1578:
1436:
3053:(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)],
2716:
2457:
1441:
1573:{\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}}}
459:
1428:
1279:
1056:
2865:
968:
2225:
1829:
1690:
862:
787:
521:
1363:
1321:
1929:
324:
2555:
658:
1194:
897:
1951:
355:
111:
1724:
998:
716:
600:
1855:
1750:
1626:
1098:
690:
626:
574:
393:
3176:
1996:
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
3181:
2632:
2359:
1373:. 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
2977:
2930:
2267:
55:, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by
3089:
1873:
is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules
2949:
2128:
136:
410:
3032:
2961:
2902:
2245:
2294:. (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is
1388:
1239:
1007:
3027:
2808:
906:
361:-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as
2953:
1970:
1599:
84:
64:
2174:
1770:
1631:
803:
728:
484:
1326:
1284:
3125:
1876:
271:
3022:
2741:
2732:
2497:
1989:
1757:
224:
2599:, with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for
2019:
1985:
1861:-module is simple in general, although it does contain the unique highest weight module of highest weight
1202:
1198:
88:
631:
3116:
28:
3160:
1166:
35:, showing how a family of representations can be obtained from holomorphic sections of certain complex
2589:
1981:
867:
261:
1934:
333:
94:
2609:, with dominant weights corresponding to nonnegative integers, and the corresponding characters
1703:
977:
695:
579:
3085:
3050:
2991:
2973:
2926:
2091:
1974:
183:
3134:
3005:
2965:
2078:
1834:
1729:
1605:
1163:
1077:
669:
605:
553:
375:
265:
205:
68:
40:
3146:
2987:
469:. It is straightforward to check that this defines a group action, although this action is
3142:
2983:
2351:
2118:
2034:
1978:
1370:
2044:
1137:
171:
121:
3170:
3015:
1993:
114:
52:
36:
3039:
1760:. However, the other statements of the theorem do not remain valid in this setting.
3067:
2711:{\displaystyle \chi _{n}{\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}}=a^{n}.}
2068:
1594:
One also has a weaker form of this theorem in positive characteristic. Namely, let
48:
2452:{\displaystyle f:G\to \mathbb {C} _{\lambda }:f(gb)=\chi _{\lambda }(b^{-1})f(g)}
2312:
gives rise to a character (one-dimensional representation) of the Borel subgroup
3043:
246:
20:
3103:, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press
3156:
2969:
1584:
532:
369:
56:
2995:
32:
1953:, Mumford gave an example showing that it need not be the case for a fixed
3138:
3101:
Representation theory of semisimple groups: An overview based on examples
2606:
2964:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
2790:
and forms an irreducible representation under the standard action of
1114:
3084:, Graduate Texts in Mathematics, vol. 235, New York: Springer,
2012:
The theorem can be stated either for a complex semisimple Lie group
3155:
This article incorporates material from Borel–Bott–Weil theorem on
2764:
is identified with the space of homogeneous polynomials of degree
3010:
The
Penrose Transform: its Interaction with Representation Theory
3074:, Acad. Roy. Belg. Cl. Sci. Mém. Coll. (in French), vol. 29
970:
is the dual of the irreducible highest-weight representation of
60:
43:
groups associated to such bundles. It is built on the earlier
1984:. These representations are realized in the spaces of global
1004:
It is worth noting that case (1) above occurs if and only if
1338:
1296:
1172:
3072:
Sur certaines classes d'espaces homogènes de groupes de Lie
473:
linear, unlike the usual Weyl group action. Also, a weight
1957:
that these modules are all zero except in a single degree
1767:
be a dominant integral weight; then it is still true that
147:
defines in a natural way a one-dimensional representation
2753:
and the space of the global sections of the line bundle
3119:(1998). "Borel–Weil–Bott theory on the moduli stack of
1236:
This gives us at a stroke the representation theory of
1140:, an integral weight is specified simply by an integer
357:
by bundle automorphisms, this action naturally gives a
16:
Basic result in the representation theory of Lie groups
2651:
2329:. Holomorphic sections of the holomorphic line bundle
2090:. The flag variety can also be described as a compact
602:
is dominant, equivalently, there exists a nonidentity
2811:
2635:
2500:
2362:
2177:
1969:
The Borel–Weil theorem provides a concrete model for
1937:
1879:
1837:
1773:
1732:
1706:
1634:
1608:
1439:
1391:
1329:
1287:
1242:
1169:
1080:
1010:
980:
909:
870:
806:
731:
698:
672:
634:
608:
582:
556:
487:
454:{\displaystyle w*\lambda :=w(\lambda +\rho )-\rho \,}
413:
378:
336:
274:
97:
722:The theorem states that in the first case, we have
2925:(second ed.). American Mathematical Society.
2859:
2710:
2549:
2451:
2219:
1945:
1923:
1849:
1823:
1744:
1718:
1684:
1620:
1572:
1423:{\displaystyle {\mathfrak {sl}}_{2}(\mathbf {C} )}
1422:
1357:
1315:
1274:{\displaystyle {\mathfrak {sl}}_{2}(\mathbf {C} )}
1273:
1188:
1092:
1051:{\displaystyle (\lambda +\rho )(\beta ^{\vee })=0}
1050:
992:
962:
891:
856:
781:
710:
684:
652:
620:
594:
568:
515:
453:
387:
349:
318:
105:
2282:, and each irreducible unitary representation of
3161:Creative Commons Attribution/Share-Alike License
2860:{\displaystyle X^{i}Y^{n-i},\quad 0\leq i\leq n}
963:{\displaystyle H^{\ell (w)}(G/B,\,L_{\lambda })}
1977:and irreducible holomorphic representations of
2288:is obtained in this way for a unique value of
2240:integral weight then this representation is a
8:
2220:{\displaystyle \Gamma (G/B,L_{\lambda }).\ }
1066:as a special case of this theorem by taking
1824:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
1685:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
857:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
782:{\displaystyle H^{i}(G/B,\,L_{\lambda })=0}
516:{\displaystyle \mu (\alpha ^{\vee })\geq 0}
190:. Since we can think of the projection map
1358:{\displaystyle \Gamma ({\mathcal {O}}(n))}
1316:{\displaystyle \Gamma ({\mathcal {O}}(1))}
465:denotes the half-sum of positive roots of
2826:
2816:
2810:
2699:
2675:
2646:
2640:
2634:
2532:
2499:
2425:
2412:
2381:
2377:
2376:
2361:
2202:
2187:
2176:
1939:
1938:
1936:
1924:{\displaystyle H^{i}(G/B,\,L_{\lambda })}
1912:
1907:
1896:
1884:
1878:
1836:
1806:
1801:
1790:
1778:
1772:
1756:is "close to zero". This is known as the
1731:
1705:
1667:
1662:
1651:
1639:
1633:
1607:
1548:
1513:
1478:
1457:
1440:
1438:
1412:
1403:
1394:
1393:
1390:
1337:
1336:
1328:
1295:
1294:
1286:
1263:
1254:
1245:
1244:
1241:
1171:
1170:
1168:
1079:
1033:
1009:
979:
951:
946:
935:
914:
908:
869:
839:
834:
823:
811:
805:
764:
759:
748:
736:
730:
697:
671:
633:
607:
581:
555:
498:
486:
450:
412:
377:
341:
335:
319:{\displaystyle H^{i}(G/B,\,L_{\lambda })}
307:
302:
291:
279:
273:
264:of holomorphic sections, we consider the
99:
98:
96:
2802:. Weight vectors are given by monomials
1598:be a semisimple algebraic group over an
162:, by pulling back the representation on
59:. One can equivalently, through Serre's
2913:
3040:A Proof of the Borel–Weil–Bott Theorem
2550:{\displaystyle g\cdot f(h)=f(g^{-1}h)}
2230:The Borel–Weil theorem states that if
2168:acts on its space of global sections,
330:acts on the total space of the bundle
245:(note the sign), which is obviously a
2958:Representation theory. A first course
1997:
1857:, but it is no longer true that this
7:
2350:may be described more concretely as
2001:
1323:is the standard representation, and
3177:Representation theory of Lie groups
2923:Representations of algebraic groups
1398:
1395:
1249:
1246:
1225:, and is canonically isomorphic to
653:{\displaystyle w*\lambda =\lambda }
39:, and, more generally, from higher
2268:irreducible unitary representation
2178:
1554:
1550:
1519:
1515:
1484:
1480:
1463:
1459:
1330:
1288:
14:
3182:Theorems in representation theory
1217:, the sections can be written as
1189:{\displaystyle {\mathcal {O}}(n)}
2881:, and the highest weight vector
1413:
1264:
1062:. Also, we obtain the classical
797:and in the second case, we have
3105:. Reprint of the 1986 original.
2841:
2491:on these sections is given by
1385:are the standard generators of
3159:, which is licensed under the
2921:Jantzen, Jens Carsten (2003).
2544:
2525:
2516:
2510:
2446:
2440:
2434:
2418:
2402:
2393:
2372:
2208:
2181:
2037:complex semisimple Lie group,
1918:
1890:
1812:
1784:
1673:
1645:
1417:
1409:
1352:
1349:
1343:
1333:
1310:
1307:
1301:
1291:
1268:
1260:
1183:
1177:
1039:
1026:
1023:
1011:
957:
929:
924:
918:
892:{\displaystyle i\neq \ell (w)}
886:
880:
845:
817:
770:
742:
504:
491:
441:
429:
368:We first need to describe the
313:
285:
1:
3046:. Retrieved on Jul. 13, 2014.
2962:Graduate Texts in Mathematics
2903:Theorem of the highest weight
2246:highest weight representation
2081:and a nonsingular algebraic
1946:{\displaystyle \mathbb {C} }
1628:. Then it remains true that
350:{\displaystyle L_{\lambda }}
106:{\displaystyle \mathbb {C} }
3028:Encyclopedia of Mathematics
2783:, this space has dimension
2731:may be identified with the
1971:irreducible representations
1074:to be the identity element
63:, view this as a result in
3198:
3099:Knapp, Anthony W. (2001),
3080:Sepanski, Mark R. (2007),
2796:on the polynomial algebra
1719:{\displaystyle w*\lambda }
1600:algebraically closed field
1582:
1213:). As a representation of
993:{\displaystyle w*\lambda }
711:{\displaystyle w*\lambda }
595:{\displaystyle w*\lambda }
546:, one of two cases occur:
395:. For any integral weight
65:complex algebraic geometry
3023:"Bott–Borel–Weil theorem"
3012:, Oxford University Press
2970:10.1007/978-1-4612-0979-9
542:Given an integral weight
27:is a basic result in the
3126:Inventiones Mathematicae
3123:-bundles over a curve".
2145:holomorphic line bundle
2008:Statement of the theorem
1990:holomorphic line bundles
1931:in general. Unlike over
1726:is non-dominant for all
2742:homogeneous coordinates
2733:complex projective line
2605:may be identified with
1758:Kempf vanishing theorem
1590:Positive characteristic
1203:homogeneous polynomials
1058:for some positive root
225:associated fiber bundle
25:Borel–Weil–Bott theorem
2861:
2712:
2551:
2453:
2221:
1947:
1925:
1851:
1850:{\displaystyle i>0}
1825:
1746:
1745:{\displaystyle w\in W}
1720:
1700:is a weight such that
1686:
1622:
1621:{\displaystyle p>0}
1574:
1424:
1359:
1317:
1275:
1190:
1108:For example, consider
1094:
1093:{\displaystyle e\in W}
1052:
994:
964:
893:
858:
783:
712:
686:
685:{\displaystyle w\in W}
654:
622:
621:{\displaystyle w\in W}
596:
570:
569:{\displaystyle w\in W}
517:
455:
389:
388:{\displaystyle -\rho }
351:
320:
107:
3139:10.1007/s002220050257
2862:
2713:
2552:
2454:
2260:. Its restriction to
2222:
1982:semisimple Lie groups
1948:
1926:
1852:
1826:
1763:More explicitly, let
1747:
1721:
1687:
1623:
1583:Further information:
1575:
1425:
1360:
1318:
1276:
1191:
1095:
1053:
995:
965:
894:
859:
784:
713:
687:
655:
623:
597:
571:
523:for all simple roots
518:
456:
390:
352:
321:
108:
29:representation theory
3006:Eastwood, Michael G.
2809:
2633:
2590:special linear group
2498:
2360:
2302:Concrete description
2276:with highest weight
2254:with highest weight
2175:
2071:. In this scenario,
1935:
1877:
1835:
1771:
1730:
1704:
1632:
1606:
1437:
1389:
1327:
1285:
1240:
1167:
1078:
1008:
978:
974:with highest weight
907:
868:
804:
729:
696:
670:
632:
606:
580:
554:
485:
411:
376:
334:
272:
95:
3117:Teleman, Constantin
3082:Compact Lie groups.
3004:Baston, Robert J.;
2318:, which is denoted
1070:to be dominant and
372:action centered at
3055:Séminaire Bourbaki
3051:Serre, Jean-Pierre
2857:
2708:
2686:
2547:
2449:
2217:
1975:compact Lie groups
1965:Borel–Weil theorem
1943:
1921:
1847:
1821:
1742:
1716:
1682:
1618:
1602:of characteristic
1570:
1568:
1420:
1355:
1313:
1271:
1186:
1151:. The line bundle
1090:
1064:Borel–Weil theorem
1048:
990:
960:
889:
854:
779:
708:
682:
650:
618:
592:
566:
513:
451:
403:in the Weyl group
385:
347:
316:
103:
45:Borel–Weil theorem
2979:978-0-387-97495-8
2932:978-0-8218-3527-2
2721:The flag variety
2216:
2092:homogeneous space
1561:
1526:
1491:
1470:
184:unipotent radical
3189:
3150:
3104:
3094:
3075:
3062:
3036:
3013:
2999:
2937:
2936:
2918:
2892:
2886:
2880:
2866:
2864:
2863:
2858:
2837:
2836:
2821:
2820:
2801:
2795:
2789:
2782:
2775:
2769:
2763:
2752:
2739:
2730:
2717:
2715:
2714:
2709:
2704:
2703:
2691:
2690:
2683:
2682:
2645:
2644:
2625:
2619:
2604:
2598:
2587:
2573:
2556:
2554:
2553:
2548:
2540:
2539:
2490:
2481:
2471:
2458:
2456:
2455:
2450:
2433:
2432:
2417:
2416:
2386:
2385:
2380:
2352:holomorphic maps
2349:
2339:
2328:
2317:
2311:
2293:
2287:
2281:
2275:
2265:
2259:
2253:
2235:
2226:
2224:
2223:
2218:
2214:
2207:
2206:
2191:
2167:
2161:
2155:
2144:
2142:
2135:
2126:
2116:
2102:
2089:
2087:
2079:complex manifold
2076:
2066:
2052:
2042:
2032:
2026:
2017:
1960:
1956:
1952:
1950:
1949:
1944:
1942:
1930:
1928:
1927:
1922:
1917:
1916:
1900:
1889:
1888:
1872:
1868:
1864:
1860:
1856:
1854:
1853:
1848:
1830:
1828:
1827:
1822:
1811:
1810:
1794:
1783:
1782:
1766:
1755:
1751:
1749:
1748:
1743:
1725:
1723:
1722:
1717:
1699:
1695:
1691:
1689:
1688:
1683:
1672:
1671:
1655:
1644:
1643:
1627:
1625:
1624:
1619:
1597:
1579:
1577:
1576:
1571:
1569:
1562:
1560:
1549:
1527:
1525:
1514:
1492:
1490:
1479:
1471:
1469:
1458:
1429:
1427:
1426:
1421:
1416:
1408:
1407:
1402:
1401:
1384:
1380:
1376:
1368:
1364:
1362:
1361:
1356:
1342:
1341:
1322:
1320:
1319:
1314:
1300:
1299:
1280:
1278:
1277:
1272:
1267:
1259:
1258:
1253:
1252:
1232:
1224:
1216:
1208:
1195:
1193:
1192:
1187:
1176:
1175:
1161:
1150:
1143:
1135:
1125:
1099:
1097:
1096:
1091:
1073:
1069:
1061:
1057:
1055:
1054:
1049:
1038:
1037:
999:
997:
996:
991:
973:
969:
967:
966:
961:
956:
955:
939:
928:
927:
898:
896:
895:
890:
863:
861:
860:
855:
844:
843:
827:
816:
815:
792:
788:
786:
785:
780:
769:
768:
752:
741:
740:
717:
715:
714:
709:
691:
689:
688:
683:
659:
657:
656:
651:
627:
625:
624:
619:
601:
599:
598:
593:
575:
573:
572:
567:
545:
538:
530:
526:
522:
520:
519:
514:
503:
502:
476:
468:
464:
460:
458:
457:
452:
406:
402:
398:
394:
392:
391:
386:
364:
360:
356:
354:
353:
348:
346:
345:
329:
325:
323:
322:
317:
312:
311:
295:
284:
283:
266:sheaf cohomology
259:
244:
234:
222:
209:
203:
189:
181:
177:
161:
157:
146:
142:
134:
130:
126:
119:
112:
110:
109:
104:
102:
82:
69:Zariski topology
41:sheaf cohomology
3197:
3196:
3192:
3191:
3190:
3188:
3187:
3186:
3167:
3166:
3115:
3112:
3110:Further reading
3098:
3092:
3079:
3066:
3049:
3021:
3003:
2980:
2950:Fulton, William
2948:
2945:
2940:
2933:
2920:
2919:
2915:
2911:
2899:
2888:
2882:
2871:
2822:
2812:
2807:
2806:
2797:
2791:
2784:
2777:
2771:
2765:
2762:
2754:
2744:
2735:
2722:
2695:
2685:
2684:
2671:
2669:
2663:
2662:
2657:
2647:
2636:
2631:
2630:
2621:
2618:
2610:
2600:
2592:
2588:be the complex
2583:
2580:
2561:
2528:
2496:
2495:
2486:
2473:
2463:
2421:
2408:
2375:
2358:
2357:
2341:
2338:
2330:
2327:
2319:
2313:
2307:
2304:
2289:
2283:
2277:
2271:
2261:
2255:
2249:
2231:
2198:
2173:
2172:
2163:
2157:
2154:
2146:
2138:
2137:
2131:
2129:integral weight
2122:
2119:Cartan subgroup
2117:is a (compact)
2104:
2094:
2083:
2082:
2072:
2054:
2048:
2038:
2028:
2022:
2013:
2010:
1967:
1958:
1954:
1933:
1932:
1908:
1880:
1875:
1874:
1870:
1869:-submodule. If
1866:
1862:
1858:
1833:
1832:
1802:
1774:
1769:
1768:
1764:
1753:
1728:
1727:
1702:
1701:
1697:
1693:
1663:
1635:
1630:
1629:
1604:
1603:
1595:
1592:
1587:
1567:
1566:
1553:
1538:
1532:
1531:
1518:
1503:
1497:
1496:
1483:
1462:
1447:
1435:
1434:
1392:
1387:
1386:
1382:
1378:
1374:
1371:symmetric power
1366:
1325:
1324:
1283:
1282:
1243:
1238:
1237:
1226:
1218:
1214:
1206:
1165:
1164:
1160:
1152:
1145:
1141:
1127:
1118:
1109:
1106:
1076:
1075:
1071:
1067:
1059:
1029:
1006:
1005:
976:
975:
971:
947:
910:
905:
904:
866:
865:
835:
807:
802:
801:
790:
760:
732:
727:
726:
694:
693:
668:
667:
630:
629:
604:
603:
578:
577:
552:
551:
543:
536:
533:length function
528:
524:
494:
483:
482:
474:
466:
462:
409:
408:
404:
400:
396:
374:
373:
362:
358:
337:
332:
331:
327:
303:
275:
270:
269:
258:
250:
236:
233:
227:
221:
213:
207:
191:
187:
179:
163:
159:
156:
148:
144:
140:
137:integral weight
132:
128:
127:which contains
124:
117:
93:
92:
89:algebraic group
80:
77:
17:
12:
11:
5:
3195:
3193:
3185:
3184:
3179:
3169:
3168:
3152:
3151:
3111:
3108:
3107:
3106:
3096:
3090:
3077:
3064:
3061:(100): 447–454
3047:
3037:
3019:
3001:
2978:
2944:
2941:
2939:
2938:
2931:
2912:
2910:
2907:
2906:
2905:
2898:
2895:
2868:
2867:
2856:
2853:
2850:
2847:
2844:
2840:
2835:
2832:
2829:
2825:
2819:
2815:
2758:
2719:
2718:
2707:
2702:
2698:
2694:
2689:
2681:
2678:
2674:
2670:
2668:
2665:
2664:
2661:
2658:
2656:
2653:
2652:
2650:
2643:
2639:
2626:have the form
2614:
2579:
2576:
2558:
2557:
2546:
2543:
2538:
2535:
2531:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2485:The action of
2460:
2459:
2448:
2445:
2442:
2439:
2436:
2431:
2428:
2424:
2420:
2415:
2411:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2384:
2379:
2374:
2371:
2368:
2365:
2334:
2323:
2303:
2300:
2228:
2227:
2213:
2210:
2205:
2201:
2197:
2194:
2190:
2186:
2183:
2180:
2162:and the group
2150:
2045:Borel subgroup
2009:
2006:
1966:
1963:
1941:
1920:
1915:
1911:
1906:
1903:
1899:
1895:
1892:
1887:
1883:
1846:
1843:
1840:
1820:
1817:
1814:
1809:
1805:
1800:
1797:
1793:
1789:
1786:
1781:
1777:
1741:
1738:
1735:
1715:
1712:
1709:
1681:
1678:
1675:
1670:
1666:
1661:
1658:
1654:
1650:
1647:
1642:
1638:
1617:
1614:
1611:
1591:
1588:
1581:
1580:
1565:
1559:
1556:
1552:
1547:
1544:
1541:
1539:
1537:
1534:
1533:
1530:
1524:
1521:
1517:
1512:
1509:
1506:
1504:
1502:
1499:
1498:
1495:
1489:
1486:
1482:
1477:
1474:
1468:
1465:
1461:
1456:
1453:
1450:
1448:
1446:
1443:
1442:
1419:
1415:
1411:
1406:
1400:
1397:
1354:
1351:
1348:
1345:
1340:
1335:
1332:
1312:
1309:
1306:
1303:
1298:
1293:
1290:
1270:
1266:
1262:
1257:
1251:
1248:
1185:
1182:
1179:
1174:
1156:
1138:Riemann sphere
1116:
1105:
1102:
1089:
1086:
1083:
1047:
1044:
1041:
1036:
1032:
1028:
1025:
1022:
1019:
1016:
1013:
1002:
1001:
989:
986:
983:
959:
954:
950:
945:
942:
938:
934:
931:
926:
923:
920:
917:
913:
901:
900:
888:
885:
882:
879:
876:
873:
853:
850:
847:
842:
838:
833:
830:
826:
822:
819:
814:
810:
795:
794:
778:
775:
772:
767:
763:
758:
755:
751:
747:
744:
739:
735:
720:
719:
707:
704:
701:
681:
678:
675:
661:
649:
646:
643:
640:
637:
617:
614:
611:
591:
588:
585:
565:
562:
559:
512:
509:
506:
501:
497:
493:
490:
477:is said to be
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
384:
381:
344:
340:
315:
310:
306:
301:
298:
294:
290:
287:
282:
278:
254:
249:. Identifying
231:
217:
152:
122:Borel subgroup
101:
76:
73:
37:vector bundles
15:
13:
10:
9:
6:
4:
3:
2:
3194:
3183:
3180:
3178:
3175:
3174:
3172:
3165:
3164:
3162:
3158:
3148:
3144:
3140:
3136:
3132:
3128:
3127:
3122:
3118:
3114:
3113:
3109:
3102:
3097:
3093:
3091:9780387302638
3087:
3083:
3078:
3073:
3069:
3068:Tits, Jacques
3065:
3060:
3057:(in French),
3056:
3052:
3048:
3045:
3041:
3038:
3034:
3030:
3029:
3024:
3020:
3017:
3011:
3007:
3002:
2997:
2993:
2989:
2985:
2981:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2947:
2946:
2942:
2934:
2928:
2924:
2917:
2914:
2908:
2904:
2901:
2900:
2896:
2894:
2891:
2885:
2879:
2875:
2854:
2851:
2848:
2845:
2842:
2838:
2833:
2830:
2827:
2823:
2817:
2813:
2805:
2804:
2803:
2800:
2794:
2787:
2780:
2774:
2768:
2761:
2757:
2751:
2747:
2743:
2738:
2734:
2729:
2725:
2705:
2700:
2696:
2692:
2687:
2679:
2676:
2672:
2666:
2659:
2654:
2648:
2641:
2637:
2629:
2628:
2627:
2624:
2617:
2613:
2608:
2603:
2596:
2591:
2586:
2577:
2575:
2572:
2568:
2564:
2541:
2536:
2533:
2529:
2522:
2519:
2513:
2507:
2504:
2501:
2494:
2493:
2492:
2489:
2483:
2480:
2476:
2470:
2466:
2443:
2437:
2429:
2426:
2422:
2413:
2409:
2405:
2399:
2396:
2390:
2387:
2382:
2369:
2366:
2363:
2356:
2355:
2354:
2353:
2348:
2344:
2337:
2333:
2326:
2322:
2316:
2310:
2301:
2299:
2297:
2292:
2286:
2280:
2274:
2269:
2264:
2258:
2252:
2247:
2243:
2239:
2234:
2211:
2203:
2199:
2195:
2192:
2188:
2184:
2171:
2170:
2169:
2166:
2160:
2153:
2149:
2141:
2136:determines a
2134:
2130:
2125:
2120:
2115:
2111:
2107:
2101:
2097:
2093:
2086:
2080:
2075:
2070:
2065:
2061:
2057:
2051:
2046:
2041:
2036:
2031:
2025:
2021:
2016:
2007:
2005:
2003:
1999:
1995:
1994:flag manifold
1991:
1987:
1983:
1980:
1976:
1972:
1964:
1962:
1913:
1909:
1904:
1901:
1897:
1893:
1885:
1881:
1844:
1841:
1838:
1818:
1815:
1807:
1803:
1798:
1795:
1791:
1787:
1779:
1775:
1761:
1759:
1739:
1736:
1733:
1713:
1710:
1707:
1679:
1676:
1668:
1664:
1659:
1656:
1652:
1648:
1640:
1636:
1615:
1612:
1609:
1601:
1589:
1586:
1563:
1557:
1545:
1542:
1540:
1535:
1528:
1522:
1510:
1507:
1505:
1500:
1493:
1487:
1475:
1472:
1466:
1454:
1451:
1449:
1444:
1433:
1432:
1431:
1404:
1372:
1346:
1304:
1255:
1234:
1230:
1222:
1212:
1204:
1200:
1196:
1180:
1159:
1155:
1148:
1139:
1134:
1130:
1124:
1122:
1112:
1103:
1101:
1087:
1084:
1081:
1065:
1045:
1042:
1034:
1030:
1020:
1017:
1014:
987:
984:
981:
952:
948:
943:
940:
936:
932:
921:
915:
911:
903:
902:
883:
877:
874:
871:
851:
848:
840:
836:
831:
828:
824:
820:
812:
808:
800:
799:
798:
776:
773:
765:
761:
756:
753:
749:
745:
737:
733:
725:
724:
723:
705:
702:
699:
679:
676:
673:
666:
662:
647:
644:
641:
638:
635:
615:
612:
609:
589:
586:
583:
563:
560:
557:
549:
548:
547:
540:
534:
510:
507:
499:
495:
488:
480:
472:
447:
444:
438:
435:
432:
426:
423:
420:
417:
414:
382:
379:
371:
366:
342:
338:
308:
304:
299:
296:
292:
288:
280:
276:
267:
263:
257:
253:
248:
243:
239:
230:
226:
220:
216:
211:
202:
198:
194:
185:
176:
173:
170:
166:
155:
151:
138:
123:
120:along with a
116:
115:maximal torus
90:
87:Lie group or
86:
74:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3154:
3153:
3130:
3124:
3120:
3100:
3081:
3071:
3058:
3054:
3026:
3009:
2957:
2922:
2916:
2889:
2883:
2877:
2873:
2869:
2798:
2792:
2785:
2778:
2772:
2766:
2759:
2755:
2749:
2745:
2736:
2727:
2723:
2720:
2622:
2615:
2611:
2601:
2594:
2584:
2581:
2570:
2566:
2562:
2559:
2487:
2484:
2478:
2474:
2468:
2464:
2461:
2346:
2342:
2335:
2331:
2324:
2320:
2314:
2308:
2305:
2295:
2290:
2284:
2278:
2272:
2262:
2256:
2250:
2244:irreducible
2241:
2237:
2232:
2229:
2164:
2158:
2151:
2147:
2143:-equivariant
2139:
2132:
2123:
2113:
2109:
2105:
2099:
2095:
2084:
2073:
2069:flag variety
2063:
2059:
2055:
2049:
2039:
2029:
2023:
2020:compact form
2014:
2011:
1998:Serre (1954)
1968:
1762:
1593:
1235:
1228:
1220:
1211:binary forms
1210:
1157:
1153:
1146:
1132:
1128:
1126:, for which
1120:
1110:
1107:
1063:
1003:
796:
721:
718:is dominant.
664:
550:There is no
541:
478:
470:
367:
255:
251:
241:
237:
228:
218:
214:
200:
196:
192:
174:
168:
164:
153:
149:
113:, and fix a
78:
49:Armand Borel
44:
24:
18:
3133:(1): 1–57.
3044:Jacob Lurie
2954:Harris, Joe
2887:has weight
2870:of weights
2306:The weight
2242:holomorphic
2018:or for its
2002:Tits (1955)
1752:as long as
663:There is a
531:denote the
247:line bundle
212:, for each
75:Formulation
21:mathematics
3171:Categories
3157:PlanetMath
2943:References
1585:Jordan map
1209:(i.e. the
1205:of degree
692:such that
628:such that
576:such that
370:Weyl group
365:-modules.
223:we get an
206:principal
85:semisimple
57:Raoul Bott
53:André Weil
33:Lie groups
3033:EMS Press
3018:by Dover)
3016:reprinted
2996:246650103
2852:≤
2846:≤
2831:−
2781:≥ 0
2677:−
2638:χ
2534:−
2505:⋅
2427:−
2414:λ
2410:χ
2383:λ
2373:→
2298:linear.)
2204:λ
2179:Γ
2035:connected
1914:λ
1808:λ
1737:∈
1714:λ
1711:∗
1669:λ
1555:∂
1551:∂
1520:∂
1516:∂
1485:∂
1481:∂
1473:−
1464:∂
1460:∂
1331:Γ
1289:Γ
1085:∈
1035:∨
1031:β
1021:ρ
1015:λ
988:λ
985:∗
953:λ
916:ℓ
878:ℓ
875:≠
841:λ
766:λ
706:λ
703:∗
677:∈
648:λ
642:λ
639:∗
613:∈
590:λ
587:∗
561:∈
508:≥
500:∨
496:α
489:μ
448:ρ
445:−
439:ρ
433:λ
421:λ
418:∗
407:, we set
383:ρ
380:−
343:λ
309:λ
260:with its
178:, where
3070:(1955),
3008:(1989),
2956:(1991).
2897:See also
2876:−
2607:integers
2569:∈
2477:∈
2467:∈
2462:for all
2238:dominant
2112:∩
2103:, where
2088:-variety
1986:sections
1831:for all
1692:for all
1430:, then
1201:are the
1199:sections
1197:, whose
864:for all
789:for all
529:ℓ
479:dominant
461:, where
326:. Since
3147:1646586
3035:, 2001
2988:1153249
2578:Example
2296:complex
1992:on the
1979:complex
1365:is its
1136:is the
1104:Example
899:, while
268:groups
210:-bundle
182:is the
67:in the
3145:
3088:
2994:
2986:
2976:
2929:
2776:. For
2593:SL(2,
2266:is an
2215:
2053:, and
2027:. Let
1144:, and
665:unique
527:. Let
135:be an
131:. Let
23:, the
3042:, by
2909:Notes
2740:with
2340:over
2236:is a
2127:. An
2077:is a
2033:be a
1865:as a
262:sheaf
204:as a
91:over
83:be a
3086:ISBN
2992:OCLC
2974:ISBN
2927:ISBN
2582:Let
2560:for
2472:and
2067:the
2000:and
1842:>
1613:>
1227:Sym(
1219:Sym(
660:; or
399:and
79:Let
61:GAGA
51:and
3135:doi
3131:134
3014:. (
2966:doi
2788:+ 1
2770:on
2620:of
2270:of
2248:of
2156:on
2121:of
2047:of
1988:of
1973:of
1696:if
1369:th
1233:.
1162:is
1149:= 1
535:on
481:if
471:not
235:on
186:of
158:of
139:of
47:of
31:of
19:In
3173::
3143:MR
3141:.
3129:.
3031:,
3025:,
2990:.
2984:MR
2982:.
2972:.
2960:.
2952:;
2893:.
2748:,
2737:CP
2574:.
2565:,
2482:.
2108:=
2058:=
2043:a
2004:.
1961:.
1381:,
1377:,
1281::
1223:)*
1115:SL
1113:=
1100:.
539:.
424::=
232:−λ
195:→
167:=
143:;
71:.
3163:.
3149:.
3137::
3121:G
3095:.
3076:.
3063:.
3059:2
3000:.
2998:.
2968::
2935:.
2890:n
2884:X
2878:n
2874:i
2872:2
2855:n
2849:i
2843:0
2839:,
2834:i
2828:n
2824:Y
2818:i
2814:X
2799:C
2793:G
2786:n
2779:n
2773:C
2767:n
2760:n
2756:L
2750:Y
2746:X
2728:B
2726:/
2724:G
2706:.
2701:n
2697:a
2693:=
2688:)
2680:1
2673:a
2667:0
2660:b
2655:a
2649:(
2642:n
2623:B
2616:n
2612:χ
2602:G
2597:)
2595:C
2585:G
2571:G
2567:h
2563:g
2545:)
2542:h
2537:1
2530:g
2526:(
2523:f
2520:=
2517:)
2514:h
2511:(
2508:f
2502:g
2488:G
2479:B
2475:b
2469:G
2465:g
2447:)
2444:g
2441:(
2438:f
2435:)
2430:1
2423:b
2419:(
2406:=
2403:)
2400:b
2397:g
2394:(
2391:f
2388::
2378:C
2370:G
2367::
2364:f
2347:B
2345:/
2343:G
2336:λ
2332:L
2325:λ
2321:χ
2315:B
2309:λ
2291:λ
2285:K
2279:λ
2273:K
2263:K
2257:λ
2251:G
2233:λ
2212:.
2209:)
2200:L
2196:,
2193:B
2189:/
2185:G
2182:(
2165:G
2159:X
2152:λ
2148:L
2140:G
2133:λ
2124:K
2114:B
2110:K
2106:T
2100:T
2098:/
2096:K
2085:G
2074:X
2064:B
2062:/
2060:G
2056:X
2050:G
2040:B
2030:G
2024:K
2015:G
1959:i
1955:λ
1940:C
1919:)
1910:L
1905:,
1902:B
1898:/
1894:G
1891:(
1886:i
1882:H
1871:λ
1867:G
1863:λ
1859:G
1845:0
1839:i
1819:0
1816:=
1813:)
1804:L
1799:,
1796:B
1792:/
1788:G
1785:(
1780:i
1776:H
1765:λ
1754:λ
1740:W
1734:w
1708:w
1698:λ
1694:i
1680:0
1677:=
1674:)
1665:L
1660:,
1657:B
1653:/
1649:G
1646:(
1641:i
1637:H
1616:0
1610:p
1596:G
1564:.
1558:x
1546:y
1543:=
1536:Y
1529:,
1523:y
1511:x
1508:=
1501:X
1494:,
1488:y
1476:y
1467:x
1455:x
1452:=
1445:H
1418:)
1414:C
1410:(
1405:2
1399:l
1396:s
1383:Y
1379:X
1375:H
1367:n
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1347:n
1344:(
1339:O
1334:(
1311:)
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1305:1
1302:(
1297:O
1292:(
1269:)
1265:C
1261:(
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1231:)
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1221:C
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1207:n
1184:)
1181:n
1178:(
1173:O
1158:n
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1142:n
1133:B
1131:/
1129:G
1123:)
1121:C
1119:(
1117:2
1111:G
1088:W
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1072:w
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1060:β
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1040:)
1027:(
1024:)
1018:+
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1000:.
982:w
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958:)
949:L
944:,
941:B
937:/
933:G
930:(
925:)
922:w
919:(
912:H
887:)
884:w
881:(
872:i
852:0
849:=
846:)
837:L
832:,
829:B
825:/
821:G
818:(
813:i
809:H
793:;
791:i
777:0
774:=
771:)
762:L
757:,
754:B
750:/
746:G
743:(
738:i
734:H
700:w
680:W
674:w
645:=
636:w
616:W
610:w
584:w
564:W
558:w
544:λ
537:W
525:α
511:0
505:)
492:(
475:μ
467:G
463:ρ
442:)
436:+
430:(
427:w
415:w
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397:λ
363:G
359:G
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314:)
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300:,
297:B
293:/
289:G
286:(
281:i
277:H
256:λ
252:L
242:B
240:/
238:G
229:L
219:λ
215:C
208:B
201:B
199:/
197:G
193:G
188:B
180:U
175:U
172:/
169:B
165:T
160:B
154:λ
150:C
145:λ
141:T
133:λ
129:T
125:B
118:T
100:C
81:G
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