α-Расслоенные модули над алгеброй Ли sl(n,C)
Вивчені властивості узагальненої групи Вейля для α-розшарованих модулів над алгеброю Лі sl(n,C).
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| Datum: | 1993 |
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| Format: | Artikel |
| Sprache: | Russian |
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Інститут математики НАН України
1993
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| Schriftenreihe: | Український математичний журнал |
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| Zitieren: | α-Расслоенные модули над алгеброй Ли sl(n,C) / В.С. Мазорчук // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1215–1224. — Бібліогр.: 5 назв. — рос. |
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irk-123456789-1645852020-02-11T01:26:18Z α-Расслоенные модули над алгеброй Ли sl(n,C) Мазорчук, В.С. Статті Вивчені властивості узагальненої групи Вейля для α-розшарованих модулів над алгеброю Лі sl(n,C). Properties of the generalized Weyl group are studied for α-stratified modules over the Lie algebra sl (n, ℂ). 1993 Article α-Расслоенные модули над алгеброй Ли sl(n,C) / В.С. Мазорчук // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1215–1224. — Бібліогр.: 5 назв. — рос. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164585 519.48 ru Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Russian |
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Статті Статті |
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Статті Статті Мазорчук, В.С. α-Расслоенные модули над алгеброй Ли sl(n,C) Український математичний журнал |
| description |
Вивчені властивості узагальненої групи Вейля для α-розшарованих модулів над алгеброю Лі sl(n,C). |
| format |
Article |
| author |
Мазорчук, В.С. |
| author_facet |
Мазорчук, В.С. |
| author_sort |
Мазорчук, В.С. |
| title |
α-Расслоенные модули над алгеброй Ли sl(n,C) |
| title_short |
α-Расслоенные модули над алгеброй Ли sl(n,C) |
| title_full |
α-Расслоенные модули над алгеброй Ли sl(n,C) |
| title_fullStr |
α-Расслоенные модули над алгеброй Ли sl(n,C) |
| title_full_unstemmed |
α-Расслоенные модули над алгеброй Ли sl(n,C) |
| title_sort |
α-расслоенные модули над алгеброй ли sl(n,c) |
| publisher |
Інститут математики НАН України |
| publishDate |
1993 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164585 |
| citation_txt |
α-Расслоенные модули над алгеброй Ли sl(n,C) / В.С. Мазорчук // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1215–1224. — Бібліогр.: 5 назв. — рос. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT mazorčukvs arassloennyemodulinadalgebrojlislnc |
| first_indexed |
2025-07-14T17:12:34Z |
| last_indexed |
2025-07-14T17:12:34Z |
| _version_ |
1837643234941075456 |
| fulltext |
YAK 519.48
B. C. MaJOP'IYK, cry11.. (ICHes. ytt-T)
a-PACCJIOEHHhIE MO.UY Jill HA.U AJirEEPOH Jill sl (n, a:)
The properties of the generalized Wey! group are studied fo r a-stratified modules over the Lie algebra
sl (n, ([).
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sl (11, <I:).
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,LJ,eHHhlX Cl-npHMHTHBHblM 3JieMeI-IT0M. TIOHj{THe TaKoro M0AYJij{ ,LJ,aH0 B [l], ,l),Jljl
CJiyYM a.nre6pbl sl (3, CC ) OHR YaCTliYH0 11.CCJie,LJ,0BaHbl B [2]. }lJij{ TaKhX M0AyJiett
H8A a.nrn6pott sl (!l , CC) n0CTp0eHa rpynna, UMe1ow.aj{ CB0t\CTBa, atta.norHYHbie CB0tt
CTBaM r pynnbl BettJij{ , CB}{JaHH0tt C KJI3CH'-leCKH.MH. M0J~YJij{MH BepMa. TioJiyYeH KpH-
Tep»tt nenpttoo1~11M0CTH M0AY Jij{ M (A, p) H noc-rpoett ana.nor EfT-pe30JihBeHThI.
1. 06o3H3'1CHHJI H npei:.oapHTCJILHJ,Ie pe3y,JILT8Tbl. O6o3HalfHM lfep3 18
a1ire6py Jltt sl (n, CC); Li = Li_U Li+ - CTaHJ~apTttaj{ CHCTeMa KopneA a.nre6phI 18;
7t = {al' ... , a,,_i} - MH0)KeCTB0 npOCTbIX K0p1rnA; W- rpynna BeAJI.lJ CHCTeMhI
K0pttett A TiycTb fJ - - no,LJ,a.nr e6pa Kaprnna, COCT0.lJW.a.lI H3 Altarona.nbttbIX MaTPHU
co CJieA0M 0; n_ H r,+ - IT0,LJ,aJire6pbI 18, C00TBeTCTBYIOW.He MH0)KeCTBaM Li_ H
Li+. TorAa 18 = n_EB fJ EB n+. 06oJnalfHM qepe3 U(l8) y1mnepca.nhuy10 o6epTb1Ba-
1ow.y10 a.nre6py aJI!'e6pbl 18 . TIOJIO)KliM
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= A(h)v, V h E fJ} . Ecnn V,_ ctc 0, TO A uaJbmaeTC.lJ BecoM M0AYJij{ V R v,_ -
C00TBeTCTBYIOW.HM BeC0BbIM IT0,ll,ITp0CTpallCTB0M. MH0)KeCTB0 Bcex BeC0B V o6o-
3HalfaeTCj{ supp(V). MOl.(yJlb V na3hIBaeTCj{ BeC0BhIM, eCJIH OH }{8Jij{eTCH npj{MQA
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BeC0Bhie '8-MOAYJIH.
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Cyw.eCTByeT K0l·ICTPYKI.J;lrn ymmepcaJibHhlX M0AyneA, nop0)KAeHIIblX npHMHTHBHblM
3JieMe1noM. )],mi: AE fJ * onpeAennM CC KaK fJ EB n+-MOJJ.YJU,, no110)KHB (h + a)z =
= ( A - p)(h)z, rAe h E fJ , a E n+, z E CC. MOAYJih
M (11. ) = U('8) ® CC
U(~Gln +)
H83blBaeTC.lJ M0AyJ1e1,,1 BepMa H HBJI.lJeTCj{ YHHBepcanhHblM 0frheKT0M B KaTeropBH
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er~1111crBem11,1A npocTort cpaKT0P M0AYJIH M (A). Han0MIIHM cneHy1ow.11e cnof!:crBa
M0AY 11et1 BepMa [4 J:
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fl yen, Sy E W - orpa)Ke1rne, coorneTCTBy1o~ee y E Li+· Tm·Ha 3KB11mu1em11u
© 13. C. MA30PLIYK. 1993
ISSN 004/-6053 . Y,q,. NOi/i . )l{_Vf' II . , 1993, Ill. 45 , N" 9 1215
1216 B. C. MA30P'-1YK
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µ ~ Syy ~ ... ~ Syk ... Sy!µ = A
(Ha f)* onpe)leJleH CTaHJ].apnlblA. 'laCTH'IHbIA. ITOP51AOK).
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A E p ++ - MHO)KeCTBO AOMHHaHTHbIX BeCOB. Tor Aa TO'-Ula CJ1el].y10ma51 ITOCJle/].OBa
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0 ➔ EB M(w'J..) ➔ ... ➔ EB M(w'J..) ➔ M('J..) ➔ L('J..) ➔ 0
weW weW
/(w)=110 /(w)=l
(rAe no = card(.!\+)). 3Ta ITOCJJe)lOBaTeJlbHOCTb Ha3bIBaeTC51 Err-pe30JlbBeHTOA.
MOAYJ151 M (A).
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\
13 E L\+ \{a}. 3JleMeHT V E VA Ha30BeM a -npHMHTHBHbIM 3JleMeHTOM seca A., eCJlH
nv = 0. TiyCTb c - KBaAparn'lttbrA. 3JreMettT Ka3lfM11pa nettTpa aJ1re6pb1 U(IS).
)lJ1J1 e E er: no110)1(11M U£ = U(IS)/(c - e). )lm, h E f.J, a En, 'J.. E f.J~ z E er:
onpeAeJ111M (h + a)z = (A - p)(h)z , npeBparns er: s f.J EB n-MoAyJrb. MoAyJib
M('J.., e) = Ur ® 0::
· u( ~EBn )
Ha3bIBaeTC51 o6o6memlb1M MOJlyJieM BepMa, nopO)l(f..leHHbIM a-npHMHTHBHbIM 3Jie
MeHTOM. Q6o3na'l11M qepe3 x13, (Y13, H13) KOpttes1>1e seKTOpb1 aiire6p n+, (n_, f.J).
Onpc11,eneHHC 1. 18 -MoiJy,ib V Ha3bUJaemcJ1 a-pacc,weHHblM, ec✓1u OH 6e3
Kpyl{eHLtJI , KaK <Xa., Ya.• Ha.>- Jt0iJy✓1b.
a-Paccnoem-11,1e 0606mennb1e MOAym,r BepMa J1BJ1HIOTCJ1 y1msepcaJ1bHbIMH
o6neKTaMH B KaTerop,rn a-paccJioeHHbIX IB-MO,D,yJ1ett , nopO)l(,D,eHHbIX a-npBMB
TBBHbL"1 3JleMeHTOM.
2. O6o6~CIIHC rpynm,1 BeitJIJI /J,JIJI CJiy'laJI KpaitHCl'O K0pm1. flyCTb c' =
=(Ha.+ 1)2 + 4Y a.Xa. - onepinop Ka311Mttpa anre6pb1 <Xa., Ya.• Ha.>. 113 pa6oTbI [3]
cne,n,yeT, 'ITO MbI Mm1ceM JaMem-1T1> napaMeTp113a[(HIO MOAYJI51 M ('J.., e) ua M ('J.., e'),
rAe e' - co6cTBeHHOe 3Ha'le1-me onepaTopa c' ua a -npHMIITHBHOM 3JleMem·e.
floJJO)l(HM p = R n /laJiee 6yt(eM o6o3Ha'laTb MOJlYJJb M ('J.., p ). (0'1eBll)lHO, 'ITO
M ('J.., p) = M ('J.., -p ).)
JleMMa 1. flycmb h j = A(H) u H ; = H cc M oiJy11b M (A, p ) a -pacc✓1oeH
mozaa u mOAbKO "WlOa, Kozaa VIE ~: p 2 # (2/ + h,.)2.
aoKll3atneAbClll80 CJle,n,yeT H3 JleMMbl 1 paOOTbl [2] .
OcnOBHbIM Pe3YJibTaTOM pa6oTbl [2] 51BJ151eTC51 CJ1e,n,y10maa TeOpeMa.
TeopeMa 1 (<l>ymop11bti1). flycmb
18 = s1(3, 0::), h1 = 'J..(H 1) = 'J..(Ha.), h2 = 'J..(H2), N± = (h 1 + 2h2 ± p )/2.
ToziJa uMeem Mecmo oiJu11113 c✓1eiJy10u{U.X c11yitaee:
1. Ec,iu N ± n N = 0, mo M (A, p) - Henplt600UM.
ISSN 0041-6053. YKp. Mam. JK.)'P"·• 1993, m. 45, N• 9
a-PACCJlOEHHblE MO,[(YJUI HA,[( AJifEEPOf:t JlJf sl (n, <I) 1217
2. Eow N ± n N = N, mo cyu~ecmoyem eOuHcmee1111btil noOM.oOy/lb M (A,', p')
MoOy/l.fl M('A., p ), nputtel\t
h{ = h1 +N, h2 = h2 -2N, p' = p +N±.
3. E C/Ill N± n N = N, , 2' mo
M('A.2 ,p 2)C M(J... 1,p 1)C M('A.,p),
nputtn.1 O/l.fl coomflemcnwy,ow,ux napalltempoo ul\teel\t
hi= h1 +N;, h~ = h2 -2N;, P; = p +~. i = 1,2.
IlyCTb 0Il51Tb (8 = SI (n, ([). IlpOH3B0JibHbIA 3JieMeHT A E ~ • M0)KH0 3a,lJ.aTb
Ha6opoM (hi, ...• h,._ 1), rJJ,e h; = 'A.(H;). 6yJJ.eM npeJJ.noJiaraTb, 'IT0 H 1 = H 0 _.
PaccMoTpHM npocTpaucrno <t" c KOOpJJ.HHaTaMH (hp ...• h,._1• p ). TiyCTb N ± =
= (h 1 + 2h2 ± p )/ 2 . OapeJJ.eJIHM CJieJJ.yI<l.llne mrneAHbie npeo6pa30BaHH51 apo
crpaucTsa <t":
EB: h 1 ➔ h1 +N+,
h2 ➔ h2-2N+,
h3 ➔ h3 +N+,
p ➔ P -N+,
0: h1 ➔ h1 + N-.
h2 ➔ h2-2N-,
h3 ➔ h3 +N-,
p ➔ P +N-
(OCTaJibHbie K00PJJ.HHaTbl He H3MeH51IOTC51). TiyCTb S 3 , ••• , S n-1 - 0Tpa)KeHH51
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npuMumufJHolM 3/leMeHmol\t.
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2. W a oeilcnwyem KOK zpynna Beil/I.fl onucaHHOil 8blme cucmeMbl KOpHeil, He
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ISSN 0041-6053. YKp. MOIII . )KJpH. , 1993. Ill. 45, N' 9
1218 B. C. MA30PYYK
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... , h,,_1, p ).
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nepnJIOCKOCTH, onucammtt B npeL{bIJlYU{ett JieMMe, TO HX HOCHTeJIH pacnOJIO)Kelihl Ha
pa3HbIX peweTKax npoCTPaHCTBa ~-. OHH He MOryT HMeTb HeTpHBHaJlbHhlX pacwupe
HHtt H CTOTOMY HX B3aHMOCBH3b 1-1ac He mnepecyeT. ,[(anee Mb! paccMaTpHsaeM MOll,y
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lllUI f HenpepbIBI-Ia H f (s) = f (TJ). \;;/ E' > 0 MU0:lKeCTB0 "mIOXHX" T0'-leK HHf'J\e He
nnoTno B OKpecTnocr.ax B (S, e') H B (TJ, e' ). 113-:m nenpepbmuocTH J. KOHe<Inocn1
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3. IlOAMOAYJIH MOl~YJUI M (A. p ). 3aMeTHM, 'ITO ecmJ B HCKOTOport T0'lKe
A E Q* q>HKCnponano 31-U\<Ienue p, TO MO)KHO f'OB0pH'lb 0 C0I'JiaCOBat-IIIOM C w; co
ITOCTaBJieHmJ KU)KJI0MY 3JleMe1rry µ 113 paccMaTpHnaeMolt nmepmIOCK0CTH 11e
KOT0poro 3Ha'-leHH>l p µ- TT pH 3TOM I")..,= p H eCJIH /..\JI}( W E W a HMeeM W (µ , /J µ) =
ISSN 004/-6053, YKp, ,\1(//11, ;,.;,yp1t,, 1993, 11!, 45 , NV 9
a-P/,CC.TTOEHHblE MOJJ,Y.TTH HAJJ, AJIT'E6POfi .TTH sl (n. <t) 1219
= (µ',p'), TO p'=pµ'·
JleMMa 3. MuoJKecmBo A E b • maKux, 'Imo M (A - µ, p '}.,...µ) C M (A, p,J,
ecmb a1ue6pau'leCKOe noo,,moJKecmoo o b*.
a01uname.AbC11l60. OnpeAe.TIHM
x,, = [Xi' X2J, K = n_EB (H,) EB (x,).
[Xi, U(K)] C U(K)EBU(K)f/ 1 EB ... EBU(K)H,,_1 EBU(K)X,,.
TaKHM o6pa30M, 11.m1 11J06oro u E U(K) cyU1ecrnyeT e11.uncTBeHHbIA na6op (u;, 0, • •.
• • • , U;, ,._1), JIHHeAHO 3aBHCHlllHA OT U TaKOA, 'ITO
[Xi. u] = lt;, o + ui, Jx,, + U;, ,Jl2 + ... + ui, ,._// ,,_[.
IlycTb 11.mI npoH3BO.TibHOro A E b*
"-
fi = ui, o + (A - P )(H2)1ti, 2 + + (A - P )(H,._1)ui, n-1•
/
g"-: u ➔ <J}(u), .. . , /~(u)),
g"- - mmeAnoe OT06pa)Ke1me U(K) ➔ U (K )"-1. TiycTb X - nOAMHO)KeCTBO
U(K) TaKoe, 'ITO 11.11H scex h E b: [h, u] = - µ (h) u, a TatOKe 311eMeHT u co11.ep)KITT
111-100 X 1 11H6o Y 1• no HH TO H Apyroe 011.HospeMenuo. O'Ies1-111.110, dim(X) < oo.
O6o3Ha'IHM 'Iepe3 v"- KanonH'leCKHA o6pa3yIOll.lHA MOAYJIH M (A, p,J. Cnpase11.
J1HBbI c11e11.yIOU1He YTBep)K)].eHHH :
M(A - µ,p'}.,...µ) C M(A.,p.,) ¢::>
3e E M (A., PV'}.,...p-µ: e ,,= 0 n 11.e = 0 ¢=>
3u E X \ {0}: ttuv"- = 0 ¢::>
3u E X \ {0}: X2uv,._ = .. . = X,,zro,., = 0 ¢::>
3u E X \ {0}: [X2, u]v,._ = ... = [X,., u}v,._ = 0
3u E X\{0}: J}(u)v,._ = ... =
,._
f,, (u)v,._ = 0
¢::>
¢::>
3u E X\{0}: "-h <u> = ... = "-f,, (u) = 0 ¢::>
g"- I x HMeeT panr MeHbllle dim (X) =>
M (A - µ, P'}.,...µ) C M (A, PD 3KBHsanenTH0 paset1CTBY uymo onpeAeJIHTeJIH, 3ne
MeHTbI KOTOporo JIHHeAHO 3aBHCHT OT A, '{TO H Tpe6oBaJIOCb AOKa3aTb.
fIOJIO)KHM
wi = <EB,S3, .. . ,S,,_1>, w~ = <G.S3 .... ,s,._,>.
3TH rpynnbl MO)KHO TpaKTOBaTb, KaK rpynnbl BeAJIH CHCTeM KOpneA THna An-2
(11.anee Mbl 3TO no.n.pa3yMesaeM). TiycTb P - pellleTKa 'IHCeJI (A,PD TaKHX, 'ITO
AJIH scex w E Wa(A, PD = w(A', p,:) $ (A', p,._ , ), r11.e (A', p,._,)-anaJior
"AOMHHaHTHOrO seca".
Teo per.ta 4. llycmb Sy - ompaJKeHue 8 W;, (A, PD E . P u
Sf.-)., PD = (µ, p µ) $ 0., p,J.
ISSN 0041-6053. YKp. Mam . JKyp1t., 1993, m. 45, N• 9
1220 B. C. MAJOPqYK
Tozoa M (µ, p~ c M (>., p,J.
l(oKll3tlmMbCmeo. Ecne S.y E { EB, S3, ... , Sn-I}, TO :no O'IeBHAHO. B c.rry,rne,
KOrAa pa3.1IO)KeHee Sy He coAep)KHT EB, :no cneAyeT 03 Teopee MOAynea BepMa.
B OCTasweMCJI c.rryqae nposeAeM AOKa3aTeJlbCTBO HHAYKQHett no AJIHHe Y.
.[{oKa)KeM nepsbtlt war HHAYKQHH. B 3TOM cnyqae Sy= S3 EB S3. Bo3MO)KHbI TPH
saperurra (pee. 1):
,._ µ' t,..' A,
S3 S3
S3
S3
S3
S3
µ µ µ' ')..,'
2 3
PHC. 1.
1. 3To O'leBHAHbl'1: c11yqatt.
2. B 3TOM cnyqae cymeCTsyeT o6pa3y10mHtt MOAYJIR M(A.', p ,:) TaKott, 'ITO AJIJI
HeKOTOporo U E U(n_): V = uv' rAe v' - KaI-IOHH'IeCKH8 06pa3y10m0li MOAYJij{
M (µ', Pµ•). Cornacuo neMMe 7.6.9 03 [4] cymecrnyeT Qenoe I TaKoe, 'ITO Y~u E
U(n_ Y}), rAe Y}v' - KauoHe'leCKett o6pa3y10mHtt MOAYJIR M (A, p,.). 0TCIOAa
nerKO nonyqaeM, 'ITO o6pa3y10m0a 3JieMeHT MOAY JOI M (µ, p µ) Jie)KHT B MOAY Jle
MO,.,p'J.).
3. ,lloKa)KeM, 'ITO JII06o8 3JieMeHT, KOTOpbllt MO)KeT 6blTb no11y'leH B µ' 03 A
qepe3 ').._', MO)KeT 6b1Tb no11yqeu qepe3 µ. lfa 3TOro CJieAyeT, 'ITO B MOAYJie
M (A.,, p '}.) B TO'IKe µ eCTb a-npHMITT'HBHbd!: 3JieMeHT. BocnOJib3yeMCJI KOMMYTaQH
OHHblMH COOTHOWeHHJIMH ~- HanpeMep, nycTb Yt1 Y}v' Jie)KHT B TO'IKe µ', TOr,u,a
yk1 yk = yk-k1 yk1 . + yk-k1 +lyyk1 ~ I + + yk yk1 =· a
I 3 3 3+1 3 I 3+1 ''' 3 I '
(3 + i o6c)3uaqaeT CYMMY Kopuea.) Oc10a.a cJie11.yeT, 'ITO av' no11yqen 03 µ.
Ilocne,u,y10m0e HHAYKQHOHHbie ware auaimre'IHhI.
TeopeMa 5. Ec11u
(A', P,.,) ~ sy1 ().,,', P,.,) 2 ... 2 s"i .. . sy1 (A', P ,J = (A, P 0
0/1.Jl Sy; E w; (W~) . mo M (A, p,.) c M (A', p ,,:).
l(oKa3ame.11bCm80 DpOBOAHTCJI auanorH'IIIO AOKaJaTeJibCTBY -reopeMhl 7 .6.13 B
[4] ua OCHOBaHHH npeAbIAymex yTBep)KAeHH8.
JleMMa 4. flycmb ~ = WTJ < TJ 01111 w -to w; ( W~). moziJa cyu{ecmeyem
w' E w; (W~) maKou, 'lmo w'TJ < TJ.
l(oKll3tlmMbCmBO IlpOBO.llHTCJI HHAYKQeea no 11 - panry aiire6pbl. Ilpe 11 = 3
3TO O'leBHAHO. k ⇒ k + 1. PaCCMO"fPHM renepnJIOCKOCTH:
ISSN 004/-6053. YKp. Mam. )l(yp1t., 1993, 111. 45, N" 9
a-PACCJIOEHHblE MOllYJIH HAll AJirEEPOtl JIH sl (n, <t) 1221
%2 = c; + x2a2 + · · · + x,,-2an-2·
EcJIH % I = %2, TO BCe CJietJ,yeT H3 npetJ.0OJIO)KeffHj{ HHAYKUHH. TiyCTb % I # %2.
Tort1.a cymecTsyeT
w E <EB. S3, ...• S,._2> ((8, S3, ...• Sn-2>)
TaKoe,'ITO s,._,wTJE %2. TIOJIO)KHM µ=w-1s,,_,wri. O<IeBHtJ,HO, w'=w-1s,._1WE
E w;(w;;:). TaK KaK % 1 = % 2 + la,._1 AJIH neKOToporo neoTpnu;aTeJibHOro l. TO
s,,_, WTJ = WTJ - la,._, . 3aMeTHM, 'ITO w-1(a,._,) - KOpeHb, npH'IeM OOJIO)tmTeJibHblR.
Tort1.a µ=Tl - tw-1(a,,_1) < fl.
JlcMMa S. llycmb d - cucme,.w KOpHeu m1ma A,., W - ee zpynna Beu1111.
EC/m 01111 HeKomopow w E W uMeeM wA. <A.mo cyufeCm8yem y E d+ maKoe,
'Imo SyA < A.
aoKa3amei lbClll60 npoBOJJ.HTCH HHAYKI~HeA 00 paury anre6pb1. IlpH II= 2 3TO
o<IeBHtJ.HO. k ⇒ k + 1. TiycTb {al' ... , a,.} -npocrble Kopnu. Onpet1.enHM
%µ = µ + x1a 1 + ... + x,,_1a,._1•
lfa yc110snH cnet1.yeT, 'ITO AJI.ll neKoToporo neorpuu;aTe11bnoro l: ·%,.,"- = %"- - la,..
Tipu l = 0 nee cneµ,yeT H3 npet1.nono)KeHH.ll HHAYKQHH. TiycTb l i'- 0. Torµ_a
cymecTBye-r
TaKoe, 'ITO SYA E % wt.." na11ee nerKO BHtJ.eTb, 'ITO SYA. < A.
lfa nony<IeHHbIX pe3yJibTclTOB HeMeµ,nem10 BbITeKaeT CJietJ,yIOIUa.ll TeOpeMa.
TcopeMa 6 (KpuTepnit 11enpunoµ_m1ocTu Moµ_ymI M (A., p )). B
o6o311a'leHu11x Hacmo11uiezo 11y11Kma ;,.woy11b M (A, p) He11puooiJ11M mozoa u mo11.bKO
mozoa, KOliJa:
1. a1111 111060l0 KOJ)HJI. "(, npeiJcmaOU/IIOlO O OUOe CYMMbl KOJJHeU U3
{ a 3 •• .• , a,,_1}, <Xy, YY, Hy>-Mooy.1b, 11opo)l(Oeft1tbll'1 a-11puMumu8flbtM 3/leMe11moM.
1tenpt1000UM.
2. a//Jl /1106020 KO]JHJI Y, npeocmaouMOlO O ouoe CY/IIMbl KOJJfleil U3
{a2, ... , a,,_i} u cooep)/((ll/{elO a 2• <Xy, Yy, Hy, Xy+a,• Yy+a.,• fly+a./-Mooy/lb ,
n o pOJKOeWtblil a-11pU.MllmllOJlb/M 3.lleMe/mZOM, 11e 11p1100011M.
4. Cny-tait cpci.11cro KOpHJ1 a. TiycTb -renepb Ha. = Hi tJ.JI.H i "# I, 11 - I.
Aua.norH'IHO npetJ.bltJ.YIUeA 'IaCTH BBet1eM B <!:" cneµ_y1omne JIHHeAHbie npeo6-
pa30BaHH.ll:
EB' : hi ➔ hi + N '+
hi-I ➔ hi-I - 2N '+
h,_2 ➔ h,_2 + N '+
EB" : hi ➔ hi + N "+
8': hi ➔ hi +N'
hi- 1 ➔ hi-I - 2N ,_
h,_2 ➔ hi-2 + N ,_
p ➔ p +N'-.
8": h- ➔ h . +N"-
' l
ISSN 0041 -6053. YKp. ,..,am. JKYP"· · 1993. 111. 45, N• 9
1222
h ➔ h'. +N"-
i+2 1+2
p ➔ p - N"+, p ➔ p +N"- ,
rne
N '± = (h, + 2h;_1 ± p ) / 2
H
N"± = (h; + 2h;+J ±p ) /2.
(0CTIU3llIHeCJI KOOpl{HHaTbl He H3MeHJUOTCll.)
Onpel{eJieune 3. I'pynna
w a= <©', G', EB" , 8", s I ' . .. ' S;_3, S;+3• . . . ' s,,_J >
B. C. MA30P'-IYK
Ha3bl8aemc.R 060611.{eHHOu zpynnou Beu11.R a11ze6pbl (l3 OAR np eocma8AeHuu,
nopoJKOeHHblX a.-npu,.mmuBHbl,',1 3;101e11moM.
IloJIO)KHM
w;+ = <©',©",S1, ...• s,_3,s,.+3, .. .• s,,_1>.
w;;-+ = (8',EB",S1, ... ,S,_3, S;+3, . . . , S,,_1>.
AHa.nornqHQ npe.ll.blJlYI.Uelt qacTH MO.lKHO noJiyqHTb CJ1eny10m0e pe3yJTbTaTbI:
Teoper.1a 7. Mooy111t M(A, p) u M (A', p ') uM e10m oouH u mom JKe i1 eH
mpa11bHblU xapaKmep mozoa u mo11bKO mozoa, Kozoa (A, p) u (A', p ') pacno
AOJKeHbl Ha OOHOU op6ume zpymibl W a·
Teoper.ta 8. Ec11u
(A', p ">.:) ~ Sy/A', p ">:) ~ . •. ~ Sy* .. . Sy/A', p ">:) = (A, p ').)
0/1.Jl Sy, E wi+ (w;;-- ), mo M (A, p ,J C M (A', p-;.: ).
TeopeMa 9. Mooy11b M (A, p) HenpuBoouM mozoa u moAbKO mozoa, Kozoa
Oil.JI AI06ow 3/leMeHma w zpynn wi+' w;-' w~+' w~- 8bl1lOAHe110 : w (A, p ) '{,
<f, (A, p ). (L/acmwmbtu nop.J100K uHoy11upyemcR c b* .)
5. Err-pe30JlbBeHT8 06o6m:cHHblX MOAYJICA BcpMa. Bb16epeM (A, p) TaK,
qTo6bl AJT.11 mo6oro w E W a , w (A, p ) < (A, p ), H o6o3ttaqHM MHO.lKeCTBO TaKHX
KopTe)Kelt qepe3 P':+ . ITycTb 8 (A. p) < EB (A, p ) H 6y.o.eM C'IHTaTb, •no a. -
KpaAttHlt KOpeHb.
JleMMa 6. J/.11.R 111060w w E Wa cyll.{ecnrnyem w ' E W~ maKoe, 'Imo
M (w (')..., p )) C M (w' (')..., p )), npu'leM /(w) = /(w').
aoK/l3llme.AbCmBO IIpOBO,ll.HTC.11 HHAYKQHelt no AJTHHe 3JTeMettTa w(l(w)). [(w) =
= I cJTenyeT H3 Bb16paHHolt napaMeTpH38QHH. k ⇒k + 1. BblnmneM npuoe.o.eHHoe
pa3JTO)KeHHe W: W = Sy! .. . S Yt+1· TorAa no npeAnOJTO.lKeHHIO HHAYKQHH
SY2 ••• SYt+l(A, p) < s~2 ... s;t+I (A, p ), rAe
ISSN 0041-6053 . Y Kp. Mam. JKYP"·· 1993, 111. 45. N• 9
a - ACCJIOEHHbIE MOJlY JIH HA)l AJirE6POit JIB sl (n, <I:) 1223
Tenepb, 3aMemm Syl na s;l H HCITOJib30BaB BbI6pa1rny10 naMH napaMeTpH3aQHIO H
nprmeJ.J,emIOCTb p a311o.>Ketmj[ w, nOJIY'-IHM Tpe6yeMoe yTBeP)KJ.J.eHHe.
M bI IlOJIY'-IHJlH, '-ITO op6wra rpynnbl w; He COJ.J,ep)KHT "BHYTPH" ce6.ll J.J.pyrHx TO
'-leK Op6HTbl r p ynnbl Wa , 06o3Ha'-IHM '-1epe3 L('A, p) eJ.J,HHCTBeHHbIA npocTOA Q)aK
T Op MOJ.J.YJlj[ M ('A, p ). Tenepb, acnOJih3YR pe3yJihTaTbl [5], nony'-IHM c11eJ.J,y10my10
TeopeMy .
TeopeMa 10. flycmb n O - Hau6011btuaJ1 iJ11u1ta ::111eMeHma 6 zpynne wi.
C ,ieiJyJOU{G.fl llOCI/C006ame11bHOCmb mott11a:
0 ➔ ® M (w('A, p )) ➔ . ..
w E w;
/(w) = ''o
... ➔ ® M (w('A, p )) ➔ M ().., p) ➔ L ().., p) ➔ 0.
weW~
/( w) = I
J(o1u13ame.,1bCl1l60. Tip HHHMaj[ BO BHHMaHHe npeJ.J,bIJ~ymyIO neMMY H [5], J.J,OCTa
TO'-IHO J.J,OKa3aTb, '-!TO MaKCHMa.TlbHbIA nOJ.J,MOJ.J.YJib M (A, p ) naKpbrnaeTC.ll CJieJ.J.yIO
W.HM MOJ.J.y11eM:
K = ® M (w(A., p )).
w eW~
/( w) = I
TioKa)KeM, '-ITO K coJ.J,ep.>KttT Bee a-npHMHTHBHbre 3JieMeHTbI MOJ.J.YJI.ll M (A, p ).
Jho6oR a-npttMHTHBH1>18 3JieMe1-rr B M ('A, p) onpeJ.J,eJIRerc.ll 3neMetrroM w E W a·
TiycTb w = Sy1 .. • Syk - np1meJ.J,eu11oe pa3110.>Ke1me w, no TOrJ.J,a cooTBeTCTByIOmHA
a-npHMHTHBHblA 3JleMeln Jle)KHT B M (S-y,,_("-, p )). Ec1m S-yk. = 0 , TO 3JieMelff Jie
)KHT B M(®(A, p )).
AHanorH'-IHO l.J.JIH CJIY'-laH cpem1ero a rrony'-IHM c11eJ.J,yt01uy10 TeopeMY.
TeopeMa 11. flycmb 11 0 - 11au6ollblUOJ1 Ollww 3lleMe11ma e zpynne w;+.
C AeiJyJOU{OJl llOC/IC006ame11b/-/OCmb n!OI./Ha:
O ➔ ® M(w(A., p)) ➔ . .. ➔ ® M(w(11., p)) ➔ M('A,p) ➔ L(11.,p) ➔ O.
we "1'~+ we w:+
[ (w) =no /(w)=l
Tiy cTb W '(w") - caMbIA J.J,JIIU-IHbllt 3JieMeHT W a<wi). l1c11011b3YH TO, '-ITO
I W a I < I W; I, 11011y .... aeM TaKHe TeopeMbJ.
TeopeMa 12. fly c mb 11 0 - 1ta1160✓1b1ua11 011 111w 3/leMe11ma o zpynne wi.
CAeiJy1oufa1111oclleOooame11b11ocmb mo 'i1ta:
.. . ➔
O ➔ M(w'(A, p)) ➔ ® M(ww'('A,p)) ➔ .. .
wet\''~
®
wewi
/(w) = 110 - 1
/(w) = I
M (ww'(A, p )) ➔ M (w" w'('A, p )) ➔ L (w" w'('A, p )) ➔ 0.
ISSN 004/ -6053. YKp. Mtlll/ . )l(_Yf'lt ., 1993, 111 . 45, N• 9
1224 B. C. MA3OP'-IYK
TeopeMa 13. llycmb 11 0 - 1-wu6011b1uaJ1 a11u1-1a 311e.Me1-1ma ll zpynne w;+.
C11eoy10iqaJ1 noc11eooeame11b1-1ocmb mo•ma:
O ➔ EBM(w'(A,p)) ➔ EB M (ww'(A,p}) ➔ ...
we w~+
/(w) = I
.. : ➔ EB M(ww'(A,p)) ➔ M(w"w'(A,p)) ➔ l(w"w'(A,JJ)) ➔ O.
wew~+
l(w) = n0 -J
Bee ynoMJIHYTble BbIWe nocne,nosaTeJibHOCTH ecTeCTBeHHO Ha3bIBaTb Err -pe-
30JibBeHTaMH COOTBeTCTBYIOll\HX o6o6w;eHHblX MOAyJietl: BepMa, nopO)KJ].eHHblX 0:
npHMHTHBHhIM 3JieMeHTOM.
3aMe'laHUe 4. B OTJIH'-IHe OT KJiaCCl-f'IeCKOro CJIY'-la.ll Err-pe30JlbBeHTY MO)KHO
HanHCaTb .nmr ABYX O'.-paCCJioeHHblX o6o6w;eHHblX MOAyJietl: BepMa H3 owmtt Op6HTbl
o6o6w;errnotl: rpynnbI Betl:JIH.
llpuMep 1. PaccMo-rpttM 6 = sl(4, ct). TiycTb nmI nepeoro c11y <1 a,1 Ha= H 1, a n11,1 BToporo
Ha= H2· Toma op6HTa rpynm,1 w a Ha ~- 6yneT HMCTb BHll, noKa33HHblR Ha pttc. 2.
')...6 'As µ 6 µ5
'A9 Ag 'A1 µ 9 µ g
')...12 ')...ll ')...10 µ12 µ11 µ10
2
PHC. 2.
EcJJH BblTIHCaTb COOTBCT<.'TBYIOll.lHC orr-po30JJbBCHTbl, TO IIOJJY'IHM
1. 0 ➔ M 0-s, P s) ➔ M 0-s• Ps) $ M ft.,.,, P1) ➔ M (J.i, P2) $ M Q.4, p4) ➔ M O,.p P 1> ➔
➔ L 0-i, p 1) ➔ 0.
2. 0 ➔ M('),.12,P1 2) ➔ M0-9,P9) $M Q.1 i, P11) ➔ M(A6,p 6) $ M Q.s,P s) ➔ MO-s,Ps) ➔
➔ L 0.5, p 5) ➔ 0.
3. 0 ➔ M(µ5 , p'5 ) ➔ M(µ2 , p;)$M(µ4 , p~) ➔ M().l i, p;) ➔ L(µ 1 , p; ) ➔ 0.
4. 0 ➔ M (µ 12, p;2) ➔ M (µ 9, p~) EB M (µ 1p p;1) ➔ M (µ8, p; ) ➔ L (µ 8, p; ) ➔ 0.
1. <1>ymopHhli1 B:M. HeKOTopoe o6o611.letrne Mony11eR BepMa H HenpHBOllHMble npe)'.ICTaBJJCHHR
aJJre6p1,1 JIH sl(3, ct)// YKp. MaT. JKypH. -1990. - 42, N28. -C. 1031 -1037.
2. <1>ymopubli1 B . M. Becoe1,,1e s1(3)-Mony1rn, nopoJKne11tt1,,1c nonynpHMHTHBHhlMH 311eMeIna,rn //
TaM lKe. -1991. -48, N22. -C. 281-285.
3. Ppo30 JO. A .. Oscue11Xo C. A .. <l>ymopHhll'I B. M. S-roMOMOp<pH3M Xapttw-'-la1ulphl H 6-Monyntt,
nopoJKnettttble nonynpHMHTHBHblMH 3JICMCHT3MH // TaM JKC. - 1990. - 42, N28. - C. I 031 - 1037.
4. aw,ou,e )/(. YHHBepcaJJbHble o6ePThfB3IOIJ.lHe aJJre6pb1. - M . : Mttp, 1978. - 407 c.
5. Bernstein I. N ., Gelfand I. M ., Gelfand S. / . Differential operators on the base affine space and a
study of 6-modules //Pub!. of 1971 Summer School in Math., Janos Bolyai Math Soc. - Budapest.
-P. 21-64.
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