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On Minuscule Representations and the Principal SL2
Benedict H. Gross
In this paper, we review the theory of minuscule coweightsλfor a simple adjoint groupGoverC, as presented by Deligne [D]. We then decompose the associated irre- ducible representationVλof the dual groupˆG, when restricted to a principalSL2. This decomposition is given by the action of a LefschetzSL2on the cohomology of the flag varietyX=G/Pλ, wherePλis the maximal parabolic subgroup ofGassociated to the coweightλ. We reinterpret a result of Vogan and Zuckerman [V-Z, Prop 6.19] to show that the cohomology ofXis mirrored by the bigraded cohomology of theL-packet of discrete series with infinitesimal characterρ, for a real formG0ofGwith a Hermitian symmetric space. We then focus our attention on those minuscule representations with a non-zero linear formt:V→Cfixed by the principalSL2, such that the subgroupˆH?ˆGfixingtacts irreducibly on the subspaceV0= ker(t). We classify them in§10; sinceˆHturns out to be reductive, we have a decomposition
V=Ce+V0
whereeis fixed byˆH, and satisfiest(e)?= 0. We studyVas a representation ofˆH, and give an
ˆH-algebra structure onVwith identitye.
The rest of the paper studies representationsπofGwhich are lifted fromH, in the sense of Langlands. We show this lifting is detected by linear forms onπwhich are fixed by a certain subgroupLofG. The subgroupLdescends to a subgroupL0→G0overR; both have Hermitian symmetric spacesDwith dimC(DL) =1
2dimC(DG). We hope this
will provide cycle classes in the Shimura varieties associated toG0, which will enable one to detect automorphic forms in cohomology which are lifted fromH. 1 It is a pleasure to thank Robert Kottwitz, Mark Reeder, Gordon Savin, and David
Vogan for their help.
2
Table of Contents
1. Minuscule coweights
2. The real formG0
3. The Weyl group
4. The flag variety
5. The representationVof the dual groupˆG
6. The principalSL2→ˆG
7. Examples
8. Discrete series and a mirror theorem
9. Discrete series forSO(2,2n)
10. A classification theorem:V=Ce+V0
11. The representationVofˆH
12. Representations ofGlifted fromH
13. The proof of Proposition 12.4
14. The real form ofL→G
15. Bibliography
3
1. Minuscule coweights
LetGbe a simple algebraic group overC, of adjoint type. LetT?B?Gbe a maximal torus, contained in a Borel subgroup, and let Δ be thecorresponding set of simple roots forT. Then Δ gives aZ-basis for Hom(T,Gm), so a coweightλin Hom(Gm,T) is completely determined by the integers?λ,α?, forαin Δ, which may be arbitrary. LetP+ be the cone of dominant co-weights, where?λ,α? ≥0 for allα?Δ. A coweightλ:Gm→Tgives aZ-gradinggλofg= Lie(G), defined by g
λ(i) ={X?g: Adλ(a)(X) =ai·X}
We sayλis minuscule providedλ?= 0 and the gradinggλsatisfiesgλ(i) = 0 for|i| ≥2. Thus (1.1)g=gλ(-1) +gλ(0) +gλ(1). The Weyl groupNG(T)/T=WofTacts on the set of minuscule coweights, and the W-orbits are represented by the dominant minuscule coweights. These have been classified Proposition 1.2([D, 1.2]).The elementλis a dominant, minuscule coweight if and only if there is a single simple rootαwith?λ,α?= 1, the rootαhas multiplicity 1 in the highest rootβ, and all other simple rootsα?satisfy?λ,α??= 0. Thus, theW-orbits of minuscule coweights correspond bijectively to simple rootsα with multiplicity 1 in the highest rootβ. Ifλis minuscule and dominant,gλ(1) is the direct sum of the positive root spacesgγ, whereγis a positive root containingαwith multiplicity 1. Hence the dimensionNofgλ(1) is given by the formula (1.3)N= dimgλ(1) =?λ,2ρ?, whereρis half the sum of the positive roots. The subgroupWλ?Wfixingλis isomorphic to the Weyl group ofTin the subalgebra g λ(0), which has root basis Δ-{α}. We now tabulate theW-orbits of minuscule coweights, 4 by listing the simpleαoccurring with multiplicity 1 inβin the numeration of Bourbaki [B]. We also tabulateN= dimgλ(1) and (W:Wλ); a simple comparison show that (W:Wλ)≥N+ 1 in all cases; we will explain this inequality later.
Table 1.4
G α(W:Wλ)N
A?αk??+1
k?k(?+ 1-k) B ?α12?2?-1 C ?α?2??(?+1) 2 D ?α12?2?-2 ?-1,α?2?-1?(?-1) 2 E
6α1,α627 16
E
7α156 27
5
2. The real formG0
We henceforth fixGand a dominant minuscule coweightλ. LetGcbe the compact real form forG, soG=Gc(C) andGc(R) is a maximal compact subgroup ofG. Let g?→ gbe the corresponding conjugation ofG. LetTc?Gcbe a maximal torus overR. We have an identification of co-character groups Hom cont(S1,Tc(R)) = Homalg(Gm,T). We viewλas a homomorphismS1→Tc(R), and define (2.1)θ= adλ(-1) inInn(G). Thenθis a Cartan involution, which gives another descentG0ofGtoR. The groupG0 has real points G
0(R) ={g?G:
g=θ(g)}, and a maximal compact subgroupKofG0(R) is given by
K={g?G:g=
gandg=θ(g)} =G0(R)∩Gc(R). The corresponding decomposition of the complex Lie algebragunder the action ofK is given byg=k+p, with (2.2)?k= Lie(K)?C=gλ(0) p=gλ(-1) +gλ(1). The torusλ(S1) lies in the center of the connected component ofK, and the elementλ(i) gives the symmetric space
D=G0(R)/K
a complex structure, with (2.3)N= dimC(D). 6 Proposition 2.4.([D, 1.2]).The real Lie groupsG0(R)andKhave the same number of connected components, which is either 1 or 2. Moreover, the following are all equivalent:
1)G0(R)has 2 connected components.
2)The symmetric spaceDis a tube domain.
3)The vertex of the Dynkin diagram ofGcorresponding to the simple rootαis fixed by
the opposition involution of the diagram.
4)The subgroupWλfixingλhas a nontrivial normalizer inW, consisting of thosew
withwλ=±λ. In fact, the subgroupWc?Wwhich normalizesWλis precisely the normalizer of the compact torusTc(R) inG0(R). WhenWλ?=Wc, it is generated byWλand the longest elementw0, which satisfiesw0λ=-λ.
As an example, letG=SO3and
λ(t) =((
t 1 t -1))
Thenθis conjugation by
λ(-1) =((
-1 1 -1)) andG0=SO(1,2) has 2 connected components. We haveK?O(2),Wc=Whas order
2 in this case, andWλ= 1. The tube domainD=G0(R)/Kis isomorphic to the upper
half plane. 7
3. The Weyl group(cf. [H])
The Weyl groupWis a Coxeter group, with generating reflectionsscorresponding to the simple roots in Δ. Recall thatρis half the sum of the positive roots andWλ?Wis the subgroup fixingλ. Proposition 3.1.Each cosetwWλofWλinWhas a unique representativeyof minimal length. The lengthd(y)of the minimal representative is given by the formula: d(y) =?λ,ρ? - ?wλ,ρ?, wherewis any element in the coset. Proof.LetR±be the positive and negative roots, letR±
λbe the subsets of positive
and negative roots which satisfy?λ,γ?= 0. ThenR+-R+
λconsists of the roots with
?λ,γ?= 1, andR--R- λconsists of the roots with?λ,γ?=-1. These sets are stable under the action ofWλonR. On the other hand, ifw?WλstabilizesR+
λ(orR-
λ) then
w= 1, asWλis the Weyl group of the root systemRλ=R+
λ?R-
Since the lengthd(y) ofyinWis given by
(3.2)d(y) = #{γinR+:y-1(γ) is inR-}, the set (3.3)Y={y?W:y(R+
λ)?R+}
gives coset representatives forWλof minimal length. Moreover, fory?Ythe sety-1(R+) containsd(y) elements ofR-
λ, and henceN-d(y) elements ofR+
λ. Hence, ifwWλ=yWλ,
we find ?wλ,ρ?=?yλ,ρ?=?λ,y-1ρ? 1
2((N-d(y))-d(y))
1
2N-d(y).
8 Since ?λ,ρ?=1 2N, we obtain the desired formula. As an example of Proposition 3.1, the minimal representative ofWλisy= 1, with d(y) = 0, and the minimal representative ofsαWλisy=sα, withd(y) = 1. Ifw0is the longest element in the Weyl group, thenw0(R±) =R?, sow20= 1, andw0ρ=-ρ. Hence Consequently, the length of the minimal representativeyofw0Wλisd(y) =N. This is the maximal value ofdonW/Wλ, and we will soon see thatdtakes all integral values in the interval [0,N]. Assumeλis fixed by the opposition involution-w0, sow0λ=-λ. ThenDis a tube domain, andWλhas nontrivial normalizerWc=?Wλ,w0?inW, by Proposition 2.4. The
2-groupWc/Wλacts on the setW/WλbywWλ?→ww0Wλ, and this action has no fixed
points. Hence we get a fixed point-free actiony?→y?on the setY, and find: (3.4)d(y) +d(y?) =N. 9
4. The flag variety
Associated to the dominant minuscule coweightλis a maximal parabolic subgroup
P, which containsBand has Lie algebra
(4.1) Lie(P) =gλ(0) +gλ(1). The flag varietyX=G/Pis projective, of complex dimensionN. The cohomology ofXis all algebraic, soH2n+1(X) = 0 for alln≥0. Let (4.2)fX(t) =? n≥0dimH2n(X)·tn be the Poincar´e polynomial ofH?(X). Then we have the following consequence of Chevalley- Bruhat theory, which also gives a convenient method of computing the values of the func- tiond:W/Wλ→Z.
Proposition 4.3. 1)We havefX(t) =?
Ytd(y).
2)IfG is the split adjoint group overZwith the same root datum asG, andPis the standard parabolic corresponding toλ, then f
X(q) = #G
(F)/P(F) for all finite fieldsF, withq= #F.
3)The Euler characteristic ofXis given by
χ=fX(1) = #(W:Wλ).
Proof.We have the decomposition
G=? YByP, 10 where we have chosen a lifting ofyfromWtoNG(T). IfUis the unipotent radical of B, thenB=UT. SinceynormalizesT,
UyP=ByP.
This gives a cell decomposition
X=?
YUy/P∩y-1Uy
where the cell corresponding toyis an affine space of dimensiond(y). This gives the first formula. The formula forfX(q) follows from the Bruhat decomposition, which can be used to prove the Weil conjectures forX. Formula 3) forfX(1) follows immediately from 1). For an example, letG=PSp2nbe of typeCn. ThenPis the Siegel parabolic subgroup, with Levi factorGLn/μ2. From the orders ofSp2n(q) andGLn(q), we find #G (F)/P(F) =(q2-1)(q4-1)...(q2n-1)(q-1)(q2-1)...(qn-1) = (1 +q)(1 +q2)...(1 +qn).
Hence we find
f
X(t) = (1 +t)(1 +t2)...(1 +tn).(4.4)
The fact thatX=G/Pis a Kahler manifold imposes certain restrictions on its cohomology. For example, ifωis a basis ofH2(X), thenωk?= 0 inH2k(X) for all Corollary 4.5.The functiond:W/Wλ→Ztakes all integral values in[0,N], and (W:Wλ)≥N+ 1. m(k) = #{y?Y:d(y) =k}. 11 We have seen thatm(0) =m(1) = 1 in all cases. By Poincar´e duality (4.6)m(k) =m(N-k). Finally, the Lefschetz decomposition into primitive cohomology shows that: weightsN-2d(y) for the maximal torus?t0 0t-1? 12
5. The representationVof the dual groupˆG
Let ˆGbe the Langlands dual group ofG, which is simply-connected of the dual root type. This group comes (in its construction) with subgroupsˆT?ˆB?ˆG, and an identification of the positive roots for ˆBin Hom(ˆT,Gm) with the positive co-roots forB in Hom(Gm,T) (cf. [G]). Hence, the dominant co-weights forTgive dominant weights for T, which are the highest weights forˆBon irreducible representations ofˆG. LetVbe the irreducible representation ofˆG, whose highest weight forˆBis the dom- inant, minuscule co-weightλ. Proposition 5.1.The weights ofˆTonVconsist of the elements in theW-orbit of
λ. Each has multiplicity 1, sodimV= (W:Wλ).
The central characterχofVis given by the image ofλinHom(ˆT,Gm)/?
ΔZα?, and
is nontrivial. roots. These are precisely the other dominant weights for
ˆToccurring inVλ. Whenλis
same multiplicity as the highest weight, which is 1. Sinceμ= 0 is dominant,λis not in the span of the co-roots, andχ?= 1. This result gives another proof of the inequality of Corollary 4.5: (W:Wλ)≥N+1. Indeed, letLbe the unique line inVλfixed byˆB. The fixer ofLis the standard parabolic Pdual toP. This gives an embedding of projective varieties:
G/ˆP ?→P(Vλ).
Since ˆG/ˆPhas dimensionN, andP(Vλ) has dimension (W:Wλ)-1, this gives the desired inequality. The real formG0defined in§2 has LanglandsL-group (5.2)LG=ˆGxGal(C/R). 13 The action of Gal(C/R) onˆGexchanges the irreducible representationVwith dominant weightλwith the dual representationV?with dominant weight-w0λ. Hence the sum V+V?always extends to a representation ofLG. The following is a simple consequence of Proposition 2.4.
Proposition 5.3.The following are equivalent:
1) We havew0λ=-λ.
2) The symmetric spaceDis a tube domain.
3) The representationVis isomorphic toV?.
4) The central characterχofVsatisfiesχ2= 1.
5) The representationVofˆGextends to a representation ofLG.
14
The principalSL2→ˆG
The group
ˆGalso comes equipped with a principal?:SL2→ˆG[G]. The co-character G m→ˆTgiven by the restriction of?to the maximal torus?t0 0t-1? ofSL2is equal to 2ρin Hom(Gm,ˆT) = Hom(T,Gm). From this, and Proposition 5.1, we conclude the following. Proposition 6.1.The restriction of the minuscule representationVto the principal SL
2inˆGhas weights
W/W λt ?wλ,2ρ? for the maximal torus ?t0 0t-1? inSL2.
On the other hand, by Proposition 3.1, we have
(6.2)?wλ,2ρ?=?λ,2ρ? -2d(y) =N-2d(y) whered(y) is the length of the minimal representativeyin the cosetwWλ. Hence the weights for the principalSL2acting onVare the integers (6.3)N-2d(y)y?Y in the interval [-N,N]. Since these are also the weights of the LefschetzSL2acting on the cohomologyH?(G/P) by§4, we obtain the following. Corollary 6.4.The representation of the principalSL2ofˆGonVis isomorphic to the representation of the LefschetzSL2on the cohomology of the flag varietyX=G/P. 15
7. Examples
We now give several examples of the preceding theory, using the notation for roots and weights of [B]. IfGis of typeA?andα=α1we haveλ=e1. The flag varietyG/Pis projective spacePN, withN=?, and the Poincar´e polynomial is 1+t+t2+···+tN. The dual group GisSLN+1, andVis the standard representation. The restriction ofVto a principalSL2 is irreducible, isomorphic toSN= SymN(C2). A similar result holds whenGis of typeB?, soα=α1andλ=e1. HereG/Pis a quadric of dimensionN= 2?-1, withP(t) = 1 +t+···+tNas before. The dual group is ˆG= Sp2?, the representationVis the standard representation, and its restriction to the principalSL2is the irreducible representationSN. Next, supposeGis of typeD?andα=α1, soλ=e1. ThenG/Pis a quadric of dimensionN= 2?-2, and we haveP(t) = 1 +t+···+ 2t?-1+···+tN. The dual group Gis Spin2?, andVis the standard representation of the quotientSO2?. Its restriction to the principalSL2is a direct sumSN+S0, whereS0is the trivial representation. A more interesting case is whenGis of typeC?, soα=α?andλ=e1+e2+···+e?
2. Here
G/Pis the Lagrangian Grassmanian of dimensionN=?(?+1)
2, and
P(t) = (1 +t)(1 +t2)...(1 +t?) was calculated in (4.4). The dual groupˆGis Spin2?+1, andVis the spin representation of dimension 2?. Its decomposition to a principalSL2is (7.1)S 1?= 1 S 3?= 2 S
6+S0?= 3
S
10+S4?= 4
S
15+S9+S5?= 5
S
21+S15+S11+S9+S3?= 6
16 As the last example, supposeGis of typeE6. ThenG/Phas dimension 16 andquotesdbs_dbs21.pdfusesText_27
SPECIAL CYCLES ON UNITARY SHIMURA VARIETIES WITH MINUSCULE