On Quantization of a Nilpotent Orbit Closure in $G_2$

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by  Kayue Daniel Wong

ON QUANTIZATION OF A NILPOTENT ORBIT CLOSURE IN G
2
KAYUEDANIELWONG
6
1 ABSTRACT. LetGbethecomplexexceptionalLiegroupoftypeG2.Amongthefivenilpo-
0 tentorbitsinitsLiealgebrag,onlythe8-dimensionalorbitO8hasnon-normalorbitclosure
2 O8. Inthismanuscript,wewillgiveaquantizationmodelofO8,verifyingaconjectureof

p Voganin1984.
e
S

0
3

] 1. INTRODUCTION
T
R Let G be a complex simple Lie group. The G-conjugates of a nilpotent element X ∈ g
. form a nilpotent orbit O ⊂ g. Following the ideas in [18] or [20], one would like to at-
h
t tachunitaryrepresentationstoallsuchorbitsalongwiththeirfiniteG-equivariantcovers.
a
m More precisely, let V be a finite G-equivariant cover of an affine Poisson G-variety con-
taining a nilpotent orbit O as an open set, with its ring of regular functions R(V), then
[
3 onewouldlike tofinda(hopefullyunitarizable) (gC,KC)-module XV suchthat wehave
v theG-moduleisomorphism
6
7 XV|KC ∼= R(V).
4
2 (note that K ≤ G is the maximal compact subgroup of G, hence its complexification K
C
0 isisomorphictoG). Inthefollowingwork,wewillcallX asaquantizationofV.
V
.
1 As hinted in [18], one needs to take special attention when the orbit closure O is not
0
6 normal. One reasonis duetothe algebro-geometricfact thatR(O) ∼= R(O)if and onlyif
1 O is normal. Following the spirit of the orbit method, one needs to give a quantization
:
v modelforV = O andV = O separatelywhenOisnotnormal.
i
X Here is a summary on the current progress of the above quantization scheme. In [3],
r Barbasch constructs such models for a large class of classical nilpotent orbits. Using a
a
completely different method in [5], Ranee Brylinski constructs a Dixmier algebra for all
classical nilpotent orbit closures. The reconciliation betweenthe two models is the main
themeofthePh.D.thesisoftheauthor[21].
Contrary to the classical setting, very little is known about the scheme for exceptional
groups. WenowfocusonthecaseforG = G . Write{α,β} bethesimplerootsofg,with
2
αbeingtheshortroot. Thefundamentalweightsofgarethereforegivenby
{ω ,ω } = {2α+β,3α+2β}.
1 2
1
2 KAYUEDANIELWONG
BytheBala-Carter classification,wehavefivenilpotentorbitsO ,O ,O ,O andO in
0 6 8 10 12
g. Following the studyof completely prime primitive ideals of Josephin [9], [10], Vogan
in[18]conjecturedaquantizationmodelforO andO forG :
8 8 2
Conjecture 1.1 ([18], Conjecture 5.6). Let λ ∈ h∗ and J(λ) be the maximal primitive ideal in
U(g)withinfinitesimalcharacterλ. Thenthe(g ,K ) ∼= (g×g,G)-modules
C C
1 1
U(g)/J( (ω +ω )), U(g)/J( (5ω −ω ))
1 2 1 2
2 2
arequantizationsofO andO respectively. Inparticular,
8 8
1 1
U(g)/J( (ω +ω ))| ∼= R(O ), U(g)/J( (5ω −ω ))| ∼= R(O ).
2 1 2 KC 8 2 1 2 KC 8
Asaconsequence,O hasnon-normalclosure.
8
Interestingly, by the classification of spherical unitary dual of complex G given by
2
Dufloin[8],U(g)/J(1(ω +ω ))isunitarizablewhileU(g)/J(1(5ω −ω ))isnot(thisfact
2 1 2 2 1 2
isalsoobservedbyVoganinp.226of[19]). Later,LevasseurandSmithin[11]provedthat
U(g)/J(1(ω +ω ))| ∼= R(O )andO is notnormal, but wereunable toprovetherest
2 1 2 KC 8 8
oftheconjecture. Themainresultofthismanuscriptisthefollowing:
Theorem1.2. AsK ∼= Gmodules,
C
1
U(g)/J( (5ω −ω ))| ∼= R(O ).
2 1 2 KC 8
Remark1.3. Thisquantizationmodelofnilpotentorbitclosureisverydifferentfromtheclassical
model given in [5]. Namely, the Brylinski model is not necessarily of the form U(g)/J(λ). In
particular, whentheclassical nilpotentorbitclosure O isnotnormal(theclassification ofallsuch
orbitclosuresisgivenin[14]),theinfinitesimalcharacteroftheBrylinskimodelλ alwaysyields
O
associated varietyAV(U(g)/J(λ )) = O′,whereO′ isstrictlysmallerthanO.
O
Infact, itcanbeshown that theBrylinski modelalways containsthe composition factor U(g)/
J(λ ). Thisispartoftheon-goingworkofBarbaschandtheauthor[4].
O
Before going to the proof of Theorem 1.2, it is worthwhile to mention the orbits other
than O in g. Indeed, Kraft in [13] confirmed that O is the only nilpotent orbit with
8 8
non-normal closure. So we just need to consider quantizations of the orbits (and their
covers)only. For thezero orbit O thequantization is trivial, and thequantization ofthe
0
minimal orbit O is U(g)/J(1(3ω +ω )), where J(1(3ω +ω )) is the Joseph ideal. The
6 3 1 2 3 1 2
10-dimensional orbit O is a special orbit with fundamental group S . It is a simple ex-
10 3
ercisetocomparetheformulasin [2]and [15]thatthesphericalunipotentrepresentation
attached to O is a quantization of O (as a bonus, the other two unipotent representa-
10
tions attached to O essentially gives quantization of all covers of O as well). Finally,
10 10
ONQUANTIZATIONOFANILPOTENTORBITCLOSUREING2 3
the quantization of the principal orbit O is well known to be the principal series rep-
12
resentation with zero infinitesimal character. In conclusion, we completed the picture of
quantizationforallnilpotentorbitsofgandtheirclosures.
2. PROOF OF THE THEOREM
AsmentionedintheIntroduction,thenon-normalityofO impliesthatR(O )( R(O ).
8 8 8
Infact,Costantiniin[7]givesthediscrepanciesintermsofG-modules:
Theorem 2.1 ([7], Theorem 5.6). Let V be the finite-dimensional irreducible representation
(a,b)
ofG withhighest weightaω +bω ,whereaandbarenon-negativeintegers. Then
2 1 2
R(O8) ∼= R(O8)⊕ M V(1,n).
n∈N∪{0}
ThefollowingLemmagivesanotherexpressionofthediscrepanciesbetweenR(O)and
R(O):
Lemma2.2. AsvirtualG-modules,
M V(1,n) = IndGT(1,0)−IndGT(0,1)−IndGT(2,0)+IndGT(1,1)+IndGT(0,2)−IndGT(2,1),
n∈N∪{0}
whereIndG(a,b)istheshorthand fortheinducedmoduleIndG(eaω1+bω2).
T T
Proof. The Lemma can be derived from the Weyl character formula. Namely, by the
W(G )-symmetryofweightsofV ,wehave
2 (a,b)
V(a,b) = X det(w)IndGT(λw)
w∈W(G2)
withλ beingtheuniqueW(G )-conjugateofw[(a,b)+(1,1)]−(1,1)lyinginthedominant
w 2
chamber. Infact,wehave
V =IndG(1,n)−IndG(1,n+3)−IndG(2,n)+IndG(2,n+2)
(1,n) T T T T
−IndG(3,n−1)+IndG(3,n)−IndG(3,n+1)+IndG(3,n+2)
T T T T
+IndG(6,n−2)−IndG(6,n)−IndG(7,n−2)+IndG(7,n−1)
T T T T
forn> 1,and
V = IndG(1,0)−IndG(1,3)−IndG(2,0)+IndG(2,2)−IndG(0,1)+IndG(3,0)
(1,0) T T T T T T
−IndG(3,1)+IndG(3,2)+IndG(0,2)−IndG(6,0)−IndG(1,2)+IndG(4,1);
T T T T T T
V = IndG(1,1)−IndG(1,4)−IndG(2,1)+IndG(2,3)−IndG(3,0)+IndG(3,1)
(1,1) T T T T T T
−IndG(3,2)+IndG(3,3)+IndG(3,1)−IndG(6,1)−IndG(4,1)+IndG(7,0).
T T T T T T
(cid:3)
TheLemmaisprovedbyaddinguptheterms.
4 KAYUEDANIELWONG
WenowstudythetwoHarish-Chandrabi-modulesU(g)/J(1(ω +ω ))andU(g)/J(1(5ω −
2 1 2 2 1
ω )):
2
Proposition2.3. AsK ∼= G-modules
C
1
U(g)/J( (ω +ω ))| = IndG(0,0)−IndG(0,1)−IndG(2,0)+IndG(1,1);
2 1 2 KC T T T T
1
U(g)/J( (5ω −ω ))| = IndG(0,0)−IndG(1,0)−IndG(0,2)+IndG(2,1).
2 1 2 KC T T T T
Proof. Tocaterforsubsequentcalculations,weleth∗ = {(x,y,z) ∈ C3|x+y+z = 0},with
shortsimplerootα = (1,−1,0)andlongsimplerootβ = (−1,2,−1). Then
1 1
λ = (ω +ω ) = (1,1/2,−3/2); λ = (5ω −ω ) = (2,−1/2,−3/2).
1 1 2 2 1 2
2 2
Thecharacter formulas ofU(g)/J(λ) for regularλ are wellknownby theworkofBar-
baschandVogan[2]: ConsiderthesubgroupW ofW(G )generatedbyrootsαsatisfying
λ 2
hα,λi
2 ∈ Z. Thentheformulaisgivenby
hα,αi
U(g)/J(λ) = X det(w)X(λ,wλ),
w∈Wλ
where X(µ,ν) = K-finite part of IndG(e(µ,ν) ⊗1) is the principal series representation
B
withcharacter(µ,ν) ∈ h ,thecomplexificationofthemaximal torushing(herewetreat
C
G as a real Lie group). In particular, theG ∼= K -typesof X(µ,ν) is equal toIndG(eµ−ν)
C T
(seeTheorem1.8of[2]formoredetailsontheprincipalseriesrepresentations).
Now apply the above recipe for λ = (1,1/2,−3/2): With the above notations, W
1 λ1
is isomorphic to W(A ×A˜ ), generated by the roots {(0,1,−1),(2,−1,−1)}. Hence the
1 1
characterformulaofU(g)/J(λ )isgivenby
1
U(g)/J(λ ) =X((1,1/2,−3/2),(1,1/2,−3/2)) −X((1,1/2,−3/2),(1,−3/2,1/2))
1
−X((1,1/2,−3/2),(−1,3/2,−1/2)) +X((1,1/2,−3/2),(−1,−1/2,3/2)).
UponrestrictingtoK ,wehave
C
U(g)/J(λ )| ∼= IndG(e(0,0,0))−IndG(e(0,2,−2))−IndG(e(2,−1,−1))+IndT(e(2,1,−3)).
1 KC T T T G
Again, by W(G )-symmetry of finite-dimensional irreducible G-modules, the above ex-
2
pressioncanbewrittenintheformasintheProposition. ThecalculationsforU(g)/J(λ )
2
(cid:3)
isidenticaltotheoneabove. Weomitthecalculations here.
ProofofTheorem 1.2. BytheresultofLevasseurandSmithin[11],
1
U(g)/J( (ω +ω ))| ∼= R(O )
2 1 2 KC 8
ONQUANTIZATIONOFANILPOTENTORBITCLOSUREING2 5
asG-modules. ThereforethefirstequationofProposition2.3gives
(1) R(O ) ∼= IndG(0,0)−IndG(0,1)−IndG(2,0)+IndG(1,1)
8 T T T T
asvirtualG-modules. ByTheorem2.1andLemma2.2,weneedtoshowthat
1
U(g)/J(2(5ω1−ω2))|KC = R(O8)−MV(1,n)
n
= R(O )−(IndG(1,0)−IndG(0,1)−IndG(2,0)
8 T T T
+IndG(1,1)+IndG(0,2)−IndG(2,1)).
T T T
ThisisreadilyseentobetruefromtheEquation(1)andthesecondequationofProposition
(cid:3)
2.3.
3. FINAL REMARKS
In [17], Sommers gives some conjectures on the multiplicities of small representations
ofR(O)fortheexceptionalgroups. Inparticular,giventhathisconjectureistrue,onecan
showthenon-normalityofsomeorbitclosures.
TodescribemoreexplicitlywhichorbitsOareconjecturedtohavenon-normalclosures,
recall thatLusztigin[12]partitionedall nilpotentorbitsin gbyspecial pieces, i.e. forall
nilpotentorbitO′,itmustbelongtoexactlyoneofthespecialpieces
S := {O′ ⊆ O|O′ * O foranyotherspecialorbitO ( O},
O spec spec
whereOrunsthroughallspecialorbitsing.
ForeachO′ ∈ S ,LusztigassignedaLevisubgroupH(O′,O)oftheLusztig’squotient
O
A(O). Forexample,thelargestorbitinthespecialpieceO ∈ S hasH(O,O) = 1,andthe
O
smallestorbitO′′ ∈ S hasH(O′′,O) = A(O).
O
By the conjecture of Sommers, if O has non-abelian Lusztig’s quotient, i.e. A(O) =
S ,S orS ,thenallO′ ∈ S withH(O′,O)notequalto1orA(O)(thatis,notequaltoO
3 4 5 O
orO′′)havenon-normalclosures.
Forexample,inthecaseofG westudiedabove,wehaveO ∈S andH(O ,O )=
2 8 O10 8 10
S ≤ S = A(O ). So O is conjectured to have non-normal closure, which has been
2 3 10 8
showntobetrue.
Wewouldliketoendourmanuscriptwiththefollowing:
Conjecture3.1. SupposeOisanilpotentorbitwithA(O) = S ,S orS ,andO′ ∈ S satisfies
3 4 5 O
H(O′,O) 6= 1, A(O). Then there exists two distinct completely prime primitive ideals J(λ ),
1
J(λ )suchthat
2
U(g)/J(λ )| ∼= R(O), U(g)/J(λ )| ∼= R(O).
1 KC 2 KC
6 KAYUEDANIELWONG
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DEPARTMENT OF MATHEMATICS, HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY, CLEAR
WATERBAY,KOWLOON,HONGKONG
E-mailaddress:[email protected]