1187:
1110:
1297:
1242:
1040:
929:
836:
853:
Key properties of local theta correspondence include its compatibility with
Bernstein-Zelevinsky induction and conservation relations concerning the first occurrence indices along Witt towers .
576:
958:
733:
704:
985:
760:
455:
792:
672:
640:
608:
522:
490:
428:
408:
349:
299:
267:
225:
1348:
1328:
369:
326:
193:
169:
149:
126:
1122:
1045:
1752:
1609:
1247:
1192:
990:
879:
1598:
Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1
1932:
1593:
797:
1601:
531:
1358:, that works for arbitrary residue characteristic. For orthogonal-symplectic or unitary dual pairs, it was proved by
1927:
49:
677:
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of
41:
129:
1724:
865:
329:
172:
80:
1576:(2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B. (eds.),
1862:
1846:
1830:
1728:
1351:
84:
934:
709:
680:
1671:
1355:
1303:
963:
738:
525:
433:
29:
1379:
372:
235:
37:
33:
1882:
1819:
1801:
1687:
1648:
1558:
1540:
868:
over a global field, assuming the validity of the Howe duality conjecture for all local places.
1851:
Festschrift in Honor of I. I. Piatetski-Shapiro on the
Occasion of His Sixtieth Birthday, Part I
1748:
1605:
1384:
376:
1905:
1874:
1811:
1740:
1707:
1679:
1659:
1638:
1626:
1585:
1550:
1307:
839:
228:
60:
1762:
1619:
765:
645:
613:
581:
495:
463:
413:
381:
334:
272:
240:
198:
1758:
1736:
1720:
1615:
1849:(1990), "Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠2",
1675:
1769:
1698:
MĂnguez, Alberto (2008), "Correspondance de Howe explicite: paires duales de type II",
1333:
1313:
861:
354:
311:
178:
154:
134:
111:
72:
1182:{\displaystyle {\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })}
1105:{\displaystyle {\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })}
1921:
1893:
1886:
1691:
68:
1815:
1823:
1589:
1569:
1528:
1363:
1359:
53:
1562:
1362:
and
Shuichiro Takeda. The final case of quaternionic dual pairs was completed by
1789:
1785:
1573:
1367:
45:
17:
1878:
1712:
1910:
1744:
1683:
1652:
1554:
1531:; Takeda, Shuichiro (2016), "A proof of the Howe duality conjecture",
1643:
1806:
1735:, Lecture Notes in Mathematics, vol. 1291, Berlin, New York:
1545:
1792:(2015), "Conservation relations for local theta correspondence",
1292:{\displaystyle {\mathcal {R}}({\widetilde {H}},\omega _{\psi })}
1237:{\displaystyle {\mathcal {R}}({\widetilde {G}},\omega _{\psi })}
1035:{\displaystyle {\mathcal {R}}({\widetilde {H}},\omega _{\psi })}
924:{\displaystyle {\mathcal {R}}({\widetilde {G}},\omega _{\psi })}
846:. The assertion that this is a 1-1 correspondence is called the
1253:
1198:
1128:
1051:
996:
885:
48:, while the global theta correspondence relates irreducible
864:
showed a version of the global Howe duality conjecture for
71:'s representation theoretical formulation of the theory of
1580:, Progr. Math., 323, Birkhäuser/Springer, pp. 175–192
1578:
Representation Theory, Number Theory, and
Invariant Theory
95:
may be viewed as an instance of the theta correspondence.
1415:
1896:(1964), "Sur certains groupes d'opérateurs unitaires",
1354:. Alberto MĂnguez later gave a proof for dual pairs of
1600:, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.:
831:{\displaystyle {\widetilde {G}}\cdot {\widetilde {H}}}
706:
and certain irreducible admissible representations of
301:. There is a classification of reductive dual pairs.
1865:(1991), "Correspondances de Shimura et quaternions",
1336:
1316:
1250:
1195:
1125:
1048:
993:
966:
937:
931:
the set of irreducible admissible representations of
882:
800:
768:
741:
712:
683:
648:
616:
584:
534:
498:
466:
436:
416:
384:
357:
337:
314:
275:
243:
201:
181:
157:
137:
114:
40:. The local theta correspondence relates irreducible
1629:(1989), "Transcending classical invariant theory",
571:{\displaystyle ({\widetilde {G}},{\widetilde {H}})}
1342:
1322:
1291:
1236:
1181:
1104:
1034:
979:
952:
923:
830:
786:
754:
735:, obtained by restricting the Weil representation
727:
698:
666:
634:
602:
570:
516:
484:
449:
422:
402:
363:
343:
320:
293:
261:
219:
187:
163:
143:
120:
328:is now a local field. Fix a non-trivial additive
1733:Correspondances de Howe sur un corps p-adique
1662:(1986), "On the local theta-correspondence",
8:
1588:(1979), "θ-series and invariant theory", in
1475:
92:
88:
59:The theta correspondence was introduced by
1772:(1984), "On the Howe duality conjecture",
1909:
1805:
1711:
1642:
1544:
1499:
1335:
1315:
1280:
1262:
1261:
1252:
1251:
1249:
1225:
1207:
1206:
1197:
1196:
1194:
1170:
1152:
1151:
1137:
1136:
1127:
1126:
1124:
1093:
1075:
1074:
1060:
1059:
1050:
1049:
1047:
1023:
1005:
1004:
995:
994:
992:
971:
965:
960:, which can be realized as quotients of
939:
938:
936:
912:
894:
893:
884:
883:
881:
817:
816:
802:
801:
799:
767:
746:
740:
714:
713:
711:
685:
684:
682:
647:
615:
583:
554:
553:
539:
538:
533:
497:
465:
441:
435:
415:
383:
356:
336:
313:
274:
242:
200:
180:
156:
136:
113:
610:by pulling back the projection map from
1487:
1416:Mœglin, Vignéras & Waldspurger 1987
1396:
1511:
1451:
1439:
67:. Its name arose due to its origin in
1833:(1980), "Correspondance de Shimura",
1427:
128:be a local or a global field, not of
7:
1463:
1403:
1189:is the graph of a bijection between
866:cuspidal automorphic representations
843:
838:. The correspondence was defined by
76:
64:
28:is a mathematical relation between
14:
1302:The Howe duality conjecture for
953:{\displaystyle {\widetilde {G}}}
728:{\displaystyle {\widetilde {H}}}
699:{\displaystyle {\widetilde {G}}}
1816:10.1090/S0894-0347-2014-00817-1
980:{\displaystyle \omega _{\psi }}
755:{\displaystyle \omega _{\psi }}
450:{\displaystyle \omega _{\psi }}
1286:
1258:
1231:
1203:
1176:
1133:
1099:
1056:
1029:
1001:
918:
890:
781:
775:
661:
655:
629:
623:
597:
591:
565:
535:
511:
505:
479:
467:
460:Given the reductive dual pair
397:
391:
288:
282:
256:
244:
214:
208:
1:
1602:American Mathematical Society
1853:, Israel Math. Conf. Proc.,
1306:local fields was proved by
857:Global theta correspondence
50:automorphic representations
1949:
1700:Ann. Sci. Éc. Norm. Supér.
305:Local theta correspondence
42:admissible representations
1330:-adic local fields with
524:, one obtains a pair of
1879:10.1515/form.1991.3.219
1117:Howe duality conjecture
872:Howe duality conjecture
848:Howe duality conjecture
173:symplectic vector space
1863:Waldspurger, Jean-Loup
1847:Waldspurger, Jean-Loup
1831:Waldspurger, Jean-Loup
1729:Waldspurger, Jean-Loup
1725:Vignéras, Marie-France
1344:
1324:
1293:
1238:
1183:
1106:
1036:
981:
954:
925:
832:
788:
756:
729:
700:
668:
636:
604:
572:
518:
486:
451:
424:
404:
365:
345:
322:
295:
263:
221:
189:
165:
145:
122:
81:Shimura correspondence
1933:Representation theory
1500:Gan & Takeda 2016
1356:general linear groups
1352:Jean-Loup Waldspurger
1350:odd it was proved by
1345:
1325:
1294:
1239:
1184:
1107:
1037:
982:
955:
926:
833:
789:
787:{\displaystyle Mp(W)}
757:
730:
701:
669:
667:{\displaystyle Sp(W)}
637:
635:{\displaystyle Mp(W)}
605:
603:{\displaystyle Mp(W)}
573:
519:
517:{\displaystyle Sp(W)}
487:
485:{\displaystyle (G,H)}
452:
425:
423:{\displaystyle \psi }
405:
403:{\displaystyle Mp(W)}
366:
346:
344:{\displaystyle \psi }
323:
296:
294:{\displaystyle Sp(W)}
264:
262:{\displaystyle (G,H)}
222:
220:{\displaystyle Sp(W)}
190:
166:
146:
123:
85:Jean-Loup Waldspurger
1835:J. Math. Pures Appl.
1604:, pp. 275–285,
1334:
1314:
1248:
1193:
1123:
1046:
991:
964:
935:
880:
798:
766:
739:
710:
681:
646:
614:
582:
532:
496:
464:
434:
430:, which we write as
414:
382:
355:
335:
312:
273:
241:
199:
179:
155:
135:
112:
22:theta correspondence
1794:J. Amer. Math. Soc.
1713:10.24033/asens.2080
1676:1986InMat..83..229K
1631:J. Amer. Math. Soc.
1533:J. Amer. Math. Soc.
1380:Reductive dual pair
373:Weil representation
236:reductive dual pair
38:reductive dual pair
26:Howe correspondence
1911:10.1007/BF02391012
1745:10.1007/BFb0082712
1684:10.1007/BF01388961
1512:Gan & Sun 2017
1440:Sun & Zhu 2015
1340:
1320:
1289:
1234:
1179:
1102:
1032:
977:
950:
921:
828:
784:
752:
725:
696:
664:
632:
600:
568:
514:
482:
447:
420:
400:
361:
341:
318:
291:
259:
217:
185:
161:
141:
118:
93:Waldspurger (1991)
89:Waldspurger (1980)
83:as constructed by
1928:Langlands program
1754:978-3-540-18699-1
1660:Kudla, Stephen S.
1611:978-0-8218-1435-2
1385:Metaplectic group
1343:{\displaystyle p}
1323:{\displaystyle p}
1270:
1215:
1160:
1145:
1083:
1068:
1013:
947:
902:
825:
810:
722:
693:
562:
547:
377:metaplectic group
371:. There exists a
364:{\displaystyle F}
321:{\displaystyle F}
188:{\displaystyle F}
164:{\displaystyle W}
144:{\displaystyle 2}
121:{\displaystyle F}
1940:
1914:
1913:
1889:
1858:
1842:
1826:
1809:
1781:
1774:Compositio Math.
1765:
1716:
1715:
1694:
1655:
1646:
1622:
1581:
1565:
1555:10.1090/jams/839
1548:
1515:
1509:
1503:
1497:
1491:
1485:
1479:
1476:Waldspurger 1990
1473:
1467:
1461:
1455:
1449:
1443:
1437:
1431:
1425:
1419:
1413:
1407:
1401:
1349:
1347:
1346:
1341:
1329:
1327:
1326:
1321:
1298:
1296:
1295:
1290:
1285:
1284:
1272:
1271:
1263:
1257:
1256:
1243:
1241:
1240:
1235:
1230:
1229:
1217:
1216:
1208:
1202:
1201:
1188:
1186:
1185:
1180:
1175:
1174:
1162:
1161:
1153:
1147:
1146:
1138:
1132:
1131:
1111:
1109:
1108:
1103:
1098:
1097:
1085:
1084:
1076:
1070:
1069:
1061:
1055:
1054:
1041:
1039:
1038:
1033:
1028:
1027:
1015:
1014:
1006:
1000:
999:
986:
984:
983:
978:
976:
975:
959:
957:
956:
951:
949:
948:
940:
930:
928:
927:
922:
917:
916:
904:
903:
895:
889:
888:
837:
835:
834:
829:
827:
826:
818:
812:
811:
803:
794:to the subgroup
793:
791:
790:
785:
761:
759:
758:
753:
751:
750:
734:
732:
731:
726:
724:
723:
715:
705:
703:
702:
697:
695:
694:
686:
673:
671:
670:
665:
641:
639:
638:
633:
609:
607:
606:
601:
577:
575:
574:
569:
564:
563:
555:
549:
548:
540:
523:
521:
520:
515:
491:
489:
488:
483:
456:
454:
453:
448:
446:
445:
429:
427:
426:
421:
409:
407:
406:
401:
370:
368:
367:
362:
350:
348:
347:
342:
327:
325:
324:
319:
300:
298:
297:
292:
268:
266:
265:
260:
229:symplectic group
226:
224:
223:
218:
194:
192:
191:
186:
170:
168:
167:
162:
150:
148:
147:
142:
127:
125:
124:
119:
1948:
1947:
1943:
1942:
1941:
1939:
1938:
1937:
1918:
1917:
1892:
1861:
1845:
1829:
1784:
1770:Rallis, Stephen
1768:
1755:
1737:Springer-Verlag
1721:MĹ“glin, Colette
1719:
1697:
1658:
1644:10.2307/1990942
1625:
1612:
1584:
1568:
1527:
1524:
1519:
1518:
1510:
1506:
1498:
1494:
1486:
1482:
1474:
1470:
1462:
1458:
1450:
1446:
1438:
1434:
1426:
1422:
1414:
1410:
1402:
1398:
1393:
1376:
1332:
1331:
1312:
1311:
1276:
1246:
1245:
1221:
1191:
1190:
1166:
1121:
1120:
1089:
1044:
1043:
1019:
989:
988:
967:
962:
961:
933:
932:
908:
878:
877:
874:
859:
796:
795:
764:
763:
742:
737:
736:
708:
707:
679:
678:
644:
643:
612:
611:
580:
579:
530:
529:
494:
493:
462:
461:
437:
432:
431:
412:
411:
380:
379:
353:
352:
333:
332:
310:
309:
307:
271:
270:
239:
238:
197:
196:
177:
176:
153:
152:
133:
132:
110:
109:
106:
101:
30:representations
12:
11:
5:
1946:
1944:
1936:
1935:
1930:
1920:
1919:
1916:
1915:
1890:
1873:(3): 219–307,
1859:
1843:
1827:
1800:(4): 939–983,
1782:
1766:
1753:
1717:
1706:(5): 717–741,
1695:
1670:(2): 229–255,
1656:
1637:(3): 535–552,
1627:Howe, Roger E.
1623:
1610:
1586:Howe, Roger E.
1582:
1566:
1539:(2): 473–493,
1523:
1520:
1517:
1516:
1504:
1492:
1480:
1468:
1456:
1444:
1432:
1420:
1408:
1395:
1394:
1392:
1389:
1388:
1387:
1382:
1375:
1372:
1339:
1319:
1288:
1283:
1279:
1275:
1269:
1266:
1260:
1255:
1233:
1228:
1224:
1220:
1214:
1211:
1205:
1200:
1178:
1173:
1169:
1165:
1159:
1156:
1150:
1144:
1141:
1135:
1130:
1101:
1096:
1092:
1088:
1082:
1079:
1073:
1067:
1064:
1058:
1053:
1031:
1026:
1022:
1018:
1012:
1009:
1003:
998:
974:
970:
946:
943:
920:
915:
911:
907:
901:
898:
892:
887:
873:
870:
862:Stephen Rallis
858:
855:
824:
821:
815:
809:
806:
783:
780:
777:
774:
771:
749:
745:
721:
718:
692:
689:
663:
660:
657:
654:
651:
631:
628:
625:
622:
619:
599:
596:
593:
590:
587:
567:
561:
558:
552:
546:
543:
537:
513:
510:
507:
504:
501:
481:
478:
475:
472:
469:
444:
440:
419:
410:associated to
399:
396:
393:
390:
387:
360:
340:
317:
306:
303:
290:
287:
284:
281:
278:
258:
255:
252:
249:
246:
216:
213:
210:
207:
204:
184:
160:
140:
130:characteristic
117:
105:
102:
100:
97:
13:
10:
9:
6:
4:
3:
2:
1945:
1934:
1931:
1929:
1926:
1925:
1923:
1912:
1907:
1903:
1899:
1895:
1891:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1825:
1821:
1817:
1813:
1808:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1764:
1760:
1756:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1709:
1705:
1701:
1696:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1664:Invent. Math.
1661:
1657:
1654:
1650:
1645:
1640:
1636:
1632:
1628:
1624:
1621:
1617:
1613:
1607:
1603:
1599:
1595:
1594:Casselman, W.
1591:
1587:
1583:
1579:
1575:
1571:
1570:Gan, Wee Teck
1567:
1564:
1560:
1556:
1552:
1547:
1542:
1538:
1534:
1530:
1529:Gan, Wee Teck
1526:
1525:
1521:
1513:
1508:
1505:
1501:
1496:
1493:
1489:
1484:
1481:
1477:
1472:
1469:
1465:
1460:
1457:
1453:
1448:
1445:
1441:
1436:
1433:
1429:
1424:
1421:
1417:
1412:
1409:
1405:
1400:
1397:
1390:
1386:
1383:
1381:
1378:
1377:
1373:
1371:
1369:
1365:
1361:
1357:
1353:
1337:
1317:
1309:
1305:
1300:
1281:
1277:
1273:
1267:
1264:
1226:
1222:
1218:
1212:
1209:
1171:
1167:
1163:
1157:
1154:
1148:
1142:
1139:
1119:asserts that
1118:
1113:
1094:
1090:
1086:
1080:
1077:
1071:
1065:
1062:
1024:
1020:
1016:
1010:
1007:
972:
968:
944:
941:
913:
909:
905:
899:
896:
871:
869:
867:
863:
856:
854:
851:
849:
845:
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1790:Zhu, Chen-Bo
1786:Sun, Binyong
1780:(3): 333–399
1777:
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1667:
1663:
1634:
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1574:Sun, Binyong
1536:
1532:
1522:Bibliography
1507:
1495:
1488:MĂnguez 2008
1483:
1471:
1459:
1447:
1435:
1423:
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1364:Wee Teck Gan
1360:Wee Teck Gan
1301:
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73:theta series
58:
54:global field
25:
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1904:: 143–211,
1894:Weil, André
1867:Forum Math.
1452:Rallis 1984
1368:Binyong Sun
1304:archimedean
844:Howe (1979)
77:Weil (1964)
65:Howe (1979)
46:local field
18:mathematics
1922:Categories
1898:Acta Math.
1841:(9): 1–132
1428:Kudla 1986
1391:References
1308:Roger Howe
840:Roger Howe
528:subgroups
69:André Weil
61:Roger Howe
1887:123512840
1857:: 267–324
1807:1204.2969
1692:122106772
1590:Borel, A.
1546:1407.1995
1464:Howe 1989
1404:Howe 1979
1282:ψ
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1374:See also
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