Principal Hodge representations

0.25 MB·English

by  C. Robles

PRINCIPAL HODGE REPRESENTATIONS
C. ROBLES
3
1
0
Abstract. WestudyHodgerepresentationsofabsolutelysimpleQ–algebraicgroupswith
2 Hodge numbers h = (1,1,...,1). For those groups that are not of type A, we give a

n classification of the R–irreducible representations; a similar classification for type A does
a not seem possible.
J

3
2

] 1. Introduction
G
A This paper is a study of Hodge structures with Hodge numbers h = (1,1,...,1). Exam-
. ples of such Hodge structures include the cohomology group H1(X,Q) of an elliptic curve,
h
the cohomology group H3(X,Q) of a mirror quintic variety [3], and the weight component
t
a W H6(X,Q) of the cohomology group of a general fibre of a family of quasi-projective
m 6
6-folds studied by Dettweiler and Reiter [2, 6]. More precisely, the article addresses the
[
question: what are the Hodge representations ( 2.1) with Hodge numbers h = (1,1,...,1)?
§
1 To simplify exposition, we call such representations ‘principal,’ cf. Remark 2.6. The prin-
v
cipal Hodge representations are necessarily of Calabi-Yau type; Calabi-Yau Hodge repre-
6
sentations of (absolutely) simple Hodge groups are classified in [8, B]. I emphasize that
0
§
6 the classification of principal Hodge representations is essentially a representation theoretic
5 question; I will not directly address the Hodge theory of complex algebraic varieties.
.
1
0 Motivation. Very roughly, the classification question is motived by the question: Does
3
a Hodge domain admit a distinguished realization as a Mumford–Tate domain. A more
1
: precise statement of the question, and its relation to this work, is given in Section 2.2?
v
i
X Contents. The main results of this paper are characterizations of the principal Hodge
r representations (V,ϕ) of a Q–algebraic, absolutely simple[i] group G in the case that (i) the
a
vector space V is an irreducible G –module, and (ii) the Lie algebra g = sl C. The
R R C r+1
6
principal Hodge representations of symplectic groups (type C) are identified in Theorem
3.1, those of orthogonal groups (types B and D) are identified in Theorems 4.1 and 4.3, and
those of the exceptional groups of type G are identified in Theorem 5.2. The exceptional
2
groups of types E and F admit no such principal Hodge representations (Corollary 2.15
r 4
and Theorem 5.1). (For the exceptional groups, we may drop the assumption (ii) that V
R
is irreducible.)
Date: January 24, 2013.
2010 Mathematics Subject Classification. 20G05, 58A14.
Key words and phrases. Hodge group, Hodge representation, Hodge structure,Mumford–Tate group.
Robles is partially supported by NSFDMS-1006353.
[i]G is absolutely simple if GC is simple.
1
2 ROBLES
I am unable to give a similar characterization in the case that g = sl C (type A).
C r+1
A comparison of the theorems above with the results and examples of 6 suggest that no
§
such characterization is available in this case: given a representation π : G Aut(V,Q),
→
the circle ϕ giving π the structure of a principal Hodge representation is essentially unique
(if it exists) for types B, C, D and G; in contrast, the circle is far from unique in type A,
cf. 6.5. Nonetheless, we may make some general observations (cf. 6.1); for example, if
§ §
G admits a principal Hodge representation (with V not necessarily irreducible), then the
R
rank r is necessarily odd (Lemma 6.4). The rank one, three and five cases are worked out
in 6.2, 6.3 and 6.4, respectively, and examples of rank seven and nine are considered in
§
6.5.
§
Acknowledgements. I thank P. Griffiths for bringing the classification question to my
attention.
2. Hodge representations
2.1. Definition. Let V be a rational vector space, w Z, and let Q = V V Q
∈ × →
be a nondegenerate bilinear form satisfying Q(u,v) = ( 1)wQ(v,u), for all u,v V. A
− ∈
polarized Hodge structure of weight w on V is given by a nonconstant homomorphism φ :
S1 Aut(V ,Q) of R–algebraic groups with the properties that φ( 1) = ( 1)w , and
R
→ − − 1
Q(v,φ(i)v) > 0 for all 0 = v V . (Here, i = √ 1.) The associated Hodge decomposition
C
V = Vp,q is give6 n by∈Vp,q = v V −φ(z)v = zp−qv, z S1 . The Hodge
C p+q=w C
⊕ { ∈ | ∀ ∈ }
numbers h = (hp,q) of φ are hp,q = dim Vp,q.
C
Let G be a Q–algebraic group, and let π : G Aut(V,Q) be a homomorphism of Q–
→
algebraic groups, such that π : g End(V,Q) is injective. Let ϕ : S1 G be a
∗ C R
→ →
nonconstant homomorphism of R–algebraic groups. Then (V,Q,π,ϕ) is a Hodge represen-
tation of G if π ϕ is a Q–polarized Hodge structure on V of weight w. Any Q–algebraic
◦
group admitting a Hodge representation is a Hodge group. A Hodge group G is necessar-
ily reductive [4, (I.B.6)], and G contains a compact, maximal torus T ϕ(S1); that is,
R
⊃
dim (T)= rank(G ), cf. [4, (IV.A.2)].
R R
In general, the bilinear form Q and representation π will be dropped from the notation,
and the Hodge representation will be denoted by (V,ϕ).
2.2. Context. HodgerepresentationswereintroducedbyGreen,GriffithsandKerrin[4]to
determine: (i) which reductive, Q–algebraic groups G admit the structure of a Mumford–
Tate group, and (ii) what one can say about various realizations of G as a Mumford–
Tate group, and the associated Mumford–Tate domains (which generalize period domains).
Mumford–Tate groups are the symmetry groups of Hodge theory: the Mumford–Tate group
G = Aut(V,Q) of a polarized Hodge structure (V,Q,φ) is precisely the Q–algebraic
φ
⊂ G
group preservingthe Hodgetensors [4, (I.B.1)]. Moreover, φ(S1) G (R), so that(V,Q,φ)
φ
⊂
is a Hodge representation of G , and the Mumford–Tate group is a Hodge group.
φ
Mumford–Tate domains arise as follows. Let Fp = Vq,w−q denote the Hodge filtra-
q≥p
⊕
tionofV = V CinducedbytheHodgestructureφ. DefineHodgenumbersfp = dim Fp
C Q C
⊗
and f = (f•). Then F• is an element of the isotropic flag variety FlagQ(V ), and the latter
f C
is a = Aut(V ,Q)–homogeneous manifold. (If the weight w is even, then = SO(V )
C C C C
G G
is an orthogonal group; if w is odd, then = Sp(V ) is a symplectic group.) The period
C C
G
PRINCIPAL HODGE REPRESENTATIONS 3
domain FlagQ(V ) is the = Aut(V ,Q)–orbit of F•. The Mumford–Tate domain
f C R R
D isDth⊂e G (R)–orbit of FG•.
φ
⊂ D
As a G (R)–homogeneous manifold D G (R)/R; here R is the centralizer of the circle
φ φ
≃
φ(S1) and contains the compact, maximal torus T, cf. [4, II.A]. Thehomogeneous manifold
G (R)/R is a Hodge domain. One may see from this discussion that one subtlety that
φ
arises in the subject is the fact that a Hodge domain will admit various realizations as a
Mumford–Tate domain (and with inequivalent G (R)–homogeneous complex structures).
φ
One would like to know if a given Hodge domain admits a distinguished realization as a
Mumford–Tatedomain.[ii] Theclassification ofprincipalHodgerepresentationsismotivated
by this question.
AHodge representation (V,φ) of Gis principalif and only if fp+1 = fp 1, cf. Definition
−
Q
2.5. Equivalently, theperioddomain = Flag (V )is afullflagvariety, andthehorizontal
f C
D
distribution (known as Griffiths’ transversality or the infinitesimal period relation) on the
period domain is bracket–generating. In this case, the centralizer R of the circle φ(S1) is
D
the torus T, cf. Remark 2.18. So the principal Hodge representation yields a distinguished
realization of the Hodge domain G (R)/T as a Mumford–Tate domain D .
φ
⊂ D
2.3. The grading element T . Fix a Hodge representation (V,ϕ). To the circle ϕ(S1)
ϕ
is naturally associated a semisimple T ig . What follows is a terse review of T ; see
ϕ R ϕ
[8, 2.3] for details. The semisimple T h∈as the property that the T –eigenvalues of V are
ϕ ϕ C
§
rational numbers (in fact, elements of 1Z), and the T –eigenspace decomposition
2 ϕ
V = V , V d=fn v V T (v) = ℓv
C ℓ ℓ C ϕ
{ ∈ | }
Mℓ∈Q
is the Hodge decomposition: that is, Vp,q = V , where ℓ = (p q)/2 1Z. Additionally,
the T –eigenvalues of g are integers, and the Tℓ –eigenspace de−compos∈iti2on
ϕ C ϕ
g = g , g d=fn ζ g [Tℓ,ζ]= ℓζ ,
C ℓ ℓ C
{ ∈ | }
Mℓ∈Z
is the Hodge decomposition associated to the weight zero Hodge structureAd π on g: that
is, gℓ,−ℓ = g . Moreover, ◦
ℓ
g (V ) V and [g ,g ] g .
ℓ q q+ℓ k ℓ k+ℓ
⊂ ⊂
In particular, the Hodge filtration F• of V , defined by Fp = V , is stabilized by the
C ≥p
parabolic subalgebra
dfn
(2.1) p = g
≥0
of g .
C
Fix a compact maximal torus T G containing the circle ϕ(S1). Let t g and
R R
t g denote the associated Lie alg⊂ebras. Then T it. Let ∆ = ∆(g ,t ) ⊂t∗ denote
C ⊂ C ϕ ∈ C C ⊂ C
[ii]Byanalogy,thedistinguishedrealization ofarationalhomogeneous varietyG/Pistheuniqueminimal
homogeneousembeddingG/P֒→PV. ThePlu¨ckerembeddingoftheGrassmannian Gr(k,Cn)≃SL(Cn)/P
k
is an example.
4 ROBLES
the roots of t . Let Λ Λ t∗ denote the root and weight lattices. The connected real
C rt ⊂ wt ⊂ C
groups G with Lie algebra g are indexed by sub-lattices Λ Λ Λ . The torus is
R R rt wt
⊂ ⊂
T = t/Λ∗, where Λ∗ d=fn Hom(Λ,2πiZ),
and the weights Λ(V ) of V are contained in Λ. (Conversely, if U is an irreducible g –
C C C
module of highest weight µ Λ, then U is a G –module.) Any T Hom(Λ, 1Z) determines
∈ C ∈ 2
a circle ϕ : S1 T such that T = T.
ϕ
→
Notice that t g p; so we may select a Borel subalgebra b so that t b p.
C 0 C
⊂ ⊂ ⊂ ⊂
Define positive roots by
∆+ d=fn ∆(b) = α ∆ gα b .
{ ∈ | ⊂ }
LetΣ = σ ,...,σ denotethesimplerootsof∆+,andlet T1,...,Tr Hom(Λ ,Z) t
1 r rt C
{ } { } ⊂ ⊂
denotethedualbasisoft ; thatis, σ (Tj) = δj. Thelattice Hom(Λ ,Z)istheset of grading
C i i rt
elements. The fact that the T –eigenvalues of g are integers implies T Hom(Λ ,Z);
ϕ C ϕ rt
that is, T is a grading element. Therefore, T is necessarily of the form∈n Ti for some
ϕ ϕ i
ni Z. The condition gσi b p= g≥0 is equivalent to
∈ ⊂ ⊂
(2.2) σ (T ) = n 0,
i ϕ i
≥
for all 1 i r.
≤ ≤
2.4. Real, complex and quaternionic representations. Suppose that V is an irre-
R
ducible G –module. Then there exists an irreducible G –module U such that one of the
R C
following holds.
◦ As a G –module U is real: V = U U∗.
R C
≃
◦ As a G –module U is quaternionic: V = U U∗ and U U∗.
R C
⊕ ≃
◦ As a G –module U is complex: V = U U∗ and U U∗.
R C
⊕ 6≃
Notice that U is complex if and only if U is not self-dual. The real and quaternionic cases
are distinguished as follows. Define
(2.3) Tcpt d=fn 2 Ti.
nXi∈2Z
Let µ Λ(U) Λ be the highest weight of U. If U U∗, then
∈ ⊂ ≃
(2.4) U is real (resp., quaternionic) if and only if µ(Tcpt) is even (resp., odd),
cf. [4, (IV.E.4)].
2.5. Principal Hodge representations. Let
m d=fn max q Q V = 0 1Z.
{ ∈ | q 6 } ∈ 2
Then the T –eigenspace decomposition of V is
ϕ C
V = V V V V ,
C m m−1 1−m −m
⊕ ⊕···⊕ ⊕
and the Hodge numbers are h = (h ,h ,...,h ,h ), where h = dim V .
m m−1 1−m −m ℓ C ℓ
Definition 2.5. The Hodge representation (V,ϕ) is principal if the Hodge numbers are
h = (1,1,...,1,1); that is h = 1 for all m ℓ m.
ℓ
− ≤ ≤
PRINCIPAL HODGE REPRESENTATIONS 5
Remark 2.6. The Hodge representation (V,ϕ) is principle if and only if T is the mono-
ϕ
semisimple element of a principal sl C sl(V ). See [7].
2 C
⊂
If (V,ϕ) is principal, then it is necessarily the case that
(2.7) the T –eigenvalues of V are multiplicity-free,
ϕ C
(2.8) and m = 1 (dim V 1) .
2 C C −
The eigenvalues are determined by the weights Λ(V ) t∗ of V . Precisely, the T –
eigenvalues of V are λ(T ) λ Λ(V ) . In particCula⊂r, (CV,ϕ) isCprincipal if and onϕly
C ϕ C
{ | ∈ }
if
(2.9) λ(T ) λ Λ(V ) = m, m 1, m 2,..., 2 m, 1 m, m
ϕ C
{ | ∈ } { − − − − − }
and (2.8) holds.
Note that the T –eigenvalues of V are multiplicity-free only if
ϕ C
(2.10) V is weight multiplicity-free.
C
Recall that Λ(U∗) = Λ(U). So, if V = U U∗, then Λ(V ) = Λ(U) Λ(U). The
C C
− ⊕ ∪ −
necessary condition (2.10) implies
(2.11) The g –module U is either real or complex.
C
(2.12) If U is complex, then Λ(U) Λ(U) = .
∩− ∅
2.6. Multiplicity-free representations. Suppose that (V,ϕ) is a principal Hodge repre-
sentation, and that V0 V is an irreducible G –submodule. Let U be the corresponding
R R
⊂
irreducible g –module, cf. 2.4. By (2.10), it is necessarily the case that
C
§
(2.13) U is weight multiplicity-free.
The weight multiplicity-free representations have been classified by [5, Theorem 4.6.3], in
the case that g is simple. Let ω ,...,ω denote the fundamental weights of g .
C 1 r C
Theorem2.14. Letg be asimple, complex Lie group. The irreducible, weightmultiplicity-
C
free representations U of g , with highest weight µ, are as follows:
C
(a) If g = sl C, then U is one of kCr+1, and µ = ω ; SymaCr+1, and µ = aω ; or
C r+1 k 1
Syma(Cr+1)∗, and µ = aωr. V
(b) If g = so C, then U is either the standard representation C2r+1, with µ = ω ; or
C 2r+1 1
the spin representation, with µ =ω .
r
(c) If g = sp C, then either U is the standard representation C2r, with µ = ω ; or
C 2r 1
r 2,3 and U rC2r, with µ = ω .
r
∈ { } ⊂
(d) If g = so C, thenVU is either the standard representation C2r, with µ = ω ; or one
C 2r 1
of the spin representations, with µ = ω ,ω .
r−1 r
(e) If g is the exceptional Lie algebra e (C), then µ = ω ,ω and dim U = 27; if g =
C 6 1 6 C C
e (C), then µ = ω and dim U = 56. If g is the exceptional Lie algebra g (C), then
7 7 C C 2
U = C7 and µ = ω . There are no weight multiplicity-free representations for the
1
exceptional Lie algebras e and f .
8 4
From (2.13) and Theorem 2.14(e) we obtain the following.
6 ROBLES
Corollary 2.15. The exceptional Lie groups of type E and F admit no principal Hodge
8 4
representations.
2.7. Weights. It will be helpful to review some well-known properties of the weights Λ(U)
of the irreducible g –module U. Assume g is semi-simple, and let µ denote the highest
C C
weight of U. Given any weight λ Λ(U), there exists an (ordered) sequence σ ,...,σ
∈ { i1 iℓ}
of simple roots such that
(i) λ = µ (σ + +σ ), and
− i1 ··· iℓ
(ii) µ (σ + +σ ) Λ(U), for all 1 j ℓ.
− i1 ··· ij ∈ ≤ ≤
By (i), any weight λ of U is of the form µ λiσ , for some 0 λi Z. So, (2.2) implies
i
µ(T ) is the largest eigenvalue of U: − ≤ ∈
ϕ
(2.16) λ(T ) µ(T ) for all λ Λ(U).
ϕ ϕ
≤ ∈
If Uµ U is the highest weight line, then the weight space of λ = µ (σ + +σ )
is Uλ = ⊂g−σiℓ g−σi1 Uµ. In particular, since π∗ : gC End(V,Q) is−injecit1ive,··fo·r eaicℓh
1 i r, th·e·r·e exists· λ Λ(U) such that λ σ Λ→(U). Therefore, both λ(T ) and
i ϕ
λ(≤T ) ≤n are eigenvalues∈of U. In particular,−T –ei∈genvalues of V are multiplicity-free
ϕ i ϕ C
−
only if n = 0 for all 1 i r. By (2.2), n 0. Therefore, if the Hodge representation
i i
6 ≤ ≤ ≥
(V,ϕ) is principal, it is necessarily the case that
(2.17) n > 0,
i
for all 1 i r.
≤ ≤
Remark 2.18. It follows from this discussion and (2.1) that the stabilizer p = g of the
≥0
Hodge filtration F• in g is the Borel b. In particular, the Hodge domain is G (R)/T, with
C φ
T a compact, maximal torus.
Let µ∗ denote the highest weight of U∗. SwappingU and U∗ if necessary, we may assume
that
(2.19) µ(T ) µ∗(T ).
ϕ ϕ
≤
Then (2.9) and (2.16) yield
(2.20) m = µ(T ).
ϕ
Define
(2.21) m∗ d=fn µ∗(T ) m.
ϕ
≤
By (2.9), any two T –eigenvalues of V differ by an integer; therefore
ϕ C
0 m m∗ Z.
≤ − ∈
If U is complex, so that V = U U∗ then (2.7) and (2.19) force
C
⊕
m∗ < m.
The notation
(λ1 λr) d=fn λ = µ λiσ
i
··· −
will be convenient. For example, µ is denoted (0 0). Set
···
λ = λi.
| |
X
PRINCIPAL HODGE REPRESENTATIONS 7
The weights λ Λ(U) : λ = 1 may be described very easily: let µ = µiω ; then,
i
{ ∈ | | }
µ σ is a weight of U if and only if µi = 0, cf. [1, Proposition 3.2.5]. If g is simple, and
i C
− 6
U is weight multiplicity-free, then µ = pω , by Theorem 2.14. Therefore, the two largest
i
T –eigenvalues of U are m = µ(T ) and m n = (µ σ )(T ). The following lemma is a
ϕ ϕ i i ϕ
− −
consequence of this discussion and equations (2.7) and (2.9).
Lemma 2.22. Assume that g is simple. Suppose that (V,ϕ) is a principal Hodge rep-
C
resentation of G, and that V is an irreducible G –submodule. Let U be the associated
R R
irreducible, weight multiplicity-free g –module (cf. 2.4) of highest weight µ = pω , and
C i
§
satisfying (2.19).
(a) If U is real (V = U), then n = 1.
C i
(b) Suppose U is complex (V = U U∗ and U = U∗). Then 0 < m m∗ Z. Define
C
1 i∗ r by µ∗ = pωi∗. Then⊕m∗ = m 16if and only if ni > 1;−and m∈∗ = m 1
≤ ≤ − −
and ni∗ = 1 if and only if ni > 2.
3. Symplectic Hodge groups
Theorem 3.1. Let G be a Hodge group with complex Lie algebra g = sp C. Assume
C 2r
that (V,ϕ) is a Hodge representation with the property that V is an irreducible G –module.
R R
Let T be the associated grading element ( 2.3), and assume the normalization (2.2) holds.
ϕ
§
Then the Hodge representation is principal if and only if one of the following holds:
(a) V = C2r is the standard representation, and T = T1+ +Tr;
C ϕ
···
(b) r = 2, V 2C4 is the irreducible G –module of highest weight ω , and T = T1+T2;
C C 2
⊂
(c) r = 3, V V 3C6 is the irreducible G –module of highest weight ω , and T =
C C 3 ϕ
3T1+T2+T⊂1. V
Proof. The representations of G are self-dual. So the representation U associated to V
C R
(cf. 2.4) is either realor quaternionic. By (2.11), if(V,ϕ) is principal,then U is necessarily
§
real, so that V = U. The condition (2.13) and Theorem 2.14(c) restrict our attention to
C
the case that U is one of the the following fundamental representations: either U = C2r
ω1
of highest weight
ω = σ + +σ + 1σ ,
1 1 ··· r−1 2 r
(cid:0) (cid:1)
for arbitrary r; or U = rC2r of highest weight
ωr
V
ω = σ +σ if r = 2,
2 1 2
ω = σ +2σ + 3σ if r = 3.
3 1 2 2 3
(A) Let’s begin with the case that µ = ω . The weights of C2r are
1
Λ(C2r) = ω ω (σ + +σ ) 1 i r
1 1 1 i
{ } ∪ { − ··· | ≤ ≤ }
ω (σ + +σ ) 2(σ + +σ ) σ 1 i r 1 .
1 i−1 i r−1 r
∪ { − ··· − ··· − | ≤ ≤ − }
Itisstraightforward tocheck thatT = T1+ +Tr is theuniquegradingelement satisfying
ϕ
···
the normalization (2.2), and yielding the consecutive, multiplicity-free eigenvalues (2.9).
8 ROBLES
(B) Next, consider the case that r = 2 and µ = ω2. Here dimCU = 5 and the weights of U
are
Λ(U) = (σ +σ ), σ , 0
1 2 1
{± ± }
= ω , ω σ , ω (σ +σ ), ω (2σ +σ ), ω 2(σ +σ ) .
2 2 2 2 1 2 2 1 2 2 1 2
{ − − − − }
It is clear that T = T1 +T2 is the unique element satisfying the normalization (2.2), and
yielding the consecutive, multiplicity-free eigenvalues (2.9).
(C) Finally, suppose that r = 3 and µ = ω3. Then dimCU = 14, and the weights of U are
Λ(U) = (000), (001), (011), (021), (022), (111), (121),
{
(122), (132), (221), (222), (232), (242), (243) .
}
Above, we utilize the notation (λ1λ2λ3) = ω (λ1σ +λ2σ +λ3σ ) introduced in 2.7.
3 1 2 3
− §
In particular, the weights of U include ω , ω σ , ω (σ +σ ) . Since the remaining
3 3 3 3 2 3
{ − − }
weights are all of the form ω (aσ +bσ +cσ ), with 0 a,b,c Z and a+b+c 3,
3 1 2 3
− ≤ ∈ ≥
this forces 1 = n = n .
3 2
The conditions (2.8) and (2.20) yield 13/2 = µ(T ) = µ(n T1+T2+T3), so that n = 3.
ϕ 1 1
Thus,thegradingelementisnecessarilyoftheformgiven in(c). GivenT = 3T1+T2+T3,is
ϕ
straight-forward tocomputeΛ(U)(T ) = 13/2,11/2,..., 11/2, 13/2 . Thus(2.9) holds.
ϕ
{ − − }
Finally, observe that in each of the cases (A), (B) and (C) above, (2.3) yields Tcpt = 0, so
that µ(Tcpt) = 0 and V = U is real by (2.4), as required. (cid:3)
C
4. Orthogonal Hodge groups
Theorem 4.1. Let G be a Q–algebraic group with complex Lie algebra g = so C.
C 2r+1
Assume that (V,ϕ) is a Hodge representation of G with the property that V is an irreducible
R
G –module. Let T be the associated grading element ( 2.3), and assume the normalization
R ϕ
§
(2.2) holds. Then the Hodge representation is principal if and only if one of the following
holds:
(a) V = C2r+1 is the standard representation, and T = T1+ +Tr;
C ϕ
···
(b) V = U is the spin representation,
C ωr
(4.2) T = 2r−2T1 + 2r−3T2 + 2r−4T3 + +2Tr−2 + Tr−1 + Tr,
ϕ
···
and (r 2)(r 1) 4Z.
− − ∈
Theorem 4.3. Let G be a Hodge group with complex Lie algebra g = so C. Assume
C 2r
that (V,ϕ) is a Hodge representation with the property that V is an irreducible G –module.
R R
Let T be the associated grading element ( 2.3), and assume the normalization (2.2) holds.
ϕ
§
Then the Hodge representation is principal if and only if one of the following holds:
(a) r 4 is even, V = U is a spin representation (µ = ω ,ω ),
C r−1 r
≥
(4.4) T = 2r−3T1 + 2r−4T2 + 2r−5T3 + +2Tr−3 + Tr−2 + Tr−1 + Tr,
ϕ
···
and (r 3)(r 2) 4Z;
− − ∈
PRINCIPAL HODGE REPRESENTATIONS 9
(b) r 5 is odd, V = U U∗, where U U∗ is a spin representation (µ = ω ,ω ) and
C r−1 r
T≥is one of ⊕ 6≃
ϕ
T = 2r−2T1 + 2r−3T2 + +4Tr−3 + 2Tr−2 + Tr−1 + 3Tr,
(4.5) Tϕ = 2r−2T1 + 2r−3T2 +··· +4Tr−3 + 2Tr−2 + 3Tr−1 + Tr.
ϕ
···
Proof of Theorem 4.1. Let U be the irreducible so C–module associated to V as
2r+1 R
in 2.4. All representations of so C are self-dual; therefore, if (V,ϕ) is principal, it is
2r+1
§
necessarily the case that U is a real representation of G , by (2.11), and V = U. By
R C
(2.13), U is weight multiplicity-free. The irreducible, multiplicity-free representations of
so C are given by Theorem 2.14(b): either U = C2r+1 is the standard representation,
2r+1
with highest weight
ω = (σ + σ + + σ ) ;
1 1 2 r
···
or U is the spin representation of highest weight
ω = 1(σ + 2σ + + rσ ) .
r 2 1 2 ··· r
We consider each case below.
The standard representation. Suppose that U = C2r+1 is the standard representation. The
weights of U are
Λ(V ) = ω ω (σ + +σ ) 1 i r
C 1 1 1 i
{ } ∪ { − ··· | ≤ ≤ }
ω (σ + +σ )+2(σ + +σ ) 1 i r .
1 1 i−1 i r
∪ { − ··· ··· | ≤ ≤ }
It is straight forward to confirm that T = T1 + + Tr is the unique grading element
ϕ
satisfying the normalization (2.2) and such that th·e··T –eigenvalues of U satisfy (2.8) and
ϕ
(2.9).
In particular, Tcpt = 0, by (2.3), so that U is real, by (2.4), as required by (2.11).
The spin representation: preliminaries. Suppose that U is the spin representation. Then
µ = ω . The weights of U are parameterized by
r
(4.6a) (U) = λ = (λ1,...,λr) Zr λ1 0,1 , and λi λi−1 0,1 , 1 < i r ;
L { ∈ | ∈ { } − ∈ { } ∀ ≤ }
specifically,
(4.6b) Λ(U) = ω λiσ λ (U) .
r i
{ − | ∈L }
It will be convenient to define (i) a filtration 0 = 0 1 2 r−1 r =
{ } F ⊂ F ⊂ F ⊂ ··· ⊂ F ⊂ F
(U),
L
(4.7a) s d=fn λ (U) 0= λ1,...,λr−s , 0 s r;
F { ∈ L | } ≤ ≤
and (ii) a decomposition (U) = 0 1 r,
L L ⊔L ⊔···⊔ L
(4.7b) s d=fn s s−1 (4=.6) λ (U) λr−s = 0, λr−s+1 = 1 , 0 s r,
L F \F { ∈ L | } ≤ ≤
with the convention that −1 = .
F ∅
The spin representation is of dimension 2r. By (2.8) and (2.20),
(4.8) 1(2r 1) = m = ω (T ).
2 − r ϕ
10 ROBLES
The spin representation: examples. Before continuing with the proof, we consider some
examples.
Example 4.9. If r = 2, then dim U = 4, and the weights λ are
C
∈ L
(00), (01), (11), (12).
This forces T = T1+T2. TheT –graded decomposition of U is U U U U .
ϕ ϕ 3/2 1/2 −1/2 −3/2
⊕ ⊕ ⊕
Thus (2.8) and (2.9) hold. Moreover, Tcpt = 0, so that ω (Tcpt) is even; thus, U is real.
r
Example 4.10. If r = 3, then dim U = 8, and the weights are
C
(011) (111) (112)
(000), (001), , .
(012) (122) (123)
This forces T = 2T1 + T2 + T3. For this grading element, ω (T ) = 17, as required by
ϕ r ϕ 2
(4.8), and the eigenvalues 1p p = 1,3,5,7 of T are all multiplicity free. Moreover,
{±2 | } ϕ
Tcpt = 2T1, so that ω (Tcpt)= 1 is odd; thus, U is quaternionic.
r
Example 4.11. If r = 4, then dim U = 16, and the weights are
C
(0111) (1111) (1112)
(0011) (0112) (1122) (1222)
(0000), (0001), , , .
(0012) (0122) (1123) (1223)
(0123) (1233) (1234)
So, in order to have multiplicity-free, T –eigenvalues, we must have n = n = 1, n = 2
ϕ 4 3 2
andn = 4. Thischoice doesindeedgive consecutive eigenvalues 1p p = 1,3,5,...,15 ,
1 {±2 | }
and in particular satisfies ω (T ) = 1(4+2 2+3 1+4 1) = 115, as required by (4.8).
4 ϕ 2 · · · 2
Moreover, Tcpt = 2(T1+T2), so that ω (Tcpt) = 1+2= 3 is odd; thus, U is quaternionic.
r
Example 4.12. If r = 5, then dim U = 32, and the weights are
C
(00000) (00111) (01111) (01112) (11111) (11112) (11122) (11123)
(00001) (00112) (01122) (01222) (11222) (11223) (12222) (12223)
, , , .
(00011) (00122) (01123) (01223) (11233) (12233) (11234) (12333)
(00012) (00123) (01233) (01234) (12234) (12334) (12344) (12345)
In order for T to have the consecutive, multiplicity-free eigenvalues required by (2.9), we
ϕ
must have T = 8T1+4T2+2T3+T4+T5. For this grading element, (2.8) holds. Moreover,
ϕ
Tcpt = 2(T1+T2+T3), so that ω (Tcpt) =1+2+3 is even; thus, U is real.
r
The spin representation: completing the proof. We now return to the proof of Theorem 4.1.
Observe that
2 = (0 0), (0 01), (0 011), (0 012) .
F { ··· ··· ··· ··· }
The T –eigenvalues for these weights are
ϕ
2(T ) = m, m n , m (n +n ), m (n +2n ) .
ϕ r r−1 r r−1 r
F { − − − }
By (4.6), all other weights of the representation satisfy
(4.13) λ(T ) m (n +n +n ) for all λ 2.
ϕ r−2 r−1 r
≤ − 6∈ F