186:
98:
428:
616:
Kawanaka, N. (1982), "Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field",
671:
302:
410:
350:
676:
618:
29:
25:
534:; Lusztig, George (1983), "Duality for representations of a reductive group over a finite field. II",
420:
292:
279:
491:; Lusztig, George (1982), "Duality for representations of a reductive group over a finite field",
651:
536:
493:
450:
343:
576:"Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra"
372:
Alvis, Dean (1979), "The duality operation in the character ring of a finite
Chevalley group",
635:
597:
575:
555:
512:
469:
445:
424:
391:
41:
627:
587:
545:
502:
459:
381:
647:
609:
567:
524:
481:
438:
403:
643:
605:
563:
520:
477:
434:
399:
275:
33:
531:
488:
448:(1980), "Truncation and duality in the character ring of a finite group of Lie type",
665:
655:
550:
507:
464:
181:{\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(-1)^{\vert J\vert }\zeta _{P_{J}}^{G}}
386:
82:
37:
414:
61:
17:
67:
Alvis–Curtis duality has order 2 and is an isometry on generalized characters.
639:
601:
559:
516:
473:
395:
592:
631:
89:
416:
Finite groups of Lie type. Conjugacy classes and complex characters.
245:
and then taking the space of invariants of the unipotent radical of
580:
Japan
Academy. Proceedings. Series A. Mathematical Sciences
361:
on the regular unipotent elements and vanishing elsewhere.
419:, Pure and Applied Mathematics (New York), New York:
101:
180:
48:) and studied by his student Dean Alvis (
73:, 8.2) discusses Alvis–Curtis duality in detail.
346:χ is (–1)χ, where Δ is the set of simple roots.
298:
374:Bulletin of the American Mathematical Society
60:) introduced a similar duality operation for
8:
151:
145:
291:The dual of the trivial character 1 is the
591:
549:
506:
463:
385:
199:of simple roots of the Coxeter system of
172:
165:
160:
144:
119:
106:
100:
57:
53:
274:. (The operation of truncation is the
70:
45:
49:
7:
81:The dual ζ* of a character ζ of a
14:
270:is the induced representation of
191:Here the sum is over all subsets
387:10.1090/S0273-0979-1979-14690-1
353:is the character taking value |
223:of ζ to the parabolic subgroup
141:
131:
1:
551:10.1016/0021-8693(83)90202-8
508:10.1016/0021-8693(82)90023-0
465:10.1016/0021-8693(80)90185-4
299:Deligne & Lusztig (1983)
236:, given by restricting ζ to
693:
574:Kawanaka, Noriaki (1981),
301:showed that the dual of a
303:Deligne–Lusztig character
619:Inventiones Mathematicae
351:Gelfand–Graev character
182:
672:Representation theory
421:John Wiley & Sons
183:
42:Charles W. Curtis
99:
22:Alvis–Curtis duality
593:10.3792/pjaa.57.461
293:Steinberg character
280:parabolic induction
177:
632:10.1007/BF01389363
537:Journal of Algebra
494:Journal of Algebra
451:Journal of Algebra
446:Curtis, Charles W.
344:cuspidal character
178:
156:
130:
52:). Kawanaka (
430:978-0-471-90554-7
203:. The character ζ
115:
92:is defined to be
28:operation on the
684:
677:Duality theories
658:
612:
595:
570:
553:
527:
510:
484:
467:
441:
411:Carter, Roger W.
406:
389:
349:The dual of the
338:
337:
315:
314:
269:
268:
218:
217:
187:
185:
184:
179:
176:
171:
170:
169:
155:
154:
129:
111:
110:
40:, introduced by
692:
691:
687:
686:
685:
683:
682:
681:
662:
661:
615:
573:
532:Deligne, Pierre
530:
489:Deligne, Pierre
487:
444:
431:
409:
371:
368:
336:
333:
332:
331:
327:
321:
313:
310:
309:
308:
288:
276:adjoint functor
267:
266:
257:
256:
255:
253:
244:
231:
216:
215:
206:
205:
204:
161:
140:
102:
97:
96:
79:
34:reductive group
12:
11:
5:
690:
688:
680:
679:
674:
664:
663:
660:
659:
626:(3): 411–435,
613:
586:(9): 461–464,
571:
544:(2): 540–545,
528:
501:(1): 284–291,
485:
458:(2): 320–332,
442:
429:
407:
380:(6): 907–911,
376:, New Series,
367:
364:
363:
362:
347:
342:The dual of a
340:
334:
323:
317:
311:
296:
287:
284:
262:
258:
249:
240:
232:of the subset
227:
211:
207:
189:
188:
175:
168:
164:
159:
153:
150:
147:
143:
139:
136:
133:
128:
125:
122:
118:
114:
109:
105:
78:
75:
13:
10:
9:
6:
4:
3:
2:
689:
678:
675:
673:
670:
669:
667:
657:
653:
649:
645:
641:
637:
633:
629:
625:
621:
620:
614:
611:
607:
603:
599:
594:
589:
585:
581:
577:
572:
569:
565:
561:
557:
552:
547:
543:
539:
538:
533:
529:
526:
522:
518:
514:
509:
504:
500:
496:
495:
490:
486:
483:
479:
475:
471:
466:
461:
457:
453:
452:
447:
443:
440:
436:
432:
426:
422:
418:
417:
412:
408:
405:
401:
397:
393:
388:
383:
379:
375:
370:
369:
365:
360:
356:
352:
348:
345:
341:
330:
326:
320:
307:
304:
300:
297:
294:
290:
289:
285:
283:
281:
277:
273:
265:
261:
252:
248:
243:
239:
235:
230:
226:
222:
214:
210:
202:
198:
194:
173:
166:
162:
157:
148:
137:
134:
126:
123:
120:
116:
112:
107:
103:
95:
94:
93:
91:
88:with a split
87:
84:
76:
74:
72:
68:
65:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
623:
617:
583:
579:
541:
535:
498:
492:
455:
449:
415:
377:
373:
358:
354:
328:
324:
318:
305:
271:
263:
259:
250:
246:
241:
237:
233:
228:
224:
220:
212:
208:
200:
196:
192:
190:
85:
83:finite group
80:
71:Carter (1985
69:
66:
62:Lie algebras
38:finite field
21:
15:
195:of the set
18:mathematics
666:Categories
366:References
221:truncation
77:Definition
30:characters
656:119866092
640:0020-9910
602:0386-2194
560:0021-8693
517:0021-8693
474:0021-8693
396:0002-9904
158:ζ
135:−
124:⊆
117:∑
108:∗
104:ζ
413:(1985),
286:Examples
254:, and ζ
648:0679766
610:0637555
568:0700298
525:0644236
482:0563231
439:0794307
404:0546315
219:is the
90:BN-pair
44: (
36:over a
26:duality
654:
646:
638:
608:
600:
566:
558:
523:
515:
480:
472:
437:
427:
402:
394:
20:, the
652:S2CID
32:of a
24:is a
636:ISSN
598:ISSN
556:ISSN
513:ISSN
470:ISSN
425:ISBN
392:ISSN
316:is ε
58:1982
54:1981
50:1979
46:1980
628:doi
588:doi
546:doi
503:doi
460:doi
382:doi
282:.)
278:of
16:In
668::
650:,
644:MR
642:,
634:,
624:69
622:,
606:MR
604:,
596:,
584:57
582:,
578:,
564:MR
562:,
554:,
542:81
540:,
521:MR
519:,
511:,
499:74
497:,
478:MR
476:,
468:,
456:62
454:,
435:MR
433:,
423:,
400:MR
398:,
390:,
64:.
56:,
630::
590::
548::
505::
462::
384::
378:1
359:q
357:|
355:Z
339:.
335:T
329:R
325:T
322:ε
319:G
312:T
306:R
295:.
272:G
264:J
260:P
251:J
247:P
242:J
238:P
234:J
229:J
225:P
213:J
209:P
201:G
197:R
193:J
174:G
167:J
163:P
152:|
149:J
146:|
142:)
138:1
132:(
127:R
121:J
113:=
86:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.