1198:
1121:
1308:
1253:
1051:
940:
847:
864:
Key properties of local theta correspondence include its compatibility with
Bernstein-Zelevinsky induction and conservation relations concerning the first occurrence indices along Witt towers .
587:
969:
744:
715:
996:
771:
466:
803:
683:
651:
619:
533:
501:
439:
419:
360:
310:
278:
236:
1359:
1339:
380:
337:
204:
180:
160:
137:
1133:
1056:
1258:
1203:
1001:
890:
1609:
Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1
1763:
1620:
808:
542:
17:
1369:, that works for arbitrary residue characteristic. For orthogonal-symplectic or unitary dual pairs, it was proved by
688:
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of
1943:
1604:
1612:
1587:(2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B. (eds.),
1938:
60:
945:
720:
691:
974:
749:
444:
52:
140:
1735:
879:
over a global field, assuming the validity of the Howe duality conjecture for all local places.
876:
340:
183:
1862:
Festschrift in Honor of I. I. Piatetski-Shapiro on the
Occasion of His Sixtieth Birthday, Part I
91:
1873:
1857:
1841:
1739:
1362:
95:
1773:
1682:
1630:
1366:
1314:
776:
656:
624:
592:
536:
506:
474:
424:
392:
345:
283:
251:
209:
40:
8:
1860:(1990), "Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠2",
1390:
383:
246:
48:
44:
1686:
1893:
1830:
1812:
1709:
MĂnguez, Alberto (2008), "Correspondance de Howe explicite: paires duales de type II",
1698:
1659:
1569:
1551:
1344:
1324:
365:
322:
189:
165:
145:
122:
1193:{\displaystyle {\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })}
1116:{\displaystyle {\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })}
1897:
1759:
1702:
1616:
1395:
387:
1826:
1916:
1885:
1834:
1822:
1751:
1718:
1690:
1670:
1649:
1637:
1596:
1561:
1318:
850:
239:
71:
1573:
1373:
and
Shuichiro Takeda. The final case of quaternionic dual pairs was completed by
1769:
1747:
1731:
1626:
1780:
872:
83:
1932:
1904:
1889:
79:
1600:
1580:
1539:
1374:
1370:
64:
1800:
1796:
1584:
1378:
56:
28:
1723:
1921:
1755:
1694:
1663:
1565:
1542:; Takeda, Shuichiro (2016), "A proof of the Howe duality conjecture",
1654:
1420:
1817:
1746:, Lecture Notes in Mathematics, vol. 1291, Berlin, New York:
1556:
1803:(2015), "Conservation relations for local theta correspondence",
1303:{\displaystyle {\mathcal {R}}({\widetilde {H}},\omega _{\psi })}
1248:{\displaystyle {\mathcal {R}}({\widetilde {G}},\omega _{\psi })}
1046:{\displaystyle {\mathcal {R}}({\widetilde {H}},\omega _{\psi })}
935:{\displaystyle {\mathcal {R}}({\widetilde {G}},\omega _{\psi })}
857:. The assertion that this is a 1-1 correspondence is called the
59:, while the global theta correspondence relates irreducible
875:
showed a version of the global Howe duality conjecture for
82:'s representation theoretical formulation of the theory of
1591:, Progr. Math., 323, Birkhäuser/Springer, pp. 175–192
1589:
Representation Theory, Number Theory, and
Invariant Theory
106:
may be viewed as an instance of the theta correspondence.
1730:
1426:
1907:(1964), "Sur certains groupes d'opérateurs unitaires",
1365:. Alberto MĂnguez later gave a proof for dual pairs of
1611:, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.:
1480:
842:{\displaystyle {\widetilde {G}}\cdot {\widetilde {H}}}
717:
and certain irreducible admissible representations of
312:. There is a classification of reductive dual pairs.
1876:(1991), "Correspondances de Shimura et quaternions",
1347:
1327:
1261:
1206:
1136:
1059:
1004:
977:
948:
942:
the set of irreducible admissible representations of
893:
811:
779:
752:
723:
694:
659:
627:
595:
545:
509:
477:
447:
427:
395:
368:
348:
325:
286:
254:
212:
192:
168:
148:
125:
51:. The local theta correspondence relates irreducible
1504:
1640:(1989), "Transcending classical invariant theory",
582:{\displaystyle ({\widetilde {G}},{\widetilde {H}})}
1516:
1492:
1353:
1333:
1302:
1247:
1192:
1115:
1045:
990:
963:
934:
841:
797:
765:
746:, obtained by restricting the Weil representation
738:
709:
677:
645:
613:
581:
527:
495:
460:
433:
413:
374:
354:
331:
304:
272:
230:
198:
174:
154:
131:
1456:
1444:
339:is now a local field. Fix a non-trivial additive
1930:
1432:
1468:
1408:
867:
1744:Correspondances de Howe sur un corps p-adique
1673:(1986), "On the local theta-correspondence",
315:
1599:(1979), "θ-series and invariant theory", in
1872:
1856:
1840:
1486:
103:
99:
70:The theta correspondence was introduced by
1783:(1984), "On the Howe duality conjecture",
882:
1920:
1816:
1722:
1653:
1555:
1538:
1510:
971:, which can be realized as quotients of
621:by pulling back the projection map from
1708:
1498:
1427:Mœglin, Vignéras & Waldspurger 1987
14:
1931:
1795:
1779:
1579:
1522:
1462:
1450:
78:. Its name arose due to its origin in
1844:(1980), "Correspondance de Shimura",
1669:
1438:
139:be a local or a global field, not of
1903:
1636:
1595:
1474:
1414:
1200:is the graph of a bijection between
877:cuspidal automorphic representations
854:
849:. The correspondence was defined by
87:
75:
39:is a mathematical relation between
24:
1264:
1209:
1139:
1062:
1007:
896:
25:
1955:
1313:The Howe duality conjecture for
964:{\displaystyle {\widetilde {G}}}
739:{\displaystyle {\widetilde {H}}}
710:{\displaystyle {\widetilde {G}}}
1827:10.1090/S0894-0347-2014-00817-1
1532:
991:{\displaystyle \omega _{\psi }}
766:{\displaystyle \omega _{\psi }}
461:{\displaystyle \omega _{\psi }}
1297:
1269:
1242:
1214:
1187:
1144:
1110:
1067:
1040:
1012:
929:
901:
792:
786:
672:
666:
640:
634:
608:
602:
576:
546:
522:
516:
490:
478:
471:Given the reductive dual pair
408:
402:
299:
293:
267:
255:
225:
219:
13:
1:
1613:American Mathematical Society
1401:
1864:, Israel Math. Conf. Proc.,
109:
7:
1384:
1317:local fields was proved by
868:Global theta correspondence
61:automorphic representations
10:
1960:
1711:Ann. Sci. Éc. Norm. Supér.
316:Local theta correspondence
53:admissible representations
1341:-adic local fields with
535:, one obtains a pair of
114:
1890:10.1515/form.1991.3.219
1128:Howe duality conjecture
883:Howe duality conjecture
859:Howe duality conjecture
184:symplectic vector space
1874:Waldspurger, Jean-Loup
1858:Waldspurger, Jean-Loup
1842:Waldspurger, Jean-Loup
1740:Waldspurger, Jean-Loup
1736:Vignéras, Marie-France
1355:
1335:
1304:
1249:
1194:
1117:
1047:
992:
965:
936:
843:
799:
767:
740:
711:
679:
647:
615:
583:
529:
497:
462:
435:
415:
376:
356:
333:
306:
274:
232:
200:
176:
156:
133:
92:Shimura correspondence
1944:Representation theory
1511:Gan & Takeda 2016
1367:general linear groups
1363:Jean-Loup Waldspurger
1361:odd it was proved by
1356:
1336:
1305:
1250:
1195:
1118:
1048:
993:
966:
937:
844:
800:
798:{\displaystyle Mp(W)}
768:
741:
712:
680:
678:{\displaystyle Sp(W)}
648:
646:{\displaystyle Mp(W)}
616:
614:{\displaystyle Mp(W)}
584:
530:
528:{\displaystyle Sp(W)}
498:
496:{\displaystyle (G,H)}
463:
436:
434:{\displaystyle \psi }
416:
414:{\displaystyle Mp(W)}
377:
357:
355:{\displaystyle \psi }
334:
307:
305:{\displaystyle Sp(W)}
275:
273:{\displaystyle (G,H)}
233:
231:{\displaystyle Sp(W)}
201:
177:
157:
134:
96:Jean-Loup Waldspurger
1846:J. Math. Pures Appl.
1615:, pp. 275–285,
1345:
1325:
1259:
1204:
1134:
1057:
1002:
975:
946:
891:
809:
777:
750:
721:
692:
657:
625:
593:
543:
507:
475:
445:
441:, which we write as
425:
393:
366:
346:
323:
284:
252:
210:
190:
166:
146:
123:
33:theta correspondence
1805:J. Amer. Math. Soc.
1724:10.24033/asens.2080
1687:1986InMat..83..229K
1642:J. Amer. Math. Soc.
1544:J. Amer. Math. Soc.
1391:Reductive dual pair
384:Weil representation
247:reductive dual pair
49:reductive dual pair
37:Howe correspondence
18:Howe correspondence
1922:10.1007/BF02391012
1756:10.1007/BFb0082712
1695:10.1007/BF01388961
1523:Gan & Sun 2017
1451:Sun & Zhu 2015
1351:
1331:
1300:
1245:
1190:
1113:
1043:
988:
961:
932:
839:
795:
763:
736:
707:
675:
643:
611:
579:
525:
493:
458:
431:
411:
372:
352:
329:
302:
270:
228:
196:
172:
152:
129:
104:Waldspurger (1991)
100:Waldspurger (1980)
94:as constructed by
1939:Langlands program
1765:978-3-540-18699-1
1671:Kudla, Stephen S.
1622:978-0-8218-1435-2
1396:Metaplectic group
1354:{\displaystyle p}
1334:{\displaystyle p}
1281:
1226:
1171:
1156:
1094:
1079:
1024:
958:
913:
836:
821:
733:
704:
573:
558:
388:metaplectic group
382:. There exists a
375:{\displaystyle F}
332:{\displaystyle F}
199:{\displaystyle F}
175:{\displaystyle W}
155:{\displaystyle 2}
132:{\displaystyle F}
16:(Redirected from
1951:
1925:
1924:
1900:
1869:
1853:
1837:
1820:
1792:
1785:Compositio Math.
1776:
1727:
1726:
1705:
1666:
1657:
1633:
1592:
1576:
1566:10.1090/jams/839
1559:
1526:
1520:
1514:
1508:
1502:
1496:
1490:
1487:Waldspurger 1990
1484:
1478:
1472:
1466:
1460:
1454:
1448:
1442:
1436:
1430:
1424:
1418:
1412:
1360:
1358:
1357:
1352:
1340:
1338:
1337:
1332:
1309:
1307:
1306:
1301:
1296:
1295:
1283:
1282:
1274:
1268:
1267:
1254:
1252:
1251:
1246:
1241:
1240:
1228:
1227:
1219:
1213:
1212:
1199:
1197:
1196:
1191:
1186:
1185:
1173:
1172:
1164:
1158:
1157:
1149:
1143:
1142:
1122:
1120:
1119:
1114:
1109:
1108:
1096:
1095:
1087:
1081:
1080:
1072:
1066:
1065:
1052:
1050:
1049:
1044:
1039:
1038:
1026:
1025:
1017:
1011:
1010:
997:
995:
994:
989:
987:
986:
970:
968:
967:
962:
960:
959:
951:
941:
939:
938:
933:
928:
927:
915:
914:
906:
900:
899:
848:
846:
845:
840:
838:
837:
829:
823:
822:
814:
805:to the subgroup
804:
802:
801:
796:
772:
770:
769:
764:
762:
761:
745:
743:
742:
737:
735:
734:
726:
716:
714:
713:
708:
706:
705:
697:
684:
682:
681:
676:
652:
650:
649:
644:
620:
618:
617:
612:
588:
586:
585:
580:
575:
574:
566:
560:
559:
551:
534:
532:
531:
526:
502:
500:
499:
494:
467:
465:
464:
459:
457:
456:
440:
438:
437:
432:
420:
418:
417:
412:
381:
379:
378:
373:
361:
359:
358:
353:
338:
336:
335:
330:
311:
309:
308:
303:
279:
277:
276:
271:
240:symplectic group
237:
235:
234:
229:
205:
203:
202:
197:
181:
179:
178:
173:
161:
159:
158:
153:
138:
136:
135:
130:
21:
1959:
1958:
1954:
1953:
1952:
1950:
1949:
1948:
1929:
1928:
1781:Rallis, Stephen
1766:
1748:Springer-Verlag
1732:MĹ“glin, Colette
1655:10.2307/1990942
1623:
1535:
1530:
1529:
1521:
1517:
1509:
1505:
1497:
1493:
1485:
1481:
1473:
1469:
1461:
1457:
1449:
1445:
1437:
1433:
1425:
1421:
1413:
1409:
1404:
1387:
1346:
1343:
1342:
1326:
1323:
1322:
1291:
1287:
1273:
1272:
1263:
1262:
1260:
1257:
1256:
1236:
1232:
1218:
1217:
1208:
1207:
1205:
1202:
1201:
1181:
1177:
1163:
1162:
1148:
1147:
1138:
1137:
1135:
1132:
1131:
1104:
1100:
1086:
1085:
1071:
1070:
1061:
1060:
1058:
1055:
1054:
1034:
1030:
1016:
1015:
1006:
1005:
1003:
1000:
999:
982:
978:
976:
973:
972:
950:
949:
947:
944:
943:
923:
919:
905:
904:
895:
894:
892:
889:
888:
885:
870:
828:
827:
813:
812:
810:
807:
806:
778:
775:
774:
757:
753:
751:
748:
747:
725:
724:
722:
719:
718:
696:
695:
693:
690:
689:
658:
655:
654:
626:
623:
622:
594:
591:
590:
565:
564:
550:
549:
544:
541:
540:
508:
505:
504:
476:
473:
472:
452:
448:
446:
443:
442:
426:
423:
422:
394:
391:
390:
367:
364:
363:
347:
344:
343:
324:
321:
320:
318:
285:
282:
281:
253:
250:
249:
211:
208:
207:
191:
188:
187:
167:
164:
163:
147:
144:
143:
124:
121:
120:
117:
112:
41:representations
23:
22:
15:
12:
11:
5:
1957:
1947:
1946:
1941:
1927:
1926:
1901:
1884:(3): 219–307,
1870:
1854:
1838:
1811:(4): 939–983,
1793:
1777:
1764:
1728:
1717:(5): 717–741,
1706:
1681:(2): 229–255,
1667:
1648:(3): 535–552,
1638:Howe, Roger E.
1634:
1621:
1597:Howe, Roger E.
1593:
1577:
1550:(2): 473–493,
1534:
1531:
1528:
1527:
1515:
1503:
1491:
1479:
1467:
1455:
1443:
1431:
1419:
1406:
1405:
1403:
1400:
1399:
1398:
1393:
1386:
1383:
1350:
1330:
1299:
1294:
1290:
1286:
1280:
1277:
1271:
1266:
1244:
1239:
1235:
1231:
1225:
1222:
1216:
1211:
1189:
1184:
1180:
1176:
1170:
1167:
1161:
1155:
1152:
1146:
1141:
1112:
1107:
1103:
1099:
1093:
1090:
1084:
1078:
1075:
1069:
1064:
1042:
1037:
1033:
1029:
1023:
1020:
1014:
1009:
985:
981:
957:
954:
931:
926:
922:
918:
912:
909:
903:
898:
884:
881:
873:Stephen Rallis
869:
866:
835:
832:
826:
820:
817:
794:
791:
788:
785:
782:
760:
756:
732:
729:
703:
700:
674:
671:
668:
665:
662:
642:
639:
636:
633:
630:
610:
607:
604:
601:
598:
578:
572:
569:
563:
557:
554:
548:
524:
521:
518:
515:
512:
492:
489:
486:
483:
480:
455:
451:
430:
421:associated to
410:
407:
404:
401:
398:
371:
351:
328:
317:
314:
301:
298:
295:
292:
289:
269:
266:
263:
260:
257:
227:
224:
221:
218:
215:
195:
171:
151:
141:characteristic
128:
116:
113:
111:
108:
9:
6:
4:
3:
2:
1956:
1945:
1942:
1940:
1937:
1936:
1934:
1923:
1918:
1914:
1910:
1906:
1902:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1836:
1832:
1828:
1824:
1819:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1775:
1771:
1767:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1720:
1716:
1712:
1707:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1675:Invent. Math.
1672:
1668:
1665:
1661:
1656:
1651:
1647:
1643:
1639:
1635:
1632:
1628:
1624:
1618:
1614:
1610:
1606:
1605:Casselman, W.
1602:
1598:
1594:
1590:
1586:
1582:
1581:Gan, Wee Teck
1578:
1575:
1571:
1567:
1563:
1558:
1553:
1549:
1545:
1541:
1540:Gan, Wee Teck
1537:
1536:
1524:
1519:
1512:
1507:
1500:
1495:
1488:
1483:
1476:
1471:
1464:
1459:
1452:
1447:
1440:
1435:
1428:
1423:
1416:
1411:
1407:
1397:
1394:
1392:
1389:
1388:
1382:
1380:
1376:
1372:
1368:
1364:
1348:
1328:
1320:
1316:
1311:
1292:
1288:
1284:
1278:
1275:
1237:
1233:
1229:
1223:
1220:
1182:
1178:
1174:
1168:
1165:
1159:
1153:
1150:
1130:asserts that
1129:
1124:
1105:
1101:
1097:
1091:
1088:
1082:
1076:
1073:
1035:
1031:
1027:
1021:
1018:
983:
979:
955:
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1801:Zhu, Chen-Bo
1797:Sun, Binyong
1791:(3): 333–399
1788:
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1714:
1710:
1678:
1674:
1645:
1641:
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1585:Sun, Binyong
1547:
1543:
1533:Bibliography
1518:
1506:
1499:MĂnguez 2008
1494:
1482:
1470:
1458:
1446:
1434:
1422:
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1375:Wee Teck Gan
1371:Wee Teck Gan
1312:
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1125:
1123:, likewise.
886:
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319:
244:
118:
84:theta series
69:
65:global field
36:
32:
26:
1915:: 143–211,
1905:Weil, André
1878:Forum Math.
1463:Rallis 1984
1379:Binyong Sun
1315:archimedean
855:Howe (1979)
88:Weil (1964)
76:Howe (1979)
57:local field
29:mathematics
1933:Categories
1909:Acta Math.
1852:(9): 1–132
1439:Kudla 1986
1402:References
1319:Roger Howe
851:Roger Howe
539:subgroups
80:André Weil
72:Roger Howe
1898:123512840
1868:: 267–324
1818:1204.2969
1703:122106772
1601:Borel, A.
1557:1407.1995
1475:Howe 1989
1415:Howe 1979
1293:ψ
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537:commuting
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1385:See also
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186:over
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