form of this theorem in positive characteristic. Namely, let
1174:
276:
216:
79:
51:
3153:
2994:
2463:{\displaystyle f:G\to \mathbb {C} _{\lambda }:f(gb)=\chi _{\lambda }(b^{-1})f(g)}
2362:
2323:
gives rise to a character (one-dimensional representation) of the Borel subgroup
2129:
2045:
1989:
1381:
3114:, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press
2055:
1148:
182:
132:
2980:
3181:
3006:
2004:
125:
47:
63:
3078:
2079:
1964:, Mumford gave an example showing that it need not be the case for a fixed
59:
3149:
3112:
Representation theory of semisimple groups: An overview based on examples
3054:
257:
31:
3167:
1595:
380:
67:
2975:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
2801:
and forms an irreducible representation under the standard action of
43:
3095:, Graduate Texts in Mathematics, vol. 235, New York: Springer,
2023:
The theorem can be stated either for a complex semisimple Lie group
3166:
This article incorporates material from Borel–Bott–Weil theorem on
2775:
is identified with the space of homogeneous polynomials of degree
3021:
The
Penrose Transform: its Interaction with Representation Theory
2617:
3085:, Acad. Roy. Belg. Cl. Sci. Mém. Coll. (in French), vol. 29
981:
is the dual of the irreducible highest-weight representation of
1125:
54:
groups associated to such bundles. It is built on the earlier
1995:. These representations are realized in the spaces of global
1015:
It is worth noting that case (1) above occurs if and only if
3083:
Sur certaines classes d'espaces homogènes de groupes de Lie
484:
linear, unlike the usual Weyl group action. Also, a weight
71:
1968:
that these modules are all zero except in a single degree
1778:
be a dominant integral weight; then it is still true that
158:
defines in a natural way a one-dimensional representation
2764:
and the space of the global sections of the line bundle
3130:(1998). "Borel–Weil–Bott theory on the moduli stack of
1247:
This gives us at a stroke the representation theory of
1151:, an integral weight is specified simply by an integer
368:
by bundle automorphisms, this action naturally gives a
27:
Basic result in the representation theory of Lie groups
2662:
2340:. Holomorphic sections of the holomorphic line bundle
2101:. The flag variety can also be described as a compact
613:
is dominant, equivalently, there exists a nonidentity
2822:
2646:
2511:
2373:
2188:
1980:
The Borel–Weil theorem provides a concrete model for
1948:
1890:
1848:
1784:
1743:
1717:
1645:
1619:
1450:
1402:
1340:
1298:
1253:
1180:
1091:
1021:
991:
920:
881:
817:
742:
709:
683:
645:
619:
593:
567:
498:
465:{\displaystyle w*\lambda :=w(\lambda +\rho )-\rho \,}
424:
389:
347:
285:
108:
733:The theorem states that in the first case, we have
2936:(second ed.). American Mathematical Society.
2870:
2721:
2560:
2462:
2230:
1956:
1934:
1860:
1834:
1755:
1729:
1695:
1631:
1583:
1434:{\displaystyle {\mathfrak {sl}}_{2}(\mathbf {C} )}
1433:
1368:
1326:
1285:{\displaystyle {\mathfrak {sl}}_{2}(\mathbf {C} )}
1284:
1199:
1103:
1062:{\displaystyle (\lambda +\rho )(\beta ^{\vee })=0}
1061:
1003:
973:
902:
867:
792:
721:
695:
663:
631:
605:
579:
526:
464:
398:
360:
329:
116:
2293:, and each irreducible unitary representation of
3179:
3172:Creative Commons Attribution/Share-Alike License
2871:{\displaystyle X^{i}Y^{n-i},\quad 0\leq i\leq n}
974:{\displaystyle H^{\ell (w)}(G/B,\,L_{\lambda })}
1988:and irreducible holomorphic representations of
3014:
2299:is obtained in this way for a unique value of
2251:integral weight then this representation is a
2959:
2231:{\displaystyle \Gamma (G/B,L_{\lambda }).\ }
1077:as a special case of this theorem by taking
2018:
1835:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
1696:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
868:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
793:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
527:{\displaystyle \mu (\alpha ^{\vee })\geq 0}
201:. Since we can think of the projection map
1600:
1369:{\displaystyle \Gamma ({\mathcal {O}}(n))}
1327:{\displaystyle \Gamma ({\mathcal {O}}(1))}
476:denotes the half-sum of positive roots of
2388:
1950:
1935:{\displaystyle H^{i}(G/B,\,L_{\lambda })}
1918:
1812:
1767:is "close to zero". This is known as the
1673:
957:
845:
770:
461:
330:{\displaystyle H^{i}(G/B,\,L_{\lambda })}
313:
275:of holomorphic sections, we consider the
110:
3090:
2813:. Weight vectors are given by monomials
1609:be a semisimple algebraic group over an
173:, by pulling back the representation on
70:. One can equivalently, through Serre's
3126:
2931:
2312:
14:
3180:
3051:A Proof of the Borel–Weil–Bott Theorem
2925:
2561:{\displaystyle g\cdot f(h)=f(g^{-1}h)}
2241:The Borel–Weil theorem states that if
2179:acts on its space of global sections,
341:acts on the total space of the bundle
256:(note the sign), which is obviously a
3109:
3060:
2969:Representation theory. A first course
2008:
1975:
1868:, but it is no longer true that this
3077:
2361:may be described more concretely as
2012:
1334:is the standard representation, and
3188:Representation theory of Lie groups
2934:Representations of algebraic groups
1409:
1406:
1260:
1257:
1236:, and is canonically isomorphic to
664:{\displaystyle w*\lambda =\lambda }
50:, and, more generally, from higher
24:
3120:
2279:irreducible unitary representation
2189:
1565:
1561:
1530:
1526:
1495:
1491:
1474:
1470:
1349:
1341:
1307:
1299:
1183:
25:
3204:
3193:Theorems in representation theory
1228:, the sections can be written as
1200:{\displaystyle {\mathcal {O}}(n)}
2892:, and the highest weight vector
1424:
1275:
1073:. Also, we obtain the classical
808:and in the second case, we have
3116:. Reprint of the 1986 original.
2852:
2502:on these sections is given by
1396:are the standard generators of
3170:, which is licensed under the
2932:Jantzen, Jens Carsten (2003).
2555:
2536:
2527:
2521:
2457:
2451:
2445:
2429:
2413:
2404:
2383:
2219:
2192:
2048:complex semisimple Lie group,
1929:
1901:
1823:
1795:
1684:
1656:
1428:
1420:
1363:
1360:
1354:
1344:
1321:
1318:
1312:
1302:
1279:
1271:
1194:
1188:
1050:
1037:
1034:
1022:
968:
940:
935:
929:
903:{\displaystyle i\neq \ell (w)}
897:
891:
856:
828:
781:
753:
515:
502:
452:
440:
379:We first need to describe the
324:
296:
85:
13:
1:
3057:. Retrieved on Jul. 13, 2014.
2973:Graduate Texts in Mathematics
2953:
2914:Theorem of the highest weight
2257:highest weight representation
2588:
2092:and a nonsingular algebraic
1957:{\displaystyle \mathbb {C} }
1639:. Then it remains true that
361:{\displaystyle L_{\lambda }}
117:{\displaystyle \mathbb {C} }
7:
3039:Encyclopedia of Mathematics
2907:
2794:, this space has dimension
2742:may be identified with the
1982:irreducible representations
1085:to be the identity element
74:, view this as a result in
10:
3209:
3110:Knapp, Anthony W. (2001),
3091:Sepanski, Mark R. (2007),
2807:on the polynomial algebra
1730:{\displaystyle w*\lambda }
1611:algebraically closed field
1593:
1224:). As a representation of
1114:
1004:{\displaystyle w*\lambda }
722:{\displaystyle w*\lambda }
606:{\displaystyle w*\lambda }
557:, one of two cases occur:
406:. For any integral weight
76:complex algebraic geometry
3034:"Bott–Borel–Weil theorem"
3023:, Oxford University Press
2981:10.1007/978-1-4612-0979-9
553:Given an integral weight
38:is a basic result in the
3137:Inventiones Mathematicae
3134:-bundles over a curve".
2919:
2156:holomorphic line bundle
2019:Statement of the theorem
2001:holomorphic line bundles
1942:in general. Unlike over
1737:is non-dominant for all
2753:homogeneous coordinates
2744:complex projective line
2616:may be identified with
1769:Kempf vanishing theorem
1601:Positive characteristic
1214:homogeneous polynomials
1069:for some positive root
236:associated fiber bundle
36:Borel–Weil–Bott theorem
18:Borel–Bott–Weil theorem
2872:
2723:
2562:
2464:
2232:
1958:
1936:
1862:
1861:{\displaystyle i>0}
1836:
1757:
1756:{\displaystyle w\in W}
1731:
1711:is a weight such that
1697:
1633:
1632:{\displaystyle p>0}
1585:
1435:
1370:
1328:
1286:
1201:
1119:For example, consider
1105:
1104:{\displaystyle e\in W}
1063:
1005:
975:
904:
869:
794:
723:
697:
696:{\displaystyle w\in W}
665:
633:
632:{\displaystyle w\in W}
607:
581:
580:{\displaystyle w\in W}
528:
466:
400:
399:{\displaystyle -\rho }
362:
331:
118:
3150:10.1007/s002220050257
2873:
2724:
2563:
2465:
2271:. Its restriction to
2233:
1993:semisimple Lie groups
1959:
1937:
1863:
1837:
1774:More explicitly, let
1758:
1732:
1698:
1634:
1594:Further information:
1586:
1436:
1371:
1329:
1287:
1202:
1106:
1064:
1006:
976:
905:
870:
795:
724:
698:
666:
634:
608:
582:
534:for all simple roots
529:
467:
401:
363:
332:
119:
40:representation theory
3017:Eastwood, Michael G.
2820:
2644:
2601:special linear group
2509:
2371:
2313:Concrete description
2287:with highest weight
2265:with highest weight
2186:
2082:. In this scenario,
1946:
1888:
1846:
1782:
1741:
1715:
1643:
1617:
1448:
1400:
1338:
1296:
1251:
1178:
1089:
1019:
989:
985:with highest weight
918:
879:
815:
740:
707:
681:
643:
617:
591:
565:
496:
422:
387:
345:
283:
106:
3128:Teleman, Constantin
3093:Compact Lie groups.
3015:Baston, Robert J.;
2329:, which is denoted
1081:to be dominant and
383:action centered at
3066:Séminaire Bourbaki
3062:Serre, Jean-Pierre
2868:
2719:
2697:
2558:
2460:
2228:
1986:compact Lie groups
1976:Borel–Weil theorem
1954:
1932:
1858:
1832:
1753:
1727:
1693:
1629:
1613:of characteristic
1581:
1579:
1431:
1366:
1324:
1282:
1197:
1162:. The line bundle
1101:
1075:Borel–Weil theorem
1059:
1001:
971:
900:
865:
790:
719:
693:
661:
629:
603:
577:
524:
462:
414:in the Weyl group
396:
358:
327:
114:
56:Borel–Weil theorem
2990:978-0-387-97495-8
2943:978-0-8218-3527-2
2732:The flag variety
2227:
2103:homogeneous space
1572:
1537:
1502:
1481:
195:unipotent radical
16:(Redirected from
3200:
3161:
3115:
3105:
3086:
3073:
3047:
3024:
3010:
2948:
2947:
2929:
2903:
2897:
2891:
2877:
2875:
2874:
2869:
2848:
2847:
2832:
2831:
2812:
2806:
2800:
2793:
2786:
2780:
2774:
2763:
2750:
2741:
2728:
2726:
2725:
2720:
2715:
2714:
2702:
2701:
2694:
2693:
2656:
2655:
2636:
2630:
2615:
2609:
2598:
2584:
2567:
2565:
2564:
2559:
2551:
2550:
2501:
2492:
2482:
2469:
2467:
2466:
2461:
2444:
2443:
2428:
2427:
2397:
2396:
2391:
2363:holomorphic maps
2360:
2350:
2339:
2328:
2322:
2304:
2298:
2292:
2286:
2276:
2270:
2264:
2246:
2237:
2235:
2234:
2229:
2225:
2218:
2217:
2202:
2178:
2172:
2166:
2155:
2153:
2146:
2137:
2127:
2113:
2100:
2098:
2090:complex manifold
2087:
2077:
2063:
2053:
2043:
2037:
2028:
1971:
1967:
1963:
1961:
1960:
1955:
1953:
1941:
1939:
1938:
1933:
1928:
1927:
1911:
1900:
1899:
1883:
1879:
1875:
1871:
1867:
1865:
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1859:
1841:
1839:
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1833:
1822:
1821:
1805:
1794:
1793:
1777:
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1762:
1760:
1759:
1754:
1736:
1734:
1733:
1728:
1710:
1706:
1702:
1700:
1699:
1694:
1683:
1682:
1666:
1655:
1654:
1638:
1636:
1635:
1630:
1608:
1590:
1588:
1587:
1582:
1580:
1573:
1571:
1560:
1538:
1536:
1525:
1503:
1501:
1490:
1482:
1480:
1469:
1440:
1438:
1437:
1432:
1427:
1419:
1418:
1413:
1412:
1395:
1391:
1387:
1379:
1375:
1373:
1372:
1367:
1353:
1352:
1333:
1331:
1330:
1325:
1311:
1310:
1291:
1289:
1288:
1283:
1278:
1270:
1269:
1264:
1263:
1243:
1235:
1227:
1219:
1206:
1204:
1203:
1198:
1187:
1186:
1172:
1161:
1154:
1146:
1136:
1110:
1108:
1107:
1102:
1084:
1080:
1072:
1068:
1066:
1065:
1060:
1049:
1048:
1010:
1008:
1007:
1002:
984:
980:
978:
977:
972:
967:
966:
950:
939:
938:
909:
907:
906:
901:
874:
872:
871:
866:
855:
854:
838:
827:
826:
803:
799:
797:
796:
791:
780:
779:
763:
752:
751:
728:
726:
725:
720:
702:
700:
699:
694:
670:
668:
667:
662:
638:
636:
635:
630:
612:
610:
609:
604:
586:
584:
583:
578:
556:
549:
541:
537:
533:
531:
530:
525:
514:
513:
487:
479:
475:
471:
469:
468:
463:
417:
413:
409:
405:
403:
402:
397:
375:
371:
367:
365:
364:
359:
357:
356:
340:
336:
334:
333:
328:
323:
322:
306:
295:
294:
277:sheaf cohomology
270:
255:
245:
233:
220:
214:
200:
192:
188:
172:
168:
157:
153:
145:
141:
137:
130:
123:
121:
120:
115:
113:
93:
80:Zariski topology
52:sheaf cohomology
21:
3208:
3207:
3203:
3202:
3201:
3199:
3198:
3197:
3178:
3177:
3123:
3121:Further reading
3103:
3032:
2991:
2961:Fulton, William
2956:
2951:
2944:
2930:
2926:
2922:
2910:
2899:
2893:
2882:
2837:
2833:
2827:
2823:
2821:
2818:
2817:
2808:
2802:
2795:
2788:
2782:
2776:
2773:
2765:
2755:
2746:
2733:
2710:
2706:
2696:
2695:
2686:
2682:
2680:
2674:
2673:
2668:
2658:
2657:
2651:
2647:
2645:
2642:
2641:
2632:
2629:
2621:
2611:
2603:
2599:be the complex
2594:
2591:
2572:
2543:
2539:
2510:
2507:
2506:
2497:
2484:
2474:
2436:
2432:
2423:
2419:
2392:
2387:
2386:
2372:
2369:
2368:
2352:
2349:
2341:
2338:
2330:
2324:
2318:
2315:
2300:
2294:
2288:
2282:
2272:
2266:
2260:
2242:
2213:
2209:
2198:
2187:
2184:
2183:
2174:
2168:
2165:
2157:
2149:
2148:
2142:
2140:integral weight
2133:
2130:Cartan subgroup
2128:is a (compact)
2115:
2105:
2094:
2093:
2083:
2065:
2059:
2049:
2039:
2033:
2024:
2021:
1978:
1969:
1965:
1949:
1947:
1944:
1943:
1923:
1919:
1907:
1895:
1891:
1889:
1886:
1885:
1881:
1880:-submodule. If
1877:
1873:
1869:
1847:
1844:
1843:
1817:
1813:
1801:
1789:
1785:
1783:
1780:
1779:
1775:
1764:
1742:
1739:
1738:
1716:
1713:
1712:
1708:
1704:
1678:
1674:
1662:
1650:
1646:
1644:
1641:
1640:
1618:
1615:
1614:
1606:
1603:
1598:
1578:
1577:
1564:
1559:
1549:
1543:
1542:
1529:
1524:
1514:
1508:
1507:
1494:
1489:
1473:
1468:
1458:
1451:
1449:
1446:
1445:
1423:
1414:
1405:
1404:
1403:
1401:
1398:
1397:
1393:
1389:
1385:
1382:symmetric power
1377:
1348:
1347:
1339:
1336:
1335:
1306:
1305:
1297:
1294:
1293:
1274:
1265:
1256:
1255:
1254:
1252:
1249:
1248:
1237:
1229:
1225:
1217:
1182:
1181:
1179:
1176:
1175:
1171:
1163:
1156:
1152:
1138:
1129:
1120:
1117:
1090:
1087:
1086:
1082:
1078:
1070:
1044:
1040:
1020:
1017:
1016:
990:
987:
986:
982:
962:
958:
946:
925:
921:
919:
916:
915:
880:
877:
876:
850:
846:
834:
822:
818:
816:
813:
812:
801:
775:
771:
759:
747:
743:
741:
738:
737:
708:
705:
704:
682:
679:
678:
644:
641:
640:
618:
615:
614:
592:
589:
588:
566:
563:
562:
554:
547:
544:length function
539:
535:
509:
505:
497:
494:
493:
485:
477:
473:
423:
420:
419:
415:
411:
407:
388:
385:
384:
373:
369:
352:
348:
346:
343:
342:
338:
318:
314:
302:
290:
286:
284:
281:
280:
269:
261:
247:
244:
238:
232:
224:
218:
202:
198:
190:
174:
170:
167:
159:
155:
151:
148:integral weight
143:
139:
138:which contains
135:
128:
109:
107:
104:
103:
100:algebraic group
91:
88:
28:
23:
22:
15:
12:
11:
5:
3206:
3196:
3195:
3190:
3163:
3162:
3122:
3119:
3118:
3117:
3107:
3101:
3088:
3075:
3072:(100): 447–454
3058:
3048:
3030:
3012:
2989:
2955:
2952:
2950:
2949:
2942:
2923:
2921:
2918:
2917:
2916:
2909:
2906:
2879:
2878:
2867:
2864:
2861:
2858:
2855:
2851:
2846:
2843:
2840:
2836:
2830:
2826:
2769:
2730:
2729:
2718:
2713:
2709:
2705:
2700:
2692:
2689:
2685:
2681:
2679:
2676:
2675:
2672:
2669:
2667:
2664:
2663:
2661:
2654:
2650:
2637:have the form
2625:
2590:
2587:
2569:
2568:
2557:
2554:
2549:
2546:
2542:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2496:The action of
2471:
2470:
2459:
2456:
2453:
2450:
2447:
2442:
2439:
2435:
2431:
2426:
2422:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2395:
2390:
2385:
2382:
2379:
2376:
2345:
2334:
2314:
2311:
2239:
2238:
2224:
2221:
2216:
2212:
2208:
2205:
2201:
2197:
2194:
2191:
2173:and the group
2161:
2056:Borel subgroup
2020:
2017:
1977:
1974:
1952:
1931:
1926:
1922:
1917:
1914:
1910:
1906:
1903:
1898:
1894:
1857:
1854:
1851:
1831:
1828:
1825:
1820:
1816:
1811:
1808:
1804:
1800:
1797:
1792:
1788:
1752:
1749:
1746:
1726:
1723:
1720:
1692:
1689:
1686:
1681:
1677:
1672:
1669:
1665:
1661:
1658:
1653:
1649:
1628:
1625:
1622:
1602:
1599:
1592:
1591:
1576:
1570:
1567:
1563:
1558:
1555:
1552:
1550:
1548:
1545:
1544:
1541:
1535:
1532:
1528:
1523:
1520:
1517:
1515:
1513:
1510:
1509:
1506:
1500:
1497:
1493:
1488:
1485:
1479:
1476:
1472:
1467:
1464:
1461:
1459:
1457:
1454:
1453:
1430:
1426:
1422:
1417:
1411:
1408:
1365:
1362:
1359:
1356:
1351:
1346:
1343:
1323:
1320:
1317:
1314:
1309:
1304:
1301:
1281:
1277:
1273:
1268:
1262:
1259:
1196:
1193:
1190:
1185:
1167:
1149:Riemann sphere
1127:
1116:
1113:
1100:
1097:
1094:
1058:
1055:
1052:
1047:
1043:
1039:
1036:
1033:
1030:
1027:
1024:
1013:
1012:
1000:
997:
994:
970:
965:
961:
956:
953:
949:
945:
942:
937:
934:
931:
928:
924:
912:
911:
899:
896:
893:
890:
887:
884:
864:
861:
858:
853:
849:
844:
841:
837:
833:
830:
825:
821:
806:
805:
789:
786:
783:
778:
774:
769:
766:
762:
758:
755:
750:
746:
731:
730:
718:
715:
712:
692:
689:
686:
672:
660:
657:
654:
651:
648:
628:
625:
622:
602:
599:
596:
576:
573:
570:
523:
520:
517:
512:
508:
504:
501:
488:is said to be
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
395:
392:
355:
351:
326:
321:
317:
312:
309:
305:
301:
298:
293:
289:
265:
260:. Identifying
242:
228:
163:
133:Borel subgroup
112:
87:
84:
48:vector bundles
26:
9:
6:
4:
3:
2:
3205:
3194:
3191:
3189:
3186:
3185:
3183:
3176:
3175:
3173:
3169:
3159:
3155:
3151:
3147:
3143:
3139:
3138:
3133:
3129:
3125:
3124:
3113:
3108:
3104:
3102:9780387302638
3098:
3094:
3089:
3084:
3080:
3079:Tits, Jacques
3076:
3071:
3068:(in French),
3067:
3063:
3059:
3056:
3052:
3049:
3045:
3041:
3040:
3035:
3031:
3028:
3022:
3018:
3013:
3008:
3004:
3000:
2996:
2992:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2957:
2945:
2939:
2935:
2928:
2924:
2915:
2912:
2911:
2905:
2902:
2896:
2890:
2886:
2865:
2862:
2859:
2856:
2853:
2849:
2844:
2841:
2838:
2834:
2828:
2824:
2816:
2815:
2814:
2811:
2805:
2798:
2791:
2785:
2779:
2772:
2768:
2762:
2758:
2754:
2749:
2745:
2740:
2736:
2716:
2711:
2707:
2703:
2698:
2690:
2687:
2683:
2677:
2670:
2665:
2659:
2652:
2648:
2640:
2639:
2638:
2635:
2628:
2624:
2619:
2614:
2607:
2602:
2597:
2586:
2583:
2579:
2575:
2552:
2547:
2544:
2540:
2533:
2530:
2524:
2518:
2515:
2512:
2505:
2504:
2503:
2500:
2494:
2491:
2487:
2481:
2477:
2454:
2448:
2440:
2437:
2433:
2424:
2420:
2416:
2410:
2407:
2401:
2398:
2393:
2380:
2377:
2374:
2367:
2366:
2365:
2364:
2359:
2355:
2348:
2344:
2337:
2333:
2327:
2321:
2310:
2308:
2303:
2297:
2291:
2285:
2280:
2275:
2269:
2263:
2258:
2254:
2250:
2245:
2222:
2214:
2210:
2206:
2203:
2199:
2195:
2182:
2181:
2180:
2177:
2171:
2164:
2160:
2152:
2147:determines a
2145:
2141:
2136:
2131:
2126:
2122:
2118:
2112:
2108:
2104:
2097:
2091:
2086:
2081:
2076:
2072:
2068:
2062:
2057:
2052:
2047:
2042:
2036:
2032:
2027:
2016:
2014:
2010:
2006:
2005:flag manifold
2002:
1998:
1994:
1991:
1987:
1983:
1973:
1924:
1920:
1915:
1912:
1908:
1904:
1896:
1892:
1855:
1852:
1849:
1829:
1826:
1818:
1814:
1809:
1806:
1802:
1798:
1790:
1786:
1772:
1770:
1750:
1747:
1744:
1724:
1721:
1718:
1690:
1687:
1679:
1675:
1670:
1667:
1663:
1659:
1651:
1647:
1626:
1623:
1620:
1612:
1597:
1574:
1568:
1556:
1553:
1551:
1546:
1539:
1533:
1521:
1518:
1516:
1511:
1504:
1498:
1486:
1483:
1477:
1465:
1462:
1460:
1455:
1444:
1443:
1442:
1415:
1383:
1357:
1315:
1266:
1245:
1241:
1233:
1223:
1215:
1211:
1207:
1191:
1170:
1166:
1159:
1150:
1145:
1141:
1135:
1133:
1123:
1112:
1098:
1095:
1092:
1076:
1056:
1053:
1045:
1041:
1031:
1028:
1025:
998:
995:
992:
963:
959:
954:
951:
947:
943:
932:
926:
922:
914:
913:
894:
888:
885:
882:
862:
859:
851:
847:
842:
839:
835:
831:
823:
819:
811:
810:
809:
787:
784:
776:
772:
767:
764:
760:
756:
748:
744:
736:
735:
734:
716:
713:
710:
690:
687:
684:
677:
673:
658:
655:
652:
649:
646:
626:
623:
620:
600:
597:
594:
574:
571:
568:
560:
559:
558:
551:
545:
521:
518:
510:
506:
499:
491:
483:
458:
455:
449:
446:
443:
437:
434:
431:
428:
425:
393:
390:
382:
377:
353:
349:
319:
315:
310:
307:
303:
299:
291:
287:
278:
274:
268:
264:
259:
254:
250:
241:
237:
231:
227:
222:
213:
209:
205:
196:
187:
184:
181:
177:
166:
162:
149:
134:
131:along with a
127:
126:maximal torus
101:
98:Lie group or
97:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3165:
3164:
3141:
3135:
3131:
3111:
3092:
3082:
3069:
3065:
3037:
3020:
2968:
2933:
2927:
2900:
2894:
2888:
2884:
2880:
2809:
2803:
2796:
2789:
2783:
2777:
2770:
2766:
2760:
2756:
2747:
2738:
2734:
2731:
2633:
2626:
2622:
2612:
2605:
2595:
2592:
2581:
2577:
2573:
2570:
2498:
2495:
2489:
2485:
2479:
2475:
2472:
2357:
2353:
2346:
2342:
2335:
2331:
2325:
2319:
2316:
2306:
2301:
2295:
2289:
2283:
2273:
2267:
2261:
2255:irreducible
2252:
2248:
2243:
2240:
2175:
2169:
2162:
2158:
2154:-equivariant
2150:
2143:
2134:
2124:
2120:
2116:
2110:
2106:
2095:
2084:
2080:flag variety
2074:
2070:
2066:
2060:
2050:
2040:
2034:
2031:compact form
2025:
2022:
2009:Serre (1954)
1979:
1773:
1604:
1246:
1239:
1231:
1222:binary forms
1221:
1168:
1164:
1157:
1143:
1139:
1137:, for which
1131:
1121:
1118:
1074:
1014:
807:
732:
729:is dominant.
675:
561:There is no
552:
489:
481:
378:
266:
262:
252:
248:
239:
229:
225:
211:
207:
203:
185:
179:
175:
164:
160:
124:, and fix a
89:
60:Armand Borel
55:
35:
29:
3144:(1): 1–57.
3055:Jacob Lurie
2965:Harris, Joe
2898:has weight
2881:of weights
2317:The weight
2253:holomorphic
2029:or for its
2013:Tits (1955)
1763:as long as
674:There is a
542:denote the
258:line bundle
223:, for each
86:Formulation
32:mathematics
3182:Categories
3168:PlanetMath
2954:References
1596:Jordan map
1220:(i.e. the
1216:of degree
703:such that
639:such that
587:such that
381:Weyl group
376:-modules.
234:we get an
217:principal
96:semisimple
68:Raoul Bott
64:André Weil
44:Lie groups
3044:EMS Press
3029:by Dover)
3027:reprinted
3007:246650103
2863:≤
2857:≤
2842:−
2792:≥ 0
2688:−
2649:χ
2545:−
2516:⋅
2438:−
2425:λ
2421:χ
2394:λ
2384:→
2309:linear.)
2215:λ
2190:Γ
2046:connected
1925:λ
1819:λ
1748:∈
1725:λ
1722:∗
1680:λ
1566:∂
1562:∂
1531:∂
1527:∂
1496:∂
1492:∂
1484:−
1475:∂
1471:∂
1342:Γ
1300:Γ
1096:∈
1046:∨
1042:β
1032:ρ
1026:λ
999:λ
996:∗
964:λ
927:ℓ
889:ℓ
886:≠
852:λ
777:λ
717:λ
714:∗
688:∈
659:λ
653:λ
650:∗
624:∈
601:λ
598:∗
572:∈
519:≥
511:∨
507:α
500:μ
459:ρ
456:−
450:ρ
444:λ
432:λ
429:∗
418:, we set
394:ρ
391:−
354:λ
320:λ
271:with its
189:, where
3081:(1955),
3019:(1989),
2967:(1991).
2908:See also
2887:−
2618:integers
2580:∈
2488:∈
2478:∈
2473:for all
2249:dominant
2123:∩
2114:, where
2099:-variety
1997:sections
1842:for all
1703:for all
1441:, then
1212:are the
1210:sections
1208:, whose
875:for all
800:for all
540:ℓ
490:dominant
472:, where
337:. Since
3158:1646586
3046:, 2001
2999:1153249
2589:Example
2307:complex
2003:on the
1990:complex
1376:is its
1147:is the
1115:Example
910:, while
279:groups
221:-bundle
193:is the
78:in the
3156:
3099:
3005:
2997:
2987:
2940:
2787:. For
2604:SL(2,
2277:is an
2226:
2064:, and
2038:. Let
1155:, and
676:unique
538:. Let
146:be an
142:. Let
34:, the
3053:, by
2920:Notes
2751:with
2351:over
2247:is a
2138:. An
2088:is a
2044:be a
1876:as a
273:sheaf
215:as a
102:over
94:be a
3097:ISBN
3003:OCLC
2985:ISBN
2938:ISBN
2593:Let
2571:for
2483:and
2078:the
2011:and
1853:>
1624:>
1238:Sym(
1230:Sym(
671:; or
410:and
90:Let
72:GAGA
62:and
3146:doi
3142:134
3025:. (
2977:doi
2799:+ 1
2781:on
2631:of
2281:of
2259:of
2167:on
2132:of
2058:of
1999:of
1984:of
1707:if
1380:th
1244:.
1173:is
1160:= 1
546:on
492:if
482:not
246:on
197:of
169:of
150:of
58:of
42:of
30:In
3184::
3154:MR
3152:.
3140:.
3042:,
3036:,
3001:.
2995:MR
2993:.
2983:.
2971:.
2963:;
2904:.
2759:,
2748:CP
2585:.
2576:,
2493:.
2119:=
2069:=
2054:a
2015:.
1972:.
1392:,
1388:,
1292::
1234:)*
1126:SL
1124:=
1111:.
550:.
435::=
243:−λ
206:→
178:=
154:;
82:.
3174:.
3160:.
3148::
3132:G
3106:.
3087:.
3074:.
3070:2
3011:.
3009:.
2979::
2946:.
2901:n
2895:X
2889:n
2885:i
2883:2
2866:n
2860:i
2854:0
2850:,
2845:i
2839:n
2835:Y
2829:i
2825:X
2810:C
2804:G
2797:n
2790:n
2784:C
2778:n
2771:n
2767:L
2761:Y
2757:X
2739:B
2737:/
2735:G
2717:.
2712:n
2708:a
2704:=
2699:)
2691:1
2684:a
2678:0
2671:b
2666:a
2660:(
2653:n
2634:B
2627:n
2623:χ
2613:G
2608:)
2606:C
2596:G
2582:G
2578:h
2574:g
2556:)
2553:h
2548:1
2541:g
2537:(
2534:f
2531:=
2528:)
2525:h
2522:(
2519:f
2513:g
2499:G
2490:B
2486:b
2480:G
2476:g
2458:)
2455:g
2452:(
2449:f
2446:)
2441:1
2434:b
2430:(
2417:=
2414:)
2411:b
2408:g
2405:(
2402:f
2399::
2389:C
2381:G
2378::
2375:f
2358:B
2356:/
2354:G
2347:λ
2343:L
2336:λ
2332:χ
2326:B
2320:λ
2302:λ
2296:K
2290:λ
2284:K
2274:K
2268:λ
2262:G
2244:λ
2223:.
2220:)
2211:L
2207:,
2204:B
2200:/
2196:G
2193:(
2176:G
2170:X
2163:λ
2159:L
2151:G
2144:λ
2135:K
2125:B
2121:K
2117:T
2111:T
2109:/
2107:K
2096:G
2085:X
2075:B
2073:/
2071:G
2067:X
2061:G
2051:B
2041:G
2035:K
2026:G
1970:i
1966:λ
1951:C
1930:)
1921:L
1916:,
1913:B
1909:/
1905:G
1902:(
1897:i
1893:H
1882:λ
1878:G
1874:λ
1870:G
1856:0
1850:i
1830:0
1827:=
1824:)
1815:L
1810:,
1807:B
1803:/
1799:G
1796:(
1791:i
1787:H
1776:λ
1765:λ
1751:W
1745:w
1719:w
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1688:=
1685:)
1676:L
1671:,
1668:B
1664:/
1660:G
1657:(
1652:i
1648:H
1627:0
1621:p
1607:G
1575:.
1569:x
1557:y
1554:=
1547:Y
1540:,
1534:y
1522:x
1519:=
1512:X
1505:,
1499:y
1487:y
1478:x
1466:x
1463:=
1456:H
1429:)
1425:C
1421:(
1416:2
1410:l
1407:s
1394:Y
1390:X
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1355:(
1350:O
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1313:(
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1303:(
1280:)
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1272:(
1267:2
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1195:)
1192:n
1189:(
1184:O
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1153:n
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1142:/
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1134:)
1132:C
1130:(
1128:2
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1099:W
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1051:)
1038:(
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1011:.
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969:)
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955:,
952:B
948:/
944:G
941:(
936:)
933:w
930:(
923:H
898:)
895:w
892:(
883:i
863:0
860:=
857:)
848:L
843:,
840:B
836:/
832:G
829:(
824:i
820:H
804:;
802:i
788:0
785:=
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773:L
768:,
765:B
761:/
757:G
754:(
749:i
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711:w
691:W
685:w
656:=
647:w
627:W
621:w
595:w
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569:w
555:λ
548:W
536:α
522:0
516:)
503:(
486:μ
478:G
474:ρ
453:)
447:+
441:(
438:w
426:w
416:W
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308:B
304:/
300:G
297:(
292:i
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267:λ
263:L
253:B
251:/
249:G
240:L
230:λ
226:C
219:B
212:B
210:/
208:G
204:G
199:B
191:U
186:U
183:/
180:B
176:T
171:B
165:λ
161:C
156:λ
152:T
144:λ
140:T
136:B
129:T
111:C
92:G
20:)
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