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For larger groups, the first results on the W -decomposition of V T were obtained in the 1970’s by Gutkin [14] and Kostant [21]. Much work has been done since (see section 1.5), but until now the character of V T , for general V ∈ Irr(G), was known on just one non-identity conjugacy class in W , namely the class cox consisting of the Coxeter elements. This result was also obtained by Kostant. He showed that tr(cox,V T ) ∈ {−1, 0, 1} and he gave a formula for the exact value, in terms of the W -action on a certain finite quotient of the character lattice of T .
In this paper we give, for any G, any V ∈ Irr(G) and any w ∈ W a formula for the character tr(w,V T ), in terms of the highest weight of V . For those w sharing certain properties with those of cox, we give a direct generalization of Kostant’s formula for tr(cox,V T ) in terms of the W - action on other finite quotients of the character lattice of T .
To explain our formulas, it is useful to rank the elements of W according to the dimension d(w) of the fixed-point set of w in T . At one extreme, the identity element 1W has d(1W ) = dim T T and Kostant’s partition function gives a formula for tr(1W ,V ), in principle. At the other extreme, d(cox)=0 and we have seen there is a simple formula for tr(cox,V T ). Thus we expect d(w) to measure the complexity of tr(w,V T ) as a function of the highest weight of V . Indeed, our T T T formula for tr(w,V ) interpolates between Kostant’s formulas for tr(1W ,V ) and tr(cox,V ) and involves a weighted partition function Pw of rank equal to d(w).
1.1 Some basic notation
We assume G is connected and all normal subgroups of G are finite. For technical reasons we also assume G is simply connected, even though any V with V T =06 factors through the quotient of G by its center. Let g and t be the complexified Lie algebras of G and T and let R be the set of roots of T in g. Choose a set R+ of positive roots in R, and let ρ be half the sum of the roots in R+. 1 Let P be the additive group indexing the characters of T ; we write eλ : T → S for the character + indexed by λ ∈ P . Let P++ be the set of dominant regular characters of T with respect to R . For µ ∈ P++, let Vµ be the irreducible representation of G with highest weight µ − ρ. (This is consistent with the SU2 example above, and greatly simplifies our formulas.) The sign character of W is denoted by ε.
1.2 Elliptic traces on zero-weight spaces
When d(w)=0 we say w is elliptic. The character of V T on the elliptic set in W has intrinsic meaning: It determines the virtual character of V T modulo linear combinations of induced repre- sentations from proper parabolic subgroups of W [29]. Coxeter elements are elliptic; in SUn there are no others. For all G 6≃ SUn there are non-Coxeter elliptic elements. For example W (E8) has
2 30 elliptic conjugacy classes.
Assume w ∈ W is elliptic. We regard w as a coset of T in the normalizer NG(T ). Because w is elliptic, the elements of w are contained in a single conjugacy class wG in G and we have
T tr(w,Vµ ) = tr(t, Vµ), (1)
G T for any element t ∈ T ∩ w . Therefore tr(w,Vµ ) can be computed from the Weyl Character + + Formula, after cancelling poles arising from the set Rt = {α ∈ R : eα(t)=1}. We obtain ([28] and section 2.1 below) T 1 tr(w,Vµ )= ε(v)evµ(t)Ht(vµ), (2) ∆(t) t vX∈W where hvµ, αˇi ∆= e (1 − e (t)), W t = {v ∈ W : v−1R+ ⊂ R+}, H (vµ)= ρ −α t t hρ , αˇi + + + t α∈RY\Rt αY∈Rt + ∗ and ρt is the half-sum of the roots in Rt . The monomials Ht are W -harmonic polynomials on t . Their geometric meaning is discussed at the end of section 2.1.
Let G be a compact Lie group dual to G. Then P may be regarded as the lattice of one-parameter subgroups of a maximal torus T of G. Let m be the order of w and let µˆ = µ(e2πi/m) ∈ T . Let
CG(tb) and CGb(µ) be the corresponding centralizers. From (2), one gets the following vanishing result. b b b b T b Theorem 1.1 If dim CG(t) < dim CG(µ) then tr(w,Vµ )=0.
For example, if µ ∈ mQ, where m isb the order of w, then µ = 1, so Thm. 1.1 implies that T tr(w,Vµ )=0. Hence if µ ∈ nQ, where n is the least common multiple of the orders of the T elliptic elements in W then Thm. 1.1 implies that the characterb of Vµ is a linear combination of induced characters from proper parabolic subgroups of W .
Though useful, formula (2) does not tell the whole story. For example if w = cox it expresses cox T tr( ,Vµ ) as a sum of |W | terms, but we know from Kostant that cancellations put the actual trace in {−1, 0, 1}. This is because Coxeter elements have an additional property shared by some but not all elliptic elements.
We say that w ∈ W is regular if the subgroup of W generated by w acts freely on R (cf. [35]). For elliptic regular elements we generalize Kostant’s formula as follows.
T Theorem 1.2 Assume w is elliptic and regular. Then tr(w,Vµ )=0 unless there exists v ∈ W such that vµ ∈ ρ + mQ, in which case hµ, αˇi tr(w,V T )= ε(v) , µ hρ, αˇi Y where the product is over the positive coroots αˇ of G for which hµ, αˇi ∈ mZ.
3 Theorem 1.2 shows that the harmonic polynomial in (2) is actually a harmonic monomial when viewed from the dual group G. The key to Thm. 1.2 is that the assumed inequality in Thm. 1.1 holds automatically when (and only when) w is both elliptic and regular. (See [33] and section 3.1 below.) b The dual group G was first used by D. Prasad in [26] to give a new interpretation of Kostant’s cox T ˇ+ result for tr( ,Vµ ). In this case m = h is the Coxeter number, Rµ = ∅ and Thm. 1.2 becomes cox T Kostant’s formulab for tr( ,Vµ ).
At the opposite extreme, if the long element w0 = −1 in W then w0 is an elliptic involution. Applying Thm. 1.2 to w0 gives the following qualitative result (see section ??).
Corollary 1.3 Assume −1 ∈ W but G is not SU2. Then there are only finitely many V ∈ Irr(G) for which V T is an irreducible representation of W .
kn n T For SUn, n ≥ 2, the symmetric powers Sym (C ) , for k =0, 1, 2,... , afford the trivial and sign and sign characters alternately. On the other hand, for SU4 the two-dimensional representation of T W is all of Vµ for just one µ (see section 5.2). Cor. 1.3 can be sharpened to classify irreducible zero weight spaces of other groups. See section ?? which includes some history of this problem.
1.3 A general character formula for zero weight spaces
◦ Now take any element w ∈ W . Let S =(Tw) be the identity component of Tw and let L = CG(S) be the centralizer of the torus S. There is an element t ∈ T which is L-conjugate to an element of the coset w. We fix such a t and consider the coset tS ⊂ T .
+ + tS tS The positive roots are partitioned as R = RtS ⊔ R1 ⊔ R2 , where + + RtS = {α ∈ R : eα ≡ 1 on tS}, tS + R1 = {α ∈ R : eα ≡ eα(t) =16 on tS}, tS + R2 = {α ∈ R : eα is nonconstant on tS}. As in the elliptic case, we set 1 ρ = α ∆= e (1 − e ) tS 2 ρ −α α∈R+ α∈RtS XtS Y1 hλ, αˇi tS −1 + + HtS(λ)= W = {v ∈ W : v RtS ⊂ R }. hρtS, αˇi α∈R+ YtS tS The set R2 = {β1,...,βr} determines a weighted partition function Pw as follows. Let Y be the character lattice of S. For each i =1,...,r, the restriction of eβi to S is a non-trivial character eνi for some νi ∈ Y . Let zi = e−βi (t). Now for ν ∈ Y , let
n1 n2 nr Pw(ν)= z1 z2 ··· zr X 4 r where the sum runs over all r-tuples (n1,...,nr) of nonnegative integers such that niνi = ν. i=1 P Theorem 1.4 With notation as above, we have
T 1 tr(w,Vµ )= ε(v)evµ(t)Pw(vµ − ρ)HtS(vµ). (3) ∆(t) tS v∈XW
This is proved similarly to (2), but now we take the constant term along S of the restriction of the character of Vµ to the coset tS. See section 4.
T If w = 1W , formula (3) becomes Kostant’s formula for dim Vµ . Some examples of intermediate partition functions Pw are found in section ??.
1.4 Explicit results
For non-elliptic w, the formula (3) is difficult to compute explicitly as a function of µ. The dif- ficulty, measured by d(w) = dim S, is in the weighted partition function Pw. The most difficult T case is dim Vµ , which is known explicitly only for small groups. For general G, it was shown in T [22] that the function µ 7→ dim Vµ is a piecewise polynomial function, but there are no explicit T formulas for these polynomials. For w =6 1, the function µ 7→ tr(w,Vµ ) also appears to be a piecewise polynomial function.
T In section ?? we give explicit formulas for tr(w,Vµ ) for all w in the groups of rank two and also T SU4; these are the groups for which explicit polynomial formulas for dim(Vµ ) are known (to me).
T For Spin8 and F4 we carry out the idea of 1.3 above find all µ for which Vµ is irreducible. These turn out to be just the known examples, coming from small representations [30].
T For E7 and E8 the same idea, with more computation, should also find all irreducible Vµ , but we do not address this here.
T For E6, −1 ∈/ W . Instead we explicitly compute tr(w,Vµ ) for the elliptic triality w, using Thm. T 1.2. When tr(w,Vµ ) =06 the method of Cor. 1.3 applies. We find that the only possibilities for an T irreducible representation of W to be a zero weight space are the five known ones (if tr(w,Vµ ) =6 T 0) and two possible additions (if tr(w,Vµ )=0). See section 6.3.
1.5 Earlier work
Zero-weight spaces have been much-studied in the past half-century. A recent survey of the prob- lem is given in [15].
For classical groups, [1], [2], [3], [12], [14] use Schur-Weyl duality express the decomposition of T Vµ in terms of induced representations of symmetric groups. For a fixed classical group, say G =
5 SU3, this method involves multiplicities in induced representations of arbitrarily large symmetric groups.
For G = G2, [23] gives a computer algorithm, but not a formula, for explicitly computing the T character of Vµ for any given µ.
T For “small" Vµ, the decomposition of Vµ is given explicitly in [14] (for SLn), [30] (for types Dn and En) and [31] (for types Bn,Cn,G2, F4).
1.6 Organization of the paper
Section 2 is purely about the Weyl character formula on torsion elements of G. These results are applied to elliptic traces in section 3. In section 4.1 we return to the Weyl character formula, now to study its values on cosets of subtori in T . These results are applied to the traces of general T elements w ∈ W in section 4.4. In section 5 we give explicit formulas for the character of Vµ , for small groups. The final section 6 concerns irreducible zero weight spaces.
Contents
1 Introduction and statement of results 1
1.1 Somebasicnotation...... 2
1.2 Elliptictracesonzero-weightspaces ...... 2
1.3 A general character formula for zero weight spaces ...... 4
1.4 Explicitresults ...... 5
1.5 Earlierwork...... 5
1.6 Organizationofthepaper ...... 6
2 The Weyl character formula and torsion elements 7
2.1 The Weyl character formula and harmonic polynomials ...... 7
2.2 Traces of torsion elements in G ...... 8
2.3 Principalweights ...... 9
2.4 Dualgroupsandtorsionelements...... 11
3 Elliptic traces on zero weight spaces 12
6 3.1 Ellipticregulartraces ...... 13
4 A general character formula for zero weight spaces 14
4.1 TheWeylCharacterFormulaandsubtori...... 15
4.2 Charactersandconstantterms ...... 17
4.3 FormalCharacters...... 18
4.4 Thecharacterformula...... 20
4.4.1 Aspecialcase...... 21
5 Small Groups 21
5.1 Rank-TwoGroups...... 21
5.1.1 SU3 ...... 21
5.1.2 Sp4 ...... 23
5.1.3 G2 ...... 25
5.2 SU4 ...... 27
6 Irreducible zero weight spaces 30
6.1 Spin8 ...... 31
6.2 F4 ...... 32
6.3 E6 ...... 34
2 The Weyl character formula and torsion elements
This section is purely about the Weyl Character formula. Notation is that of section 1.2.
2.1 The Weyl character formula and harmonic polynomials
It is known, from [28] or as a special case of Thm. 4.5 below, that for µ ∈ P++ we have 1 tr(t, Vµ)= ε(v)evµ(t)Ht(vµ). (4) ∆(t) t vX∈W
7 v The function µ 7→ Ht(µ) and its W -translates Ht (µ) := Ht(vµ) belong to the vector space H ∗ v t of W -harmonic polynomials on t . In [24] Macdonald showed that {Ht : v ∈ W } spans an irreducible representation of W , now called the truncated induction of the sign character of Wt to W . Here Wt is the Weyl group of the centralizer CG(t). By a theorem of Borel (see [27, section 5] for a simple proof), the graded vector space H is v canonically isomorphic to the homology of the flag manifold B = G/T . The translates Ht , for v ∈ W t, correspond to the fundamental classes of the connected components of the fixed-point submanifold Bt ⊂ B. In this interpretation, equation (4), combined with the Borel-Weil theorem, is a special case of the Atiyah-Segal fixed-point theorem [4, Theorem (3.3)]. However the proof of (4) given in [28] or Thm. 4.5 below, is elementary; it boils down to the Weyl dimension formula for CG(t).
2.2 Traces of torsion elements in G
In this section we study the values of (4) on torsion elements of G.
Proposition 2.1 Let m be a positive integer. Suppose t ∈ T has Ad(t) of order m, and let y be a coset of mQ in P .
+ ˇ+ (a) If |Rt | < |Ry |, then tr(t, Vµ)=0 for all µ ∈ y ∩ P++.
+ ˇ+ (b) If |Rt | = |Ry | then there is a constant Ct,y ∈ C such that
tr(t, Vµ)= Ct,y hµ, αˇi, ˇ+ αˇY∈Ry
for all µ ∈ y ∩ P++
Proof: For any v ∈ W , the function µ 7→ evµ(t) is constant on each coset of mQ in P . Hence, for each coset y ∈ P/mQ, we can define a function τy : W → C by τy(v)= evµ(t) for any µ ∈ y. We set 1 K = ε(v)τ (v)Hv, y |W | y t t v∈W X v ∗ + where Ht (µ)= Ht(vµ). Thus Ky is a harmonic polynomial on t , of degree |Rt | and depending only on the coset y ∈ P/mQ. From equation (4) we have 1 tr(t, V )= · K (µ), for all µ ∈ y. (5) µ ∆(t) y One checks that τy(vu)= τy(v) for all u ∈ Wy. (6)
8 Equation (6) implies that under the action of W on H we have
u Ky = ε(u)Ky, for all u ∈ Wy.
It follows that Ky is divisible by the polynomial
α.ˇ (7) ˇ+ αˇY∈Ry + ˇ+ + ˇ+ Hence if |Rt | < |Ry | we have Ky = 0 and if |Rt | = |Ry | then Ky is a constant multiple of the polynomial (7). This proves the Proposition.
2.3 Principal weights
We continue with t ∈ T having Ad(t) of order m. Now we consider certain families of weights.
An weight µ ∈ P is m-principal if the W -orbit of µ meets ρ + mQ. If y ∈ P/mQ is the coset of an m-principal weight we can compute the constant Ct,y appearing in Proposition 2.1, as follows.
Lemma 2.2 Assume y = vρ + mQ for some v ∈ W . Then we have
Ky(vρ)= ε(v)∆(t).
Proof: One checks that for any v ∈ W we have
Kvρ+mQ(vρ)= ε(v)Kρ+mQ(ρ), so we may assume v =1.
From Weyl’s identity (cf. (21) below) applied to Wt it follows that
∆(t)= ε(v)evρ(t)H(vρ)= Kρ+mQ(ρ), t vX∈W as desired.
Set ˇ+ ˇ+ Rm = {αˇ ∈ R : hρ, αˇi = m}
Proposition 2.3 Let t ∈ T have Ad(t) of order m. Assume that
+ ˇ+ (a) |Rt | = |Rm|, and
(b) µ ∈ P++ is m-principal.
9 Then hvµ, αˇi tr(t, V )= ε(v) , (8) µ hρ, αˇi ˇ+ αˇY∈Rm where v is any element of W for which vµ ∈ ρ + mQ.
Proof: From equation (5) we have
K (µ) tr(t, V )= y . µ ∆(t)
ˇ+ ˇ+ For each α ∈ Ry let ǫα = ±1 be such that ǫα · vα ∈ R . Sending α 7→ ǫα · vα is a bijection ˇ+ ˇ+ + ˇ+ Ry → Rm, so we have |Rt | = |Ry |. Applying Lemma 2.2 and equation (7), we get
−1 ε(v)∆(t)= Ky(vρ)= Ct,y hv ρ, αˇi = Ct,y hρ, vαˇi, ˇ+ ˇ+ αˇY∈Ry αˇY∈Ry so ε(v)∆(t) C = . t,y hρ, vαˇi αˇ∈Rˇy Q It follows that hµ, αˇi hvµ,ǫ · vαˇi hvµ, αˇi tr(t, V )= ε(v) = ε(v) α = ε(v) . µ hρ, vαˇi hρ, ǫ · vαˇi hρ, αˇi ˇ+ ˇ+ α ˇ+ αY∈Ry αY∈Ry αY∈Rm
Remark. From Lemma 2.5 below, the stabilizer of ρ + mQ in P/mQ is the reflection subgroup + Wm = hrα :α ˇ ∈ Rmi. Since the polynomial
αˇ + αˇY∈Rm transforms under the sign character of Wm, it follows that the right side of (8) is independent of the choice of v, as it must be.
The set Tm := {t ∈ T : Ad(t) has order m and |Rt| = |Rˇm|} may consist of several W -orbits. These must be separated by characters of G. However, Prop. 2.3 implies that if µ ∈ P++ is m-principal then the character of Vµ is constant on Tm.
+ Certain elements of Tm are obtained as follows. Let 2ˇρ be the sum of the coroots αˇ ∈ Rˇ . If 1 η ∈ S has order 2m, then the element t = 2ˇρ(η) belongs to Tm. For such elements, Prop. 2.3 gives the following character value.
10 Corollary 2.4 Let m> 1 be an integer. Assume that µ + mQ is m-principal and that t = 2ˇρ(η), where η ∈ S1 has order 2m. Then we have hvµ, αˇi tr(t, V )= ε(v) , µ hρ, αˇi ˇ+ αˇY∈Rm where vµ ∈ ρ + mQ.
2.4 Dual groups and torsion elements
It will be helpful to study the W -action on P/mQ in terms of the dual group of G.
The simply connected group G has root datum D(G)=(P,R, Q,ˇ Rˇ). Let G be a simple compact Lie group with dual root datum D(G)=(Q,ˇ R,P,Rˇ ). b We fix a maximal torus T in G and identify P with the lattice of one-parameter subgroups of T . b Let π : Gsc → G be the simplyb b connected covering map. The group Gsc has root datum b D(G )=(P,ˇ R,ˇ Q, R), b b sc b where Pˇ is the lattice of one-parameter subgroups in a maximal torus of the adjoint group of G. b The kernel of π is the center Zsc of Gsc and we have a canonical isomorphism Zsc ≃ P/Q. The −1 1 preimage torus Tsc = π (T ) may be identified with Q ⊗ S and the exact sequence b b b 1 −→ Z −→ G −→π G −→ 1 (9) b b sc sc restricts to a W -equivariant exact sequence on maximal tori: b b b π 1 −→ Zsc −→ Tsc −→ T −→ 1. (10)
Now let ζ ∈ S1 have finite order m. Evaluationb b at ζ givesb an isomorphism from P/mP to the m −1 torsion subgroup T [m] := {τ ∈ T : τ = 1}. The preimage Tsc[m] := π (T [m]) fits into the exact sequence −1 π b 1 −→ Zbsc −→ π (T [m]) −→ T [m]b−→ 1. b (11) This sequence is W -equivariantly isomorphic to the canonical exact sequence b b b 0 −→ P/Q −→m P/mQ −→ P/mP −→ 0. (12)
For µ ∈ P , let Wµ = Wy be the stabilizer of the coset y = µ + mQ under the W -action on P/mQ, ˇ+ ˇ+ and let Rµ = {αˇ ∈ R : hµ, αˇi ∈ mZ}. This set of roots depends only on the coset y = µ + mQ. We sometimes write ˇ+ ˇ+ Ry = Rµ .
× ˇ+ Let ζ ∈ C have order m as above. Then Rµ is a set of positive roots for he connected centralizer C◦ (µ(ζ)) and W is the Weyl group of C◦ (µ(ζ)). Gb µ Gb
11 ˇ+ Lemma 2.5 The group Wµ is generated by {rα :α ˇ ∈ Rµ }.
−1 Proof: This follows from the isomorphism P/mQ ≃ π (T [m]) and the fact that Gsc is simply connected. b b
3 Elliptic traces on zero weight spaces
T We apply the results of the previous section to compute tr(w,Vµ ) when w ∈ W is elliptic. Ellip- ticity of w is equivalent to any of the following: Tw is finite; w fixes no nonzero element of P ; all elements of the coset w ⊂ N are T -conjugate; w itself is contained in a single G-conjugacy class. The unique G-conjugacy class containing w is denoted by wG.
One consequence of ellipticity is the following.
Lemma 3.1 Assume w ∈ W is elliptic. Let t ∈ T ∩ wG and let z belong to the center Z of G. Then there exists v ∈ W such that tv = tz.
Proof: Since t ∈ wG, there exist g ∈ G and n ∈ w such that t = gng−1. Since z ∈ Z we also have zt = gzng−1. But zn ∈ w and w is elliptic so there exists s ∈ T such that zn = sns−1. Hence zt = gsns−1g−1 = hth−1, where h = gsg−1, so zt and t are G-conjugate elements of T . It follows that zt and t are W -conjugate, as claimed.
Another consequence is that, since an elliptic element w ∈ W fixes no nonzero weight in P , we have T G tr(w,Vµ ) = tr(w ,Vµ). From equation (4) it follows that
T 1 tr(w,Vµ )= ε(v)evµ(t)H(vµ). (13) ∆(t) 1 vX∈W for any t ∈ T ∩ wG.
T By [35, Theorem 8.5], the character values tr(w,Vµ ) are rational for all dominant regular weights µ. From Proposition 2.1 we obtain
Proposition 3.2 Assume w ∈ W is elliptic. Let t ∈ T ∩ wG have Ad(t) of order m and let y ∈ P/mQ.
ˇ T (a) If |Rt| < |Ry| then tr(w,Vµ )=0 for all µ ∈ y ∩ P++.
(b) If |Rt| = |Rˇy| then there is a constant Ct,y ∈ Q such that for all µ ∈ y ∩ P++ we have
T tr(w,Vµ )= Ct,y · hµ, αˇi. ˇ αˇY∈Ry 12 3.1 Elliptic regular traces
We now assume w ∈ W is elliptic and regular, and will prove Thm. 1.2 of the introduction.
Let G be the dual group of G, as in section 2.4. We need the following consequence of [33]: Let g ∈ G have order m. Then b |Rˇ| dim C b(g) ≥ , (14) b b G m 2πi/m with equality if and only if g is G-conjugate to ρb(e ).
b Proposition 3.3 Assume wb∈ W is elliptic and regular of order m and let t ∈ T ∩ wG. Then + ˇ+ Ad(t) has order m and for all y ∈ P/mQ we have |Rt |≤|Ry |, with equality if and only if y is m-principal.
Proof: It suffices to prove the result for one t ∈ T ∩ wG. Let t = 2ˇρ(η), where η ∈ C× has order 2m. Then Ad(t) has order m and from [34, Prop.8] we have t ∈ wG.
Since Ad(t) has order m it follows that Ad(g) has order m for every g ∈ wG. Let n ∈ w. Since w is elliptic regular and Ad(n) has order m, the group hni freely permutes the root spaces of T in g and have no nonzero invariants in t. Therefore we have |R| dim C (t) = dim C (n)= . G G m
Let ζ ∈ S1 have order m. Given any ν ∈ P , the element ν(ζ) ∈ T [m] depends only on the coset y = ν + mP ∈ P/mP . Asin(12), we have a W -equivariant isomorphism b P/mP → T [m], y 7→ yˆ = ν(ζ), for any ν ∈ y.
b Now take y ∈ P/mQ and let k be the order of y + mP in the group P/mP . Clearly k divides m. Applying (14) gives the first inequality in the following.
|Rˇ| |Rˇ| |R| dim C b(ˆy) ≥ ≥ = = dim C (t). (15) G k m m G
Since G and G have the same rank, we obtain the inequality |Rt|≤|Rˇy|.
b G If y = vρ + mQb for some v ∈ W , then applying [34, Prop. 8] to G shows that yˆ ∈ w . Since w is ˇ ˇ elliptic regular of order m, we have dim CGb(ˆy)= |R|/m. Now (15) implies that |Rt| = |Ry|. b Conversely, if |Rt| = |Rˇy| then both inequalities in (15) become equalities. Thus, k = m and b G b ˇ dim CG(ˆy) = |R|/m. The condition for equality in (14) implies that yˆ ∈ w . It follows that yˆ is W -conjugate to ρ(ζ), which means the image y′ of y in P/mP is W -conjugate to ρ + mP . We may therefore assume y′ = ρ + mP and it remains to prove that y is W -conjugate to ρ + mQ.
13 In the exact sequence (11)
−1 π 1 −→ Zsc −→ π (T [m]) −→ T [m] −→ 1,
v −1 the stabilizer Wyˆ = {v ∈ W :y ˆ b=y ˆ} acts onb the fiber π b (ˆy). Since w is elliptic, this action of −1 Wyˆ on π (ˆy) is transitive, by Lemma 3.1 applied to the group Gsc. Passing to the exact sequence (12) m 0 −→ P/Q −→ P/mQ −→ P/mP b−→ 0, (16) this means that Wyˆ acts transitively on the fiber in P/mQ above ρ + mP . Since y and ρ + mQ belong to this fiber, the proof is complete.
Combining Cor. 2.4, Prop. 3.2 (a) and Prop. 3.3 we have proved Theorem 1.2 in the introduction, restated here.
Theorem 3.4 Let w ∈ W be elliptic and regular of order m and let y ∈ P/mQ.
1. If y is not in the W -orbit of ρ + mQ then for all µ ∈ y ∩ P++ we have
T tr(w,Vµ )=0.
2. If vy = ρ + mQ for some v ∈ W then for all µ ∈ y ∩ P++ we have hvµ, αˇi tr(w,V T )= ε(v) . (17) µ hρ, αˇi ˇ+ αˇY∈Rm
Remark. We have observed that the polynomial + transforms by the sign character under αˇ∈Rˇm αˇ the reflection subgroup W = hr :α ˇ ∈ Rˇ+ i, so that the product m α m Q ε(v) hvµ, αˇi ˇ+ αˇY∈Rm is independent of the choice of v ∈ W for which vµ ∈ ρ + mQ. In fact there is a unique such v −1 ˇ+ ˇ+ for which v Rm ⊂ R . With this choice, (17) becomes hµ, αˇi tr(w,V T )= ε(v) . (18) µ hρ, αˇi ˇ+ αˇY∈Rµ ˇ+ ˇ+ where Rµ = {αˇ ∈ R : hµ, αˇi ∈ mZ}. Version (18) has the computational advantage that each term in the product is positive.
4 A general character formula for zero weight spaces
T The general formula for tr(w,Vµ ) interpolates between the cases where dim Tw =0 and Tw = T .
14 4.1 The Weyl Character Formula and subtori
We return to the Weyl Character Formula, now to analyze its restriction to cosets of subtori in T . At the moment there are no further specifications on these cosets.
Fix an element t ∈ T and a subtorus S ⊂ T . We recall more notation from the introduction: The set of positive roots R+ is partitioned as
+ + tS tS R = RtS ⊔ R1 ⊔ R2 , where
+ + RtS = {α ∈ R : eα ≡ 1 on tS}, tS + R1 = {α ∈ R : eα ≡ eα(t) =16 on tS}, tS + R2 = {α ∈ R : eα is nonconstant on tS}. Also, we set
1 hλ, αˇi tS −1 + + ρtS = α, HtS (λ)= , W = {v ∈ W : v RtS ⊂ R } 2 hρtS, αˇi α∈R+ α∈R+ XtS YtS
tS ∆= eρ (1 − e−α), S0 = {s ∈ S : eα(ts) =16 ∀α ∈ R2 }. tS αY∈R1 + tS Let WtS be the subgroup of W generated by {rα : α ∈ RtS}. The product mapping WtS ×W → W is a bijection.
Lemma 4.1 Let tS be a coset of the subtorus S ⊂ T . For all µ ∈ P++ and s ∈ S0 we have
ε(v) · HtS(vµ) evµ−ρ(ts) tr(ts,Vµ)= · . tS (1 − e−α(t)) (1 − e−β(ts)) v∈W tS tS X α∈R1 β∈R2 Q Q
′ 1 Proof: Let T be the covering torus of T with character group 2 P (cf. [6, VI.3.3]).
′ 1 ′ 1 We again write eλ : T → S for the character of T corresponding to λ ∈ 2 P .
′ 1 ′ The Weyl group W acts on T via its action on 2 P , and W acts on the character ring C[T ] as w ′ (w · f)(t)= f(t ). Let A and AtS be the operators on C[T ] given by
A(f)= ε(w)w · f, AtS(f)= ε(w)w · f. wX∈W wX∈WtS Then A(f)= ε(uv)(uv · f)= ε(v)AtS(v · f). tS u∈W tS v∈XW XtS v∈XW
15 In particular for µ ∈ P we have
A(eµ)= ε(v)AtS(evµ). (19) tS v∈XW
′ When restricted to T , the character χµ of Vµ may be regarded as a function on T , via the covering map T ′ → T . It is given by the traditional expression of Weyl’s Character Formula:
A(eµ) χµ = . (20) A(eρ) On the right side of (20) the numerator and denominator are functions on T ′ whose zeros must be cancelled in order to evaluate χµ.
From Weyl’s identity (applied to both W and WtS), we have
tS tS A(eρ)= D · DtS = D · AtS(eρtS ), (21) where
tS DtS = eα/2 − e−α/2 ,D = eα/2 − e−α/2 , eρtS = eα/2 α∈R+ α∈RtS ⊔RtS α∈R+ YtS Y1 2 YtS are all elements of C[T ′].
′ ′ Recall t ∈ T has been fixed. Now we also fix s ∈ S0 and let x ∈ T be a lift of ts. Also let z ∈ T tS 1 be arbitrary. For all u ∈ WtS, v ∈ W and µ ∈ 2 P we have
euvµ(xz)= evµ(x) · euvµ(z). It follows that AtS (evµ)(xz)= evµ(x) · AtS (evµ)(z). From (21) we have tS A(eρ)(xz)= D (xz) · DtS(xz). + For all α ∈ RtS we have eα(st)=1, so eα/2(x)= ±1. It follows that eρtS (x)= ±1 and that
DtS(xz)= eρtS (x) · DtS(z)= eρtS (x)AtS (eρtS )(z), so tS A(eρ)(xz)= eρtS (x) · D (xz) · AtS (eρtS )(z). (22)
So far x is fixed but we have made no restriction on z. Since the set of elements in T ′ on which no ′ ′ root = 1 is dense in T , we can choose a sequence zn → 1 in T such that eα(xzn) =6 1 for all n and all α ∈ R.
From (20) and (22) we have that
1 AtS(evµ) χµ(xzn)= tS ε(v)evµ(x) (zn). eρtS (x) · D (xzn) tS AtS(eρtS ) v∈XW 16 Letting n → ∞ we get, from the Weyl dimension formula applied to the (connected) centralizer CL(t), where L = CG(S), that 1 χµ(x)= tS ε(v)HtS(vµ)evµ(x). eρtS (x) · D (x) tS v∈XW Since tS eρtS · D = eρ · (1 − e−α), tS tS α∈RY1 ⊔R2 it follows that ε(v)HtS(vµ) χµ(x)= evµ−ρ(x). tS (1 − e−α(x)) v∈W tS tS X α∈R1 ⊔R2 Q Since µ and all vµ − ρ lie in P we may replace x by its projection st ∈ T . Decomposing the tS tS product according to R1 ⊔ R2 we obtain the formula in Lemma 4.1 .
4.2 Characters and constant terms
◦ Given w ∈ W , let S = (Tw) be the identity component of the fixed-point group Tw := {t ∈ T : tw = t}, and let Y be the character lattice of S.
The rank of Y equals the rank of the character lattice Pw = {λ ∈ P : wλ = λ} of the quotient torus T/[w, T ], where [T,w]= {tw · t−1 : t ∈ T }.
For s ∈ S ∩ [T,w] one checks that sm = 1, where m is the order of w. This implies that the composite mapping (S ֒→ T −→ T/[T,w] (23 has finite kernel, hence is surjective. It follows that the restriction map P → Y is injective on Pw. Recall that we regard w as a subset of N. Fix n ∈ w. From the surjectivity of (23) it follows that every element of w is T -conjugate to an element of nS. Indeed, given any t ∈ T , there exist s ∈ S and z ∈ T such that s =(zw · z−1) · t, so we have
z(nt)z−1 = n(zw · z−1)t = ns.
Let C[S] be the character ring of S. Every element ϕ ∈ C[S] uniquely expressed as a finite linear combination ϕ = ϕˆ(ν)eν Xν∈Y C C C with coefficients ϕˆ(ν) ∈ . The constant-term functional S : [S] → is given by R ϕ(s) ds =ϕ ˆ(0). ZS
17 Lemma 4.2 Let V be a finite-dimensional representation of G and let w ∈ W . Then the trace of w on the zero weight space V T is given by
tr(w,V T )= tr(ns,V ) ds,
ZS for any choice of n ∈ w.
λ λ wλ Proof: For λ ∈ P , let V = {v ∈ V : tv = eλ(t)v ∀t ∈ T }. We have nV = V . It follows that for all s ∈ S we have
λ λ tr(ns,V )= tr(ns,V )= tr(n, V ) · eλ(s). λX∈Pw λX∈Pw Since the restriction map P → Y is injective on Pw it follows that
tr(ns,V ) ds = tr(n, V T ) = tr(w,V T ).
ZS
g Since n ∈ CG(S), there exists g ∈ CG(S) such that n ∈ T . We set t := ng ∈ T.
Since g centralizes S, we have (ns)g = ts for all s ∈ S. It follows that
tr(ns,V ) = tr(ts,V ), as functions on S. From Lemma 4.2 we have
◦ Proposition 4.3 Let w ∈ W and let S = (Tw) be the connected fixed-point subtorus of w in T . Then there exists t ∈ T which is CG(S)-conjugate to an element of w and for any such t the trace of w on the zero weight space V T is given by
tr(w,V T )= tr(ts,V ) ds.
ZS
4.3 Formal Characters
In this section we extend the functional S to certain rational functions on S, using the theory of formal characters [15, 22.5]. R Let S be a compact (or algebraic) torus with character lattice Y and coordinate algebra C[S]. The Fourier expansion
ϕ = ϕˆ(ν)eν , for ϕ ∈ C[S] Xν∈Y 18 defines a C-algebra isomorphism (Fourier transform) ϕ 7→ ϕˆ from C[S] to the group algebra C[Y ], where C[S] has the pointwise product and C[Y ] has the convolution product: φ ∗ ψ(ν)= φ(λ)ψ(µ). (24) λ+Xµ=ν C C Via this isomorphism we identify the constant term functional S : [S] → with the evaluation functional φ 7→ φ(0) : C[Y ] → C. R
Let us be given nonzero elements ν1,...,νr ∈ Y , not necessarily distinct. Let Γ ⊂ Y be the semigroup generated by {ν1,...,νr}. We let C((Y )) be the set of functions φ : Y → C supported on finitely many sets of the form λ − Γ for some λ ∈ Y . The convolution product (24) extends to C((Y )) and C[Y ] is a subalgebra of
C((Y )). We extend the functional S to C((Y )) by setting
R φ = φ(0). ZS For any complex number z and i =1,...,r the Fourier transform of 1 − ze−νi becomes invertible in C((Y )); its inverse is the function whose support is contained in −Nνi and whose value at −nνi, for n ∈ N, is zn. More generally, for r
h = (1 − zie−νi ) ∈ C[S] (25) i=1 Y the convolution-inverse of hˆ given by the weighted partition function
ˆ−1 n1 n2 nr h (−ν)= P(ν)= z1 z2 ··· zr (26)
r r X −1 where the sum runs over all (n1,...,nr) ∈ N such that niνi = ν. Note that hˆ is supported i=1 ˆ−1 on −Γ and that (ˆeν ∗ h )(0) = P(ν). P
Lemma 4.4 Let h ∈ C[S] have the form (25) and suppose f,g ∈ C[S] are such that h · f = g. Then f(s) ds = gˆ(ν)P(ν), S Z Xν∈Y where P(ν) is given by (26).
Proof: We have gˆ = f · h = fˆ∗ hˆ in C[Y ] hence also in C((Y )). Since hˆ is invertible in C((Y )) it follows that −1 −1 d fˆ =g ˆ ∗ hˆ = gˆ(ν)ˆeν ∗ hˆ , Xν∈Y so we have ˆ ˆ−1 f(s) ds = f(0) = gˆ(ν)(ˆeν ∗ h )(0) = gˆ(ν)P (ν). S ν ν Z X X
19 4.4 The character formula
We apply the results (and notation) of Section 4.1 to the situation of 4.2.
tS Enumerate R2 = {β1,...,βr}, let νi ∈ Y be the restriction of βi to S, and let zi = e−βi (t). Asin (26), for ν ∈ Y we set n1 n2 nr Pw(ν)= z1 z2 ··· zr r Xr where the sum runs over all (n1,...,nr) ∈ N such that niνi = ν. If there are no such r-tuples i=1 (ni) then Pw(ν)=0. P
T Theorem 4.5 The trace of w ∈ W on the zero weight space Vµ is given by
T 1 tr(w,Vµ )= ε(v)HtS(vµ)evµ−ρ(t)Pw(vµ − ρ), (27) (1 − e−α(t)) tS tS v∈W α∈R1 X Q ◦ where S =(Tw) and t ∈ T is CG(S)-conjugate to an element of w (see Proposition 4.3).
Proof: From Proposition 4.3 we have
T tr(w,Vµ )= tr(ts,Vµ) ds. ZS From Lemma 4.1 we have on S0 the relation
ε(v) · HtS(vµ) evµ−ρ(ts) tr(ts,Vµ)= · . tS (1 − e−α(t)) (1 − e−β(ts)) v∈W tS tS X α∈R1 β∈R2 Q Q For v ∈ W tS we set HtS(vµ)evµ−ρ(t) cv = ε(v) · . (1 − e−α(t)) tS α∈R1 Q In Prop. 4.4 we take f,g,h ∈ C[S] defined by
f(s) = tr(ts,Vµ), g(s)= cvevµ−ρ(s), h(s)= (1 − e−β(ts)). tS tS v∈XW βY∈R2
Since f · h = g, we have from Lemma 4.4 that
T ˆ tr(w,Vµ )= tr(ts,Vµ) ds = f(0) ( Prop. 4.3) ZS −1 −1 =g ˆ ∗ hˆ (0) = cveˆvµ−ρ ∗ hˆ (0) = cvPˆw(vµ − ρ). tS tS v∈XW v∈XW Theorem 1.4 is proved.
20 4.4.1 A special case
+ + Let J be a subset of the simple roots in R , let RJ be the positive roots spanned by J and WJ = + ◦ hrα : α ∈ RJ i. Suppose w is a Coxeter element in WJ . Then S =(∩α∈J ker eα) . Let GJ be the derived group of CG(S). We may choose n ∈ w ∩ GJ , and let t be a GJ -conjugate of n. Then t is a regular element of GJ , so we have + tS + tS + + RtS = ∅, R1 = RJ , R2 = R − RJ , HtS ≡ 1. (28) In this situation (27) becomes
T 1 tr(w,Vµ )= ε(v)evµ−ρ(t)Pw(vµ − ρ). (29) (1 − e−α(t)) + v∈W α∈RJ X Q
5 SmallGroups
T In this section we work out the complete character formula for tr(w,Vµ ) in the cases where Kostant’s partition function has a closed quasi-polynomial expression (that I am aware of). This means G is one of SU3, Sp4, G2, SU4.
We use the following notation. For a ∈ Z, let (a)2 = 0 if a is even, (a)2 = 1 if a is odd. For n ∈{3, 4, 6} let 1 if a ≡ 1 mod n (a)n = −1 if a ≡ −1 mod n 0 if gcd(a, n) > 1.
5.1 Rank-Two Groups
For the rank-two groups, let α and β be simple roots, where α is a short root for Sp4 and is a long root for G2. α0 is the highest root and β0 is the highest short root. Let ωα and ωβ be the corresponding fundamental weights, so that ρ = ωα + ωβ. For µ ∈ P++, we have
µ = aωα + bωβ (30) T where a, b are positive integers. The character of W afforded by Vµ will be expressed in terms of quasi-polynomial functions of (a, b).
Recall that N = {0, 1, 2, 3,... }.
5.1.1 SU3
T Here Vµ =6 0 if and only if a ≡ b mod 3. Since V and its dual have isomorphic zero weight spaces, we may assume that a ≤ b. The conjugacy classes in W = S3 are represented by 1=1W ,
21 a reflection r, and a Coxeter element cox. We find
T dim Vµ = a T b+1 tr(r, Vµ )=(a)2 · (−1) cox T tr( ,Vµ )=(a)3
The computations are summarized as follows. Kostant’s partition function P is given by P(mα + T nβ) = 1 + min(n, m), for n, m ∈ N. From this one easily computes the above value for dim Vµ . cox T 2 The value for tr( ,Vµ ) is obtained from Kostant’s formula. Setting ζ = η , we get
ζa − ζ−a tr(cox,V T ) = tr(ˇρ(ζ),V )= =(a) . µ µ ζ − ζ−1 3
ˇ 1 ˇ −2 For r = rα, we have S = λ(S ), where λ(z) = diag(z,z,z ), and we may take t =ρ ˇ(−1) = diag(−1, 1, −1). We have (cf. 28)
+ tS tS RtS = ∅, R1 = {α}, R2 = {β, ρ}.
We have Y = Zν, where eν(λˇ(z)) = z, z1 = β(t)= −1, z2 = ρ(t)=1, ν1 = ν2 =3ν and
1 − (n/3)2 if n ∈ 3N Pr(nν)= (0 if n∈ / 3N From 29 we get 1 tr(r, V T )= ε(v)e (t)P (vµ − ρ) µ 2 vµ r v∈W X 1 a +2b − 3 a +2b − 3 = (−1)a+b 1 − − (−1)b 1 − 2 3 3 2 2 a +2b =(−1)b+1(a) =(−1)b+1(a) . 2 3 2 2 T Using the representation theory of SU2, one can also obtain tr(r, Vµ ) by calculating multiplicities of weights kα for k ∈ N. For SU3 this is feasible because the partition function P is so simple. Remarks:
T T 1) Vµ is a multiple of the regular representation Reg of W if and only if a ∈ 6Z, in which case Vµ is a/6 copies of Reg.
T 3d 3 ∗ 2) Vµ is irreducible if and only if a ∈ 1, 2. If a =1, then Vµ = Sym (C ) where b =1+3d, and T d T T we have Vµ ≃ ε . If a = 2 then Vµ ≃ ̺. Thus, each irrep of W is of the form Vµ for infinitely many µ ∈ P++.
22 5.1.2 Sp4
In this section we explicate our character formula for G = Sp4. The simple roots are α, β where α T 1 is short. Writing µ = aωα + bωβ, we have Vµ =06 iff a is odd. For η ∈ S of order eight, let
ηa − η−a 1 if a ≡ 1, 3 mod8 (a)8 = −1 = η − η (−1 if a ≡ 5, 7 mod8. The conjugacy classes in W are represented by
2 1W , rα, rβ, cox , cox.
T For µ = aωα + bωβ the values of tr(w,Vµ ) are shown in the following table.
T w tr(w,Vµ )
1 1 (ab +(b) ) W 2 2 (−1)b+1 r · [b +(a) (b) ] α 2 4 2
(a)4 if b ∈ 2+4N rβ (b)4 if a + b ∈ 2+4N 0 otherwise (a) cox2 4 [b + a(b) ] 2 2
−(a)8(a)4 if b ∈ 2+4N cox (a)8(b)4 if a + b ∈ 2+4N 0 otherwise
The calculations are summarized as follows.
T cox cox2 The formula for dim Vµ is an exercise in [7]. The traces of and come from Theorem 3.4.
1 For w = rα we have S = βˇ0(S ), where βˇ0 is the highest coroot. Then Y = Zν with ν(βˇ0(z)) = z, and we can take t =α ˇ(i), where i2 = −1. Then
+ tS tS RtS = ∅, R1 = {α}, R2 = {β,α + β, 2α + β}.
tS Each root in R2 restricts to 2ν on S. The partition function is given by
n1+n3 Prα (nν)= (−1) , (n1X,n2,n3) 23 3 summed over {(n1, n2, n3) ∈ N : 2n1 +2n2 +2n3 = n}, so
(−1)n/2 (1 + ⌊n/4⌋) if n ∈ 2N Prα (nν)= (0 if n∈ / 2N. In the sum 1 tr(r ,V T )= ε(v)e (t)P (vµ − ρ), α µ ∆(t) vµ rα vX∈W only the terms for v =1,rα,rβ,rαrβ are nonzero and we get
T b tr(rα,Vµ )=(a)4 Prα (a +2b − 3) − (−1) P(a − 3) , which simplifies to the formula above. The trace of rβ is obtained similarly. Remarks.
1. The irreps of Sp4 with nonzero weight spaces are those factoring through SO5. We see that 2 T the long element w0 = cox has nonzero trace on V for every irrep V of SO5. This means the virtual character V T is never a linear combination of induced representations from proper parabolic subgroups of W (and in particular is never the regular representation).
In fact, this holds for any irrep V of SO2n+1, n ≥ 1. Indeed, P/2Q = A ⊔ (ρ + A), where A = Q/2Q. The W -stabilizer of ρ in ρ+A is the symmetric group Sn, so W is transitive on ρ+A. T Hence tr(w0,V ) =06 by Thm. 3.4.
T T T 2. Back to Sp4. Comparing dim Vµ and tr(w0,Vµ ) shows that w0 is a scalar on Vµ if and only if a = 1 or b = 1. In the former case Vµ is the degree b − 1 harmonic polynomials on the five- T dimensional orthogonal representation and w0 is trivial on Vµ . In the latter case Vµ is the degree T a − 1 polynomials on the four-dimensional symplectic representation and w0 acts on Vµ by the scalar (a)4.
T 3. The trivial representation of W appears in Vµ with multiplicity b a if b ∈ 4N 4 4 j k 1 a b a − a if b ∈ N +( )4(1 ( )8) 2+4 4 4 htriv,V T i = j k µ 1 a (b +(a) ) if a + b ∈ 4N 4 4 4 l m 1 a (b +(a)4) +(b4)(1+(a)8) if a + b ∈ 2+4N 4 4 l m The reflection representation appears with multiplicity 1 hrefl,V T i = [a − (a) ][b − (b) (a) ]. µ 8 4 2 4
24 5.1.3 G2
In this section we explicate our character formula for G of type G2.
For relatively prime positive integers m, n, and q ∈ Q, let Pmn(q) be the number of solutions (x, y) ∈ N × N of the equation mx + ny = q. For any integer k, we have the constant-term formula zk k − m − n m −m n −n dz = Pmn . 1 (z − z )(z − z ) 2 ZS Now Pmn(q)=0 unless q ∈ N, in which case Pmn(q) is given by Popoviciu’s formula [25] (see also [5, 1.3]) q qm′ qn′ P (q)= − − +1, mn mn n m where m′, n′ are any integers such that mm′ + nn′ =1, and {r} = r − ⌊r⌋ denotes the fractional part of a rational number r.
For G2 we will need just two of these partition functions, P12 and P23, which have the more explicit formulas q q q P (q)=1+ , P =1+ − . 12 2 23 2 3 j k j k l m The conjugacy classes in W are represented by
3 2 1W , rα, rβ, cox , cox , cox.
T For µ = aωα + bωβ the values of tr(w,Vµ ) are shown in the following table.
25 T w tr(w,Vµ )
a3b a2b2 ab3 ab 2a 3 e + + + +(b) · +(a) · (b) · 12 8 36 6 3 9 2 2 8 3a +2b − 5 3a + b − 5 b − 5 r (−1)a+b P +(−1)a · P − P α 23 2 23 2 23 2 2a + b − 3 a + b − 3 a − 3 r (−1)a+1 P − P +(−1)b · P β 12 2 12 2 12 2 1 cox3 [(a) · a(3a +2b) − (a + b) · (a + b)(3a + b)+(b) · (2a + b)b] 8 2 2 2 0 (a, b) ≡ (0, 0) mod2 1 −a(3a +2b) (a, b) ≡ (0, 1) mod2 = 8 −b(2a + b) (a, b) ≡ (1, 0) mod2 (a + b)(3a + b) (a, b) ≡ (1, 1) mod2 1 cox2 [(a) · (3a +2b) − (a + b) · (3a + b)+(2a + b) · b] 9 3 3 3 −a ab ≡ 0 mod3 (b) = 3 2a + b ab ≡ 1 mod3 3 −(a + b) ab ≡ 2 mod3
cox −(a)6 · (3a +2b)6 +(a + b)6 · (3a + b)6 − (2a + b)6 · (b)6 +1 (a, b) ≡ (±1, ±1), (±2, ±1), (3, ±2) mod6 = −1 (a, b) ≡ (±1, ±2), (±2, ∓1), (3, ±1) mod6 0 otherwise
T As far as I know, the dimension of Vµ for G2 was first given explicitly in [2]. The formulas therein evidently take positive integer values without any sign cancellations. On the other hand, they have nine different cases, according to congruences of (a, b) mod (2, 6). A more compact expression T for dim Vµ was obtained by Vergne, using coordinates (m, n) where µ − ρ = mα + nβ (see [22]). T If we change coordinates to (a, b) as in (30) then Vergne’s formula for dim Vµ simplifies to the one given above.
The character values on the reflections come from Theorem 4.5.
The elliptic character values are expressed as harmonic quasi-polynomials as in (2) and also as piecewise monomials determined by the action of W on P/mP , as in Thm. 1.2.
The latter is analyzed by recalling that W is the isometry group of the quadratic form q : P → Z
26 given by q(µ)=3a2 +3ab + b2, for which q(ρ)=7. For each m ∈{2, 3, 6}, reduction modulo m gives a quadratic form qm : P/mP → Z/mZ and a homomorphism from W to the isometry group cox6/m T O(qm) of qm on P/mP . For all m, we have tr( ,Vµ ) =06 only if qm(µ)= qm(ρ)=1. cox3 T The form q2 is anisotropic; W is transitive on nonzero vectors in P/2P and tr( ,Vµ ) =6 0 if and only if q2(µ)=1.
The form q3 is degenerate, with radical spanned by ωα +3P . The map W → O(q3) realizes W as the Borel subgroup of GL(P/3P ) stabilizing the radical line. The orbit W ρ consists of the six vectors in P/3P with q3(µ)=1; these are the cosets of the short roots.
For m = 6 the vectors with q6 = 1 form two W -orbits, represented by ρ +6P and β +6P . It cox T follows that tr( ,Vµ ) =06 if and only if q6(µ)=1 and µ is not in the coset of a root. Remarks.
T 1) Vµ is a multiple of the regular representation of W if and only if (a, b) ≡ (0, 0) mod (2, 6).
2) Let W2 ≃ WA1×A1 be a subgroup of W generated by reflections about a pair of orthogonal roots. T Then Vµ is induced from a character of W2 if and only if b ∈ 3Z.
T 3) Vµ is irreducible for W only for the trivial, seven-dimensional and adjoint representations. cox3 T Indeed, from the monomial formula for tr( ,Vµ ) we find that the list of µ ∈ P++ for which T cox3 tr(w0,Vµ ) belongs to the set {χ( ): χ ∈ Irr(W )} = {±1, ±2}, is just
(a, b)=(1, 1), (1, 2), (2, 1), (3, 2).
T The first three cases are those with Vµ irreducible. In the last case µ = 3ωα +2ωβ, we have cox3 T cox2 T tr( ,Vµ ) = −2 and tr( ,Vµ ) = +1. As these are not the values of any χ ∈ Irr(W ), it T follows that Vµ cannot be irreducible for W .
5.2 SU4
For G = SU4, let α,β,γ be simple roots with α and γ orthogonal to each other. Let ωα,ωβ,ωγ be the corresponding fundamental weights, so that ρ = ωα + ωβ + ωγ.
T For a dominant regular weight µ, the character of Vµ is expressed as a function of coordinates (a, b, c) where a, b, c are positive integers such that
µ = aωα + bωβ + cωγ.
T Replacing Vµ by its dual does not change the W -module structure of Vµ , so we may assume that
a ≤ c.
Set c − a c + a d := , f := b − d, s := . 2 2
27 T Then Vµ =06 if and only if f ∈ 1+2Z.
Index the conjugacy classes in W = S4 by partitions [λ1λ2 ··· ] of 4. For example [1111] is the identity element and [4] = cox.
The character values are shown in the following table.
T w tr(w,Vµ ) 1 ab(a + b) f ≤ 1 [1111] 2 2 (a (bc +1 − d ) f > 1 1 a · [d] − b · [s] f ≤ −1 [211] 2N 2N 2 ((1 + s(f)4) · [d]2N − (1 + d(f)4) · [s]2N f > −1 1 [22] (−1)db · [c] − (−1)ca · [d] 2 1+2N 2N [31] (a)3 · [2c + f]3N − [f]3N
1 s b + s d [4] (−1)aδ +(−1)bδ − (−1)cδ 2 2 2 2 where for any A ⊂ Z and x ∈ Q we use the notation
1 if x ∈ A (−1)x if x ∈ N [x]A = δ(x)= (0 if x 6∈ A (0 if x 6∈ N.
T The dimension of Vµ was obtained in [22, Thm. 6.1], using coordinates different from ours. Our standing condition a ≤ c puts us in case (2) (if f ≤ 1) or (4) (if f > 1) of [loc. cit.].
T We give here just the computation for the trace of a reflection on Vµ , leaving the other (easier) cases to the reader.
2 We take w = rβ, S = {diag(x,y,y,z): xy z = 1}, t = diag(−1, −1, 1, 1) =α ˇ(−1). Then + tS tS RtS = ∅, R1 = {β}, R2 = {α, γ, α + β, β + γ, α + β + γ}. For s ∈ S we set χ(s)= x/y, η(s)= y/z, (χη)(s)= x/z. From Theorem 4.5 we have 1 tr(r ,V T )= − ε(v)(−1)hvµ,αˇiP (vµ − ρ). (31) β µ 2 w vX∈W Here the partition function Pw is given by
r Pw(ν)= (−1) , (Xp,q,r) 28 where the sum is over those (p,q,r) ∈ N3 such that
2pχ +2qη + r(χ + η)= ν. (32)
If ν is the restriction to S of vµ − ρ then (32) means that
2p + r = A, 2q + r = C, (33) where vµ − ρ = Aα + Bβ + Cγ. In particular, r ≡ A ≡ C mod 2. Setting n = min{A, C} we have (−1)r =(−1)n and we find that
Pw(vµ − ρ)= P (n) · [A − C]2Z, where n P (n)=(−1)n 1+ . 2 j k and [x]2Z =1 if x ∈ 2Z, zero otherwise.
−1 We next find those v ∈ W for which (33) has a solution. Write v as a permutation v1v2v3v4, −1 −1 sending i 7→ vi. The values of v ωˇα and v ωˇγ are determined by v1 and v4, respectively. When v1 = 4 we have A < 0 so there is so solution to (33). Likewise, when v4 = 1, 2 we have C < 0 (since a ≤ c, in the case v4 =2), so there is again no solution to (33).
The remaining possibilities are as follows: For each pair (v1, v4) there are two v’s, with opposite values of ε(v). In each row the top v has sign ε(v) = +1 and the bottom v has sign ε(v) = −1, and m =(f − 3)/2.
−1 −1 v1 v4 n = min{A, C} |A − C| v hµ, v αˇi 1 4 m + s d 1234 a 1324 a + b 1 3 m s 1423 a + b + c 1243 a 2 4 m + d s 2314 b 2134 −a 2 3 m d 2143 −a 2413 b + c 3 4 m − f b + s 3124 −a − b 3214 −b
Label the rows of this table by the pairs (i, j)=(v1, v4). For each (i, j) let mij, Aij, Cij be the corresponding values of m, A, C and set
Mij = P (mij) · [Aij − Cij]2Z.
After further simplification, (31) becomes
T a b tr(rβ,Vµ )=(−1) M13 + M24 − M14 − M23 − (−1) · M34 a b =(−1) [P (m)+ P (m + d)] · [s]2Z − [P (m + s)+ P (m)] · [d]2Z − (−1) P (m − f) · [s − d]1+2Z .
29 T Breaking into cases for s,d even/odd, we get the values for tr(rβ,Vµ ) in the table above. Remarks:
T 1. Vµ is a multiple of the regular representation when a, b, c ∈ 2Z and either a or c is in 3Z. For T T example, if (a, b, c) = (2, 4, 12) then dim Vµ = 24 so Vµ is exactly the regular representation of W .
T 2. The table of µ ∈ P++ for which Vµ is irreducible is as follows.
T (a, b, c) Vµ (1, 3, 1) ̺2 (1, 1, 4k + 1) εk (2, 1, 4k + 2) εk ⊗ ̺ (1, 2, 4k + 3) εk+1 ⊗ ̺
Here, k ∈ N. ̺ and refl2 are the reflection and two-dimensional irreducible representations of S4, respectively. As Kostant and Gutkin discovered for Sn, each irrep of S4 appears as the zero weight 4 4 space of exactly one constituent of ⊗ C . In fact, all but refl2 appear in infinitely many higher tensor powers as well.
3. We have d =0 exactly when Vµ is self-dual. In this case µ =(a, b, a) with b odd, so s = a and f = b. The formulas simplify to
4 T 1 tr(1 ,Vµ )= 2 a(ab + 1) T 1 tr(211,Vµ )= 2 [(a)2 + a · (b)4] T 1 a+1 tr(22,Vµ )= 2 (−1) [a + b · (a)2] T tr(31,Vµ )=(a)3([2a + b]3N − [b]3N) 1 if a + b ∈ 2+4N T tr(4,Vµ )= −1 if a ∈ 2+4N 0 otherwise
6 Irreducible zero weight spaces
T It was shown already in [14] and [21] that every irrep of the symmetric group Sn appears as V for G some irrep of SLn. Since, from [21] we have tr(cox ,Vµ) ∈ {−1, 0, +1}, it follows that the trace of an n-cycle on any irrep of Sn also lies in {−1, 0, +1}.
When they first met, Kostant asked Lusztig if he could prove this last fact for Sn directly. Lusztig observed that since an n-cycle generates its own centralizer in Sn, one need only find n irreps of Sn for which tr(cox) =06 . The exterior powers of the reflection representation fit the bill. In any irreducible Weyl group W with Coxeter number h, it is still true that a Coxeter element generates its own centralizer and one can again find h irreps (no longer exterior powers of the
30 reflection representation) of W on which cox has nonzero trace. Thus for any simple G we have:
(1) The trace of coxG on any irrep of G lies in {−1, 0, +1};
(2) The trace of cox on any irrep of W lies in {−1, 0, +1}.
One may ask if (1) ⇒ (2) as it did for the symmetric group. That is, does every irrep of W appear T as V for some irrep of G? We have seen the answer is negative, for G = Sp4 and G2. In this section we will see it is also negative for D4, F4 and E6. So the question, raised in [16], becomes: which irreps of W appear as V T for some irrep V of G?
Theorem 3.4 leads to a method for answering this question in the case that −1 ∈ W . This means w0 = −1 acts by a scalar ±1 on any irreducible representation of W . Since w0 is an elliptic T involution, Thm. 3.4 implies that Vµ can only be irreducible if µ +2Q belongs the the W -orbit of −1 ˇ+ ˇ+ ˇ+ ρ +2Q in P/2Q, in which case there is v ∈ W such that v R2 = Rµ ⊂ R and
hµ, αˇi dim V T = ε(v) . (34) µ hρ, αˇi ˇ+ αˇY∈Rµ
Since µ ∈ P++, each factor hµ, αˇi is strictly positive and increases when we add a fundamental weight to µ. Hence each coset in P/2Q contains only finitely many µ’s for which the product in (34) is the dimension of an irreducible representation of W . Thus one arrives at a finite list M of ′ T possible µ s for which Vµ can be irreducible. Computation of other character values for µ ∈ M T can show that Vµ is also reducible for certain µ ∈ M.
T For Spin8 and F4 this eliminates all but the known irreducible Vµ as we will see. For E6 we use the elliptic triality to do the same, up to two possible exceptions.
6.1 Spin8
We label the Dynkin diagram of type D4 as
1 2 3 4 and a weight µ = aω1 + bω2 + cω3 + dω4 ∈ P will be written as
a b c µ = d .
We have ρ ∈ Q. It follows that the W -orbit of ρ in P/2Q lies in Q/2Q. Now µ ∈ Q if and only if T a ≡ c ≡ d mod 2. We will show that the only µ ∈ Q ∩ P++ with Vµ irreducible are the known ones (cf. [30]):
1 1 1 3 1 1 1 1 3 1 1 1 1 2 1 µ = 1 1 1 3 1 . (35)
31 These Vµ are the trivial, spherical harmonics of degree two on the three eight dimensional orthog- onal representations of G, and the adjoint representation, respectively.
The set ˇ+ ˇ+ Rµ = {αˇ ∈ R : hµ, αˇi ∈ 2Z} depends only on the coset a b c µ +2P =: d ∈ P/2P. 2P The W -orbit of ρ +2P in P/2P consists of the nonzero cosets
1 1 1 1 0 1 0 1 0 1 , 1 , 0 . (36) 2P 2P 2P The sign character ε is trivial on the stabilizer in W of each the cosets (36). It follows that the sign T of tr(w0,Vµ ) is constant on the fiber in P/2Q above each of these cosets.
T The values of tr(w0,Vµ ) are shown in the table below
5 T µ +2P 2 tr(w0,Vµ )
1 1 1 1 (a + b)(b + c)(b + d)(a + b + c + d) 2P
1 0 1 1 −b(a + b + c)(a + b + d)(b + c + d) 2P
0 1 0 0 −acd(a +2b + c + d) 2P In each of these cosets, we find that besides the µ listed in (35), there is only one other µ ∈ P++ (up to diagram symmetry) for which
T tr(w0,Vµ ) ∈{χ(w0): χ ∈ Irr(W )} = {1, 2, 3, ±4, 6, ±8}
T T and also | tr(w0,Vµ )| = dim Vµ , namely
5 1 1 µ = 1 .
T T For this µ we have tr(w0,Vµ ) = dim(Vµ )=6. However, Vµ is the degree-four spherical harmon- ics, whose zero weight space can be written down explicitly and seen to be reducible for W .
6.2 F4
T For the group G of type F4, we use our formula for tr(w0,Vµ ) to determine all irreps Vµ for which T Vµ is irreducible for W . We show this occurs only for the known cases dim Vµ =1, 26, 52.
32 We label the Dynkin diagram for F4 as
12 ⇒ 34.
We have P = Q and the quotient X := P/2Q = P/2P is an F2-vector space with basis ω¯i := ωi +2P , i =1,..., 4. We write µ =(a,b,c,d) for µ = aω1 +bω2 +cω3 +dω4 ∈ P and (a,b,c,d)2 for the coset µ +2P ∈ X.
We have 2P ⊂ Qℓ ⊂ P, where Qℓ is the lattice spanned by the long roots (note Qℓ was called Q in section 6.1). The W -action on X preserves the subspace Y := Qℓ/2P and gives an isomorphism
W/{±1} −→∼ GL(X,Y )= {g ∈ GL(X): gY = Y }.
The latter group has three orbits in X, of sizes 1, 3, 12, the latter being X \ Y = W (ρ +2P ). It T follows that if µ =(a,b,c,d) then tr(w0,Vµ ) =06 if and only if c and d are not both even. In this case there is v ∈ W such that µ ∈ vρ +2P and we have
hµ, αˇi ε(v) tr(w ,V T )= ε(v) · = · hµ, αˇi. 0 µ hρ, αˇi 215 · 32 · 5 ˇ+ ˇ+ αˇY∈Rµ αˇ∈YvRµ ˇ+ + where we choose v so that vR2 = Rµ . We list all positive coroots αˇ1,..., αˇ24 as linear forms Ai(µ)= hµ, αˇii, as follows.
A1 = a A2 = b A3 = c A4 = d A5 = a + b A6 = b + c A7 = c + d A8 = a + b + c A9 =2b + c A10 = b + c + d A11 =2b + c + d A12 = a + b + c + d A13 = a +2b + c A14 =2b +2c + d A15 = a +2b + c + d A16 =2a +2b + c A17 = a +2b +2c + d A18 =2a +2b + c + d A19 = a +3b +2c + d A20 =2a +2b +2c + d A21 =2a +3b +2c + d A22 =2a +4b +2c + d A23 =2a +4b +3c + d A24 =2a +4b +3c +2d
We have ε(v) tr(w ,V T )= A A ··· A , 0 µ 215 · 32 · 5 i1 i2 i10 ˇ+ where vR2 = {Ai1 , Ai2 , ··· Ai10 }.
These are shown in the table below, where the cosets in X \ Y are labelled by (a,b,c,d)2, with
33 (c,d)2 =6 (0, 0)2.
15 2 T µ ≡ (a,b,c,d)2 v 2 · 3 · 5 · tr(w0,Vµ )
(1111)2 1W A5A6A7A11A12A13A17A18A21A23
(1011)2 r1 −A2A7A8A10A11A13A17A18A19A23
(0111)2 r2 −A1A6A7A8A11A15A18A19A21A23
(1010)2 r3 −A2A4A8A12A13A14A15A20A21A22
(1101)2 r4 −A3A5A8A9A10A15A16A17A21A24
(1110)2 r1r3 A4A5A6A10A13A14A15A19A20A22
(1001)2 r1r4 A2A3A6A9A12A15A16A17A19A24
(0010)2 r3r2 A1A2A4A5A14A17A19A20A21A22
(0101)2 r2r4 A1A3A9A10A12A13A16A19A21A24
(0110)2 r2r1r3 −A1A4A6A8A10A12A14A17A20A22
(0011)2 r3r2r1r3 A1A2A5A7A10A11A12A15A18A23
(0001)2 r4r3r2r1r3 −A1A2A3A5A6A8A9A13A16A24 In each of the above cosets we next find all µ for which T tr(w0,Vµ ) ∈{χ(w0): χ ∈ Irr(W )} = {1, 2, ±4, 6, −8, 9, 12, −16}.
In fact from all of the the twelve cosets there are only five such µ. In these cases Vµ is small enough T to compute dim Vµ , as shown in the next table. T T µ tr(w0,Vµ ) dim Vµ (1, 1, 1, 1) 1 1 (2, 1, 1, 1) −4 4 (1, 1, 1, 2) 2 2 (2, 2, 1, 1) 4 228 (1, 1, 1, 2) 12 12
The first three cases, where dim Vµ =1, 52, 26 are known to have irreducible zero weight spaces. T In the case µ = (2, 2, 1, 1), Vµ is clearly reducible. In the last case µ = (1, 1, 1, 2) we note that W has a unique 12-dimensional irreducible character χ12, and χ12(cox)=1. On the other hand, the ˇ ˇ cox T co-root β =2ˇα1 +4ˇα2 +3ˇα3 +ˇα4 has hµ, βi = 12, which means that tr( ,Vµ )=0, by Kostant T [21] (or Prop. 3.2 above). It follows that V(2,2,1,1) is reducible. This completes the determination T of all µ for which Vµ is irreducible, for the case G = F4.
6.3 E6
T cox4 In this section we compute tr(w,Vµ ) where w = is the elliptic regular element of order three. We will see that this almost determines the irreducible zero weight spaces for E6.
34 T Since ρ ∈ Q, all µ ∈ P++ for which Vµ =06 belong to Q. Hence we consider the action of W on Q/3Q. Let q be the W -invariant quadratic form q : Q → Z such that q(α)=2 for every α ∈ R.
Passing to Q/3Q gives a quadratic form q3 : Q/3Q → F3 with radical 3P/3Q and nondegenerate quotient Q/3P . This gives an isomorphism ϕ : W × {−I} sim→ O(Q/3P ).
One computes q(ρ) =78, so ρ +3P is isotropic. Since w0ρ = −ρ, it follows that W is transitive on the set of nonzero isotropic vectors in Q/3P . Arguing as in the last step of the proof of Prop. 3.3, it follows that W is also transitive on the nonzero isotropic vectors in Q/3Q.
From this discussion and Theorem 4.5 we obtain
cox4 T Proposition 6.1 For w = we have tr(w,Vµ ) =06 if and only if q3(µ +3Q)=0. In this case there exists v ∈ W for which vµ ∈ ρ +3Q and we have ε(v) tr(w,V T )= hµ, αˇi. (37) µ 35 · 63 · 9 ˇ+ αˇY∈Rµ
T 12 3 45 To compute tr(w,Vµ ) explicitly, we label the Dynkin diagram as 6 and write a weight µ = aω1 + bω2 + cω3 + dω4 + eω5 + fω6 ∈ P as
ab c de µ = f . As linear forms on P , we list the coroots as
A1 = a B1 = c C1 = e A2 = b B2 = f C2 = d A3 = a + b B3 = c + f C3 = d + e A4 = b + c B4 = b + c + d C4 = c + d A5 = a + b + c B5 = b + c + d + f C5 = c + d + e A6 = b + c + f B6 = a + b + c + d + e C6 = c + d + f A7 = a + b + c + d B7 = b +2c + d + f C7 = b + c + d + e A8 = a + b + c + f B8 = a + b + c + d + e + f C8 = c + d + e + f A9 = a + b + c + d + f B9 = a + b +2c + d + e + f C9 = b + c + d + e + f A10 = a + b +2c + d + f B10 = a +2b +2c +2d + e + f C10 = b +2c + d + e + f A11 = a +2b +2c + d + f B11 = a +2b +3c +2d + e + f C11 = b +2c +2d + e + f A12 = a +2b +2c + d + e + f B12 = a +2b +3c +2d + e +2f C12 = a + b +2c +2d + e + f
ˇ+ Note that Rµ depends only on the F3-line in Q/3P containing µ + 3P . Let X be the set of q3-isotropic lines in Q/3P . We have |X| = 40, but we can reduce the calculation further. Let ϑ be the involution of P/3Q arising from the nontrivial symmetry of the Dynkin diagram. Then X = Y ⊔Z where Y = {y1,...,y16} consists of the ϑ-fixed lines and Z = {z1, ϑz1, ··· , z12, ϑz12} are the remaining lines.
T The table of tr(w,Vµ ) is structured as follows. In each isotropic line in Q/3P , we give a vector x = ciωi mod 3P where +, 0, − correspond to ci = +1, 0, −1 ∈ F3. Next we give an element P 35 ˇ+ ˇ+ T v ∈ W such that vR3 ⊂ R and x = vρ +3P . Finally we give the numerator of tr(w,Vµ ) for T any µ ∈ x ∩ P++. If µ ∈ −x ∩ P++ then tr(w,Vµ ) is the negative of the rightmost column in row T T x evaluated at µ. If µ ∈ zi then tr(w,Vϑµ) is obtained from tr(w,Vµ ) by interchanging Ai ↔ Ci.
36 3 10 T x µ +3P = x v 2 · 3 · tr(w,Vµ ) ++ + ++ y1 + 1W A5A6A10B4B8B10C10C6C5 + − − − + y2 − r3 −A3A6A9B4B9B11C3C6C9 + + − + + y3 − r6 −A4A8A10B5B6B10C4C8C10 ++ 0 ++ y4 + r2r1 A7A8A12B1B5B7C7C8C12 −− −−− y5 + . r4r1 A5A9A12B3B4B7C5C9C12 − − + − − y6 + r5r1 A4A8A10B5B6B10C4C8C10 +0 + 0+ y7 + r3r6 A2A8A9B6B7B11C2C8C9 + − + − + y8 + r6r3 A3A4A7B5B9B12C3C4C7 + + − + + y9 + r1r2 A4A9A11B3B6B9C4C9C11 − − 0 − − y10 + r2r4 A6A7A11B1B8B9C6C7C11 0+ 0 +0 y11 − r4r2r3 A1A6A8B1B10B11C1C6C8 + 0 − 0 + y12 − r6r3r6 −A2A5A7B7B8B12C2C5C7 − + + + − y13 0 r5r3r2 −A3A10A12B2B4B5C3C10C12 +0+0+ y14 0 r1r3r2 −A2A10A11B2B6B8C2C10C11 0 + − + 0 y15 + r6r4r2r3 A1A4A5B3B10B12C1C4C5 00+00 y16 0 r3r6r4r2r3 −A1A2A3B2B11B12C1C2C3 − − + ++ z1 + r1 −A4A7A8B7B9B10C9C6C5 −− − + + z2 + r2 −A5A8A11B5B8C4C7C10C12 − 0 − − + z3 − r1r3 A2A7A8B5B8B11C3C6C10 − − 0 ++ z4 − r2r6 A8A11B1B4B6C6C9C10C12 − ++0+ z5 0 r3r2 A3A11B2B7B9C2C7C9C12 0+ + − + z6 − r2r3 A1A9A12B3B5B11C3C4B8 − − 0 ++ z7 + r1r2r1 −A6A9A10A12B1B4B6C9C11 0 − − 0 + z8 0 r2r3r2 −A1A10A12B2B7B10C2C5C8 − 0 + +0 z9 − r4r1r3 −A2A5A6A9B3B8B11C1C11 0 + 0 − + z10 + r6r2r3 −A1A7A12B1B4B12C3C5C6 − 0 + − + z11 + r6r1r3 −A2A5A9B4B6B12C3C4C10 − 0 0 +0 z12 + r6r4r1r3 A2A4A7A8B1B6B12C1C11
We turn now to irreducible zero weight spaces for E6.
As an abstract group, we have W ≃ SO5(3), via the twisted mapping εϕ. The Steinberg repre- ′ sentation 81p of SO5(3) and its twist 81p = ε ⊗ 81p are the only irreducible representations of W whose character vanishes on w. We conclude a dichotomy:
37 T Proposition 6.2 If Vµ is irreducible for W then exactly one of the following holds.
T (i) q3(µ)=0 and tr(w,Vµ ) =06 .
T ′ (ii) q3(µ) =06 and Vµ ≃ 81p or 81p.
I do not know if case (ii) ever occurs, but it seems unlikely.
T Up to duality, there are five known representations Vµ for which Vµ is irreducible. These are small representations (cf. [30]).
T T µ Vµ tr(w,Vµ ) 11 1 11 1 1p 1 11 1 11 2 6p −3 21 1 12 (38) 1 20p 2 41 1 11 11 1 14 1 1 24p 6 22 1 11 11 1 22 1 1 64p −8
− The representation 64p is another Steinberg representation, via the isomorphism W ≃ O6 (2). However, because −1 ∈/ W , there is no analogue of Prop. 6.2 for m =2.
T Using the table of tr(w,Vµ ) and the method used for D4 and F4, we find that (38) is the complete T list of irreducible zero weight spaces with tr(w,Vµ ) =06 . We conclude
Proposition 6.3 The only irreducible representations of W (E6) afforded by a zero weight space ′ of the compact Lie group E6 are 1p, 6p, 20p, 24p, 64p and possibly 81p, 81p.
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