Graded multiplicities in the exterior algebra of the little adjoint module
aa r X i v : . [ m a t h . R T ] J un Graded multiplicities in the exterior algebra of thelittle adjoint module
Ibukun Ademehin
School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
Abstract
As a first application of the double affine Hecke algebra with unequal pa-rameters on Weyl orbits to representation theory of semisimple Lie alge-bras, we find the graded multiplicities of the trivial module and of thelittle adjoint module in the exterior algebra of the little adjoint module of asimple Lie algebra g with a non-simply laced Dynkin diagram. We provethat in type B, C or F these multiplicities can be expressed in terms ofspecial exponents of positive long roots in the dual root system of g . Keywords:
Exterior algebra, Invariants, Graded multiplicity
1. Introduction
Let g be a simple Lie algebra of rank r over C . Let V λ be the finite di-mensional irreducible g -module with highest weight λ. The graded mul-tiplicity of V µ in the exterior algebra V V λ is the polynomial GM [ V V λ : V µ ] ( q ) = X i ≥ d i q i , where d i is the multiplicity of V µ in V i V λ . When the exterior algebra isclearly identified we write GM µ ( q ) . Email address: [email protected] (Ibukun Ademehin)
Preprint submitted to Elsevier May 22, 2019 ecomposing the exterior algebra of the adjoint module V g has beengiven some consideration since the last century following the more classi-cal problem of decomposing the symmetric algebra S g . Known results in V g include the following. Kostant [10] used the Hopf-Koszul-Samelsontheorem, which asserts that the skew invariants ( V g ) g form an exterioralgebra over the graded subspace of primitive invariants h P , · · · , P r i , todeduce the graded multiplicity of ( V g ) g expressed in terms of the expo-nents of g , which is the Poincar´e polynomial of the De Rham cohomologyof the Lie group G associated with g . Kostant also obtained in [10] theungraded multiplicity of g in V g . In [2] Bazlov proved Joseph’s conjec-ture on the graded multiplicity of g in V g . The formula for this gradedmultplicity suggests that the isotypic component of g is a free moduleover the subring V h P , · · · , P r − i of the skew invariants. This indeed isthe case, as was proved by De Concini, Papi and Procesi [8] who used theChevalley transgression theorem to explicitly obtain basis vectors for thefree module.In [15] Panyushev obtained a classification of orthogonal irreducible g -modules V whose skew invariants is again an exterior algebra (this clas-sification includes the little adjoint module when the root system of g isof type B, C or F ), and also proved the g -module isomorphism of V V to the reduced Spin of V. Besides these results not much is known about the decomposition of V V when V is not the adjoint module. This motivates our study of thering structure of the exterior algebra of the little adjoint module. Let g be of type B, C, F or G. Let V θ s be the little adjoint modulewith highest weight the highest short root θ s of g and character χ θ s = r s + P α ∈ R s e α , where r s is the number of short simple roots of R, the rootsystem of g and R s is the set of short roots in R. Recall that in type B r , V θ s is the standard (2 r + 1) -dimensional module of so r +1 ( C ) , [9, Chap. 19.4] . To describe V θ s in type C r , let { ω i } ri =1 be the set of fundamental weightsof g = sp r ( C ) and let V ω = C r be the standard module of sp r ( C ) . Let φ : V C r → C be the contraction defined as φ ( v ∧ w ) = Q ( v, w ) , Q is the skew-symmetric form associated with sp r ( C ) . Then V θ s = V ω is the kernel of φ contained in V C r = V θ s ⊕ V , [9, Chap. 17.2] . In type F , V θ s is the standard module of the automorphism group ofthe -dimensional × Hermitian traceless Jordan octonion matrices,[5, Sect. 6.2.3]. In type G , V θ s is the standard module of the Lie group G , which is the automorphism group that preserves an alternating cubicform on the -dimensional imaginary octonions, [1, Sect. 4]. In the present paper, we prove the formulae for the graded multiplici-ties of the trivial module and of the little adjoint module in V V θ s , as The-orem 3.4 and Theorem 4.7. We state these below.Let R be of type B, C or F. Let { h i } r s i =1 be the set of special exponentsof ( R + s )ˇ which form the partition dual to the partition arising from thepositive long coroots in R ˇ with respect to the special heights of thesecoroots (note that the coroots of short roots of R have the long length in ( R + s )ˇ , [3, Sect. 1.1]). Then Theorem A (Theorem 3.4) . The graded multiplicity of the trivial module V in V V θ s is given by GM ( q ) = r s Y i =1 (1 + q h i +1 ) . (1) Theorem B (Theorem 4.7) . The graded multiplicity of the little adjoint modulein its exterior algebra V V θ s is GM θ s ( q ) = r s − Y i =1 (1 + q h i +1 ) r s X i =1 ( q h i − (2 h − + q h i ) . (2)When R is of type G the graded multiplicity of V and V θ s respec-tively in V V θ s are GM ( q ) = (1 + q )(1 + q ) , (3) GM θ s ( q ) = (1 + q )( q + q + q ) . (4)3 .3. The method of proof To obtain GM ( q ) in formulae (1) and (3) we use Cherednik’s innerproduct on characters [12] evaluated on χ V V θs and χ . This reduces thegraded multiplicity GM ( q ) to a ratio of polynomials in terms of the multi-parameter Poincar´e polynomial W ( q k ) [11] and Cherednik’s generalisa-tion of the constant term of Macdonald’s weight function ∆ . In the gen-eralised W ( q k ) and ct (∆) we use different integer coefficients for theheights of short and long simple roots and of their coroots. This arisesfrom the unique integer labelling k ( α ) on the different Weyl orbits of R which occurs in the formulae ct (∆) and W ( q k ) appearing in χ V V θs andin its inner products with χ . The positive integer k ( α ) relates the inde-terminates q and t occurring in W ( q k ) and ct (∆) . In section where weprove the formula for GM ( q ) , we set t α = q k ( α ) , see [12]. Cancellationsoccur in the ratio of W ( q k ) and ct (∆) which reduce simplifying GM ( q ) to a task of counting special heights of coroots in ( R + s )ˇ . We do this case bycase when R is of B, C, F or G. We note that in type
B, C or F, GM ( q ) can be expressed in terms of the partition dual to the partition arising fromthe coroots in ( R + s )ˇ with respect to the special heights of these coroots. Remark 1.1.
The formulae for GM ( q ) in type B, C and F are givenwithout proof and with no reference to the special exponents in Panyu-shev’s classification of orthogonal irreducible g -modules with an exterioralgebra of skew invariants. See [15, Table 1].To prove the formulae for GM θ s ( q ) in (2) and (4) we use the action ofthe operator Y θ ˇ from the double affine Hecke algebra HH q, t s, l on a subsetof the group algebra Q q, t s, l [ P ] generated by the integral weight lattice P over the field Q ( q ± d , t ± s, l ) , where θ is the highest root of R, and we followBazlov’s treatment of GM [ V g : g ] ( q ) in [2] (Bazlov in calculating GM [ V g : g ] ( q ) used the label k ( α ) = 1 for all α ∈ R, but in our case of GM λ ( q ) in V V θ s we maintain our unique integer labelling k ( α ) on the Weyl orbits of R : we use k ( α ) = 2 for α ∈ R s and k ( α ) = 1 for α ∈ R l ) . The unique la-belling k ( α ) on the W -orbits of R necessitates our use of Y θ ˇ ∈ HH q, t s, l andthe double parameter t s, l for the indeterminate t in Q q, t [ P ] . The outcomeof these is a non-trivial generalisation of Bazlov’s results on the action of Y θ ˇ on Q q, t [ P ] in [2]. We use the action of Y θ ˇ (which is unitary with re-spect to Cherednik’s inner product ( , ) on Q q, t s, l [ P ]) on e θ s , as well as4ome combinatorial properties of a subset of R associated with Y θ ˇ to cal-culate ( e α , for all α ∈ R s . From these, the task of simplifying GM θ s ( q ) is reduced to counting the special heights of the positive short roots of R .We do this case by case for the different root systems. In type B, C or F again, we note that formula (2) obtained for GM θ s ( q ) , expressible in termsof the special exponents of ( R + s )ˇ , suggests that the isotypic component of V θ s is a free module over a subring of the skew invariants in V V θ s . Moti-vated by De Concini, Papi and Procesi’s result on the isotypic componentof g in V g [8], we conclude the paper with a conjecture on the structureof the isotypic component of V θ s in the exterior algebra V V θ s . Note thatto obtain GM θ s ( q ) in Sect. 4 we set t α = q − k ( α )2 , see [13, Sect. 4] and [2]. Let R be a root system of g of type B, C, F or G, i.e. R s = ∅ . Wefix a basis of simple roots
Φ = { α i } ri =1 for R and the corresponding set offundamental weights { ω i } ri =1 . Let Q and P be the root and the integral weight lattice of R spannedby simple roots { α i } ri =1 and fundamental weights { ω i } ri =1 respectively.Let P + be the set of dominant integral weights of P and let W be theWeyl group of R generated by the simple reflections s i = s α i . We use subscripts s and l to mark objects related to short and long rootsrespectively. For instance, θ and θ s are the highest root and the highestshort root of R respectively. We denote by the ordered pair ( k s , k l ) thelabel k ( α ) ∈ Z > of short and long roots respectively.
2. Preliminaries
Let χ λ ∈ Z [ P ] be the character of V λ such that χ λ = P µ ∈P ( V λ ) m µλ e µ , where P ( V λ ) is the set of weights of V λ and m µλ is the dimension of theweight space V µλ of µ in V λ . The following is from [12, 5.1]. Let Q q, t [ P ] bethe group algebra of P generated by formal exponentials e λ , λ ∈ P , overthe field Q q, t of rational functions in q ± d and t ± , where d = | P / Q | . Let f = P λ ∈P f λ e λ be an element of the group algebra Q q, t [ P ] and let the5ar and ∗ involutions on Q q, t [ P ] be defined as ¯: e λ e − λ , q q, t t, ∗ : e λ e − λ , q q − , t t − , extended by Q -linearity over Q q, t [ P ] . Let the symmetric and non-degene-rate inner product due to Macdonald on Q q, t [ P ] be defined as h f , h i = 1 | W | ct ( f ¯ h ∇ ) , (5)where the constant term ct ( f ¯ h ∇ ) ∈ Q q, t of f ¯ h ∇ is the coefficient of e in f ¯ h ∇ , ∇ = Y α ∈ R ∞ Y i =0 − q i e α − q k ( α )+ i e α , and the integer labelling k ( α ) on the W -orbits of R relates the indeter-minates q and t, here as t α = q k ( α ) (see [12, (5.1.1)]). Observe that ∇ becomes a finite product when k ( α ) ∈ Z > : ∇ = Y α ∈ R k ( α ) − Y i =0 (1 − q i e α ) . We define the inner product due to Cherednik on Q q, t [ P ] as ( f , h ) = ct ( f h ∗ ∆) , where ∆ = Y α ∈ R ∞ Y i =0 (1 − q i e α )(1 − q i +1 e α )(1 − q k ( α )+ i e α )(1 − q k ( α )+ i +1 e α ) . Here also, when k ( α ) ∈ Z > , ∆ becomes a finite product: ∆ = Y α ∈ R k ( α ) − Y i =0 (1 − q i e α )(1 − q i +1 e α ) . f, h are W -invariant in Q q, t [ P ] , then h f , ¯ h ∗ i = 1 W ( q k ) ( f , h ) , (6)where W ( q k ) is the Poincar´e multi-parameter polynomial W ( q k ) = X w ∈ W q P α ∈ R ( w ) k ( α ) = Y α ∈ R + − q ( ρ k , α ˇ)+ k ( α ) − q ( ρ k , α ˇ) , (7)[11, Sect. 2], R ( w ) = { α ∈ R + | w ( α ) ∈ R − } and ρ k is the double parame-ter special weight (11).Let h , i denote h , i when ( k s , k l ) = (1 , . The irreducible charac-ters χ λ , λ ∈ P + are orthonormal with respect to h , i , [12, Sect. 5.3.15]and therefore form an orthonormal basis for Z [ P ] . Hence, if J = L di =0 J i is a graded g -module with graded character χ J ( q ) = P di =0 q i χ Ji , then thegraded multiplicity of a g -module V λ in J is GM λ ( q ) = h χ J ( q ) , χ λ i . (8)Recall χ θ s = r s + P α ∈ R s e α [15, 2.9], where r s = R s ∩ Φ) . It is easy toshow that χ V V θs ( − q ) = (1 − q ) r s Y α ∈ R s (1 − qe α ) = (1 − q ) r s ∇ , ∇ , where in ∇ , and ∇ we use the ordered pair ( k s , k l ) = (2 , and ( k s , k l ) = (1 , respectively. By (5) and (8) the graded multiplicity of V λ in V V θ s evaluated at − q becomes GM λ ( − q ) = (1 − q ) r s h , χ λ i , . (9)Since χ V V θs ( q ) ∈ Z q [ Q ] , GM λ ( q ) = 0 when λ / ∈ Q . The problem offinding the graded multiplicities of the irreducible characters of V V θ s istherefore reduced to calculating h , χ λ i , , for λ ∈ P + ∩ Q . In the nextsections we will calculate GM λ ( q ) for the two smallest dominant elementsof P + ∩ Q , i.e λ = 0 and θ s , to prove our results (1) – (4).7 . Calculating GM ( q ) By (5), (6) and (9) the graded multiplicity of the trivial module V in V V θ s is given by GM ( − q ) = (1 − q ) r s ct (∆ , ) | W | W ( q k ) . (10)To simplify this we consider the following. ρ k Set ρ s = P α ∈ R + s α = P α i ∈ R s ω i and ρ l = P α ∈ R + l α = P α i ∈ R l ω i . See[7, (3.1.1)]. The double parameter special weight ρ k is defined as ρ k = k s ρ s + k l ρ l = 12 X α ∈ R + k ( α ) α = r X i =1 k ( α i ) ω i . (11)See [7, (3.2.2)]. We extend ( ρ k , ) over Q + ˇ by Z -linearity and say α ˇ ∈Q + ˇ has special height ( ρ k , α ˇ) . The following theorem due to Cherednik [7, (3.3.2)] provides a gener-alisation of Macdonald’s constant term of ∆ by allowing different valuesfor k ( α ) on the different W -orbits of R. Theorem 3.1 (Macdonald’s Constant Term) . The constant term of Macdon-ald’s weight function ∆ is ct (∆) = Y α ∈ R + ∞ Y i =1 (1 − q ( ρ k , α ˇ) + i ) (1 − t α q ( ρ k , α ˇ) + i )(1 − t − α q ( ρ k , α ˇ) + i ) , where t α = q k ( α ) . Observe that when k ( α ) ∈ Z ≥ , ct (∆) becomes a finite product: ct (∆) = Y α ∈ R + k ( α ) Y i =1 − q ( ρ k , α ˇ) + i − q ( ρ k , α ˇ) +1 − i . (12)8y (7), (11), (12) and using ( k s , k l ) = (2 , , GM ( − q ) in (10) simplifiesto GM ( − q ) = (1 − q ) r s Y α ∈ R + s − q ( ρ k , α ˇ) + 1 − q ( ρ k , α ˇ) − . (13)Let Z ˇ ⊆ ( R + s )ˇ . We denote by H Z ˇ ( n ) the set { α ˇ ∈ Z ˇ | ( ρ k , α ˇ) = n } . Let h Z ˇ ( n ) = H Z ˇ ( n ) . Observe that α ˇ ∈ H R + s ˇ ( n − gives the factor − q n in (13), while α ˇ ∈ H R + s ˇ ( n + 1) gives (1 − q n ) − . Therefore, since θ s ˇ is thehighest coroot in ( R + s )ˇ [16, Lemma 5.1.4], GM ( − q ) becomes GM ( − q ) = (1 − q ) r s ( ρ k , θ s ˇ)+1 Y n =1 (1 − q n ) h R + s ˇ ( n − − h R + s ˇ ( n +1) . (14)Simplifying GM ( q ) is therefore reduced to a task of counting the specialheights of positive long roots in the dual root system R ˇ (recall that thecoroots of the roots in R s have the long length in R ˇ , [3, Sect. 1.1]). We dothis case by case when R is of type B, C, F or G. B r Recall that the sets of positive short roots and their coroots in type B r are R + s = { X i ≤ m ≤ r α m , ≤ i ≤ r } , ( R + s )ˇ= { X i ≤ m 1) = h R + s ˇ ( n + 1) for ≤ n ≤ r. We substitute this in (14) and obtain in type B r GM ( q ) = 1 + q r +1 . (15) Remark 3.2. Formula (15) implies that the only skew invariants of the Lie9roup SO r +1 ( C ) in its standard module are the scalars and the volumeform. C r Recall that the set of the positive short roots in type C r is given as R + s = J ∪ K = { X i ≤ m 4) = { α ˇ + · · · + α ˇ p +1 + 2 α ˇ p +2 + · · · + 2 α ˇ r , α ˇ + · · · + α ˇ p +2 α ˇ p +1 + · · · + 2 α ˇ r , · · · , α ˇ p +12 + α ˇ p +32 + 2 α ˇ p +52 + · · · +2 α ˇ r } ,H K ˇ (4 r − p − 2) = { α ˇ + · · · + α ˇ p + 2 α ˇ p +1 + · · · + 2 α ˇ r , α ˇ + · · · + α ˇ p − +2 α ˇ p + · · · + 2 α ˇ r , · · · , α ˇ p +12 + 2 α ˇ p +32 + · · · + 2 α ˇ r } ,H eK ˇ (2 r ) = { α ˇ + · · · + α ˇ r − + 2 α ˇ r , α ˇ + · · · + α ˇ r − + 2 α ˇ r − + 2 α ˇ r , · · · , α ˇ r + 2 α ˇ r +22 · · · + 2 α ˇ r } . Hence, h oK ˇ ( n ) = h eK ˇ ( n ) = p + 12 and h eK ˇ (2 r ) = r (18)whenever n ∈ { p + 2 , p + 4 , r − p − , r − p − } , r ∈ Z + 1 and p =1 , , · · · , r − , respectively r ∈ Z and p = 1 , , · · · , r − . Table 1 is obtained from (17) and (18). The reader can easily check thatthe cardinalities in columns – in Table 1 match the cardinalities of J ˇ , K ˇ and R + s ˇ in (16).Observe that for any rank rh R + s ˇ ( n − 1) = h R + s ˇ ( n + 1) + 1 when n = 4 p + 1 , p = 1 , · · · , r − , (19)otherwise, when n = 1 h R + s ˇ ( n − 1) = h R + s ˇ ( n + 1) . (20)11 h J ˇ ( n ) h oK ˇ ( n ) h eK ˇ ( n ) h oR + s ˇ ( n ) h eR + s ˇ ( n )2 r − r − r − r − r − r − r − r − r − r − r − r − ... ... ... ... ... ... r − r − r − r +32 r +22 r − r − r − r +12 r +22 r − r − r − r +12 r r r − r r − r r + 2 0 r − r − r − r − r + 4 0 r − r − r − r − ... ... ... ... ... ... r − r − Table 1: The multiplicity of the special heights of the positive long coroots in type C r We substitute (19) and (20) into (14) and GM ( q ) in type C r simplifies to GM ( q ) = r − Y p =1 (1 + q p +1 ) . (21) F Let σ k = P ri =1 k ( α i ) ω i ˇ , where ω ˇ i are the coweights of the fundamen-tal weights of P dual to the simple roots of R. With ( k s , k l ) = (2 , , ( σ k , α ) = ht l α + 2 ht s α, where ht m α = P α i ∈ Φ m λ i , m ∈ { s, l } , when α = P ri =1 λ i α i . From [4, Sect. VI: 4.9, Plate VIII] the positive short roots and their co-roots in type F are as in Table 2.We substitute Table 2 into (14) and obtain GM ( q ) in type F as GM ( q ) = (1 + q )(1 + q ) . (22)12 ∈ R + s α ˇ ( σ k , α ) ( ρ k , α ˇ) α α ˇ α α ˇ α + α α ˇ + α ˇ α + α α ˇ + α ˇ α + α + α α ˇ + 2 α ˇ + α ˇ α + α + α α ˇ + α ˇ + α ˇ α + α + α + α α ˇ + 2 α ˇ + α ˇ + α ˇ α + 2 α + α α ˇ + 2 α ˇ + α ˇ α + α + 2 α + α α ˇ + 2 α ˇ + 2 α ˇ + α ˇ α + 2 α + 2 α + α α ˇ + 4 α ˇ + 2 α ˇ + α ˇ α + 2 α + 3 α + α α ˇ + 4 α ˇ + 3 α ˇ + α ˇ 11 14 α + 2 α + 3 α + 2 α α ˇ + 4 α ˇ + 3 α ˇ + 2 α ˇ 13 16 Table 2: The special heights of the short roots and of their coroots in type F G The positive short roots and their coroots in the root system of type G are as given in Table 3. See [4, Sect. VI: 4.13, Plate IX]. α ∈ R + s α ˇ ( σ k , α ) ( ρ k , α ˇ) α α ˇ α + α α ˇ + 3 α ˇ α + α α ˇ + 3 α ˇ Table 3: The special heights of the positive short roots and the positive long coroots intype G We substitute Table 3 into (14) and get GM ( q ) in type G as GM ( q ) = (1 + q )(1 + q ) . (23)The reader can easily check that the degree of GM ( q ) for the differentroot systems is indeed dim V θ s = r s + R s . This is exactly what we expectsince if dim V = d, then V V = V V ⊕ · · · ⊕ V d V and V d V ⊂ ( V V ) g since V d V is isomorphic to the trivial module of g . GM ( q ) obtained with the odd powers of q when R is of type B, C or F suggests that the skew invariants ( V V θ s ) g forman exterior algebra over primitive invariant generators in V V θ s in thesecases of R. This indeed is the case as V θ s in type B, C or F is included inPanyushev’s classification of orthogonal g -modules with an exterior al-gebra of skew invariants; see [15, Table 1]. In type G , let ( V V θ s ) g = span { T , T , T , T } , where T i is the invariant generator of degree i in V V θ s . By [15, Lemma 1.3] ( V V θ s ) g does not form an exterior algebra overprimitive generators. Indeed, since T , T , T ∈ Z ( V V θ s ) , the centre of V V θ s , the algebra of skew invariants ( V V θ s ) g is commutative when R isof type G . GM ( q ) and the special exponents of ( R + s )ˇ In what follows we define the special exponents of ( R + s )ˇ when R is oftype B, C, F or G and give as one of our main results: GM ( q ) expressedin terms of these special exponents. Definition 3.3. Let R be of type B, C, F or G . The special exponents of ( R + s )ˇ , h ≤ · · · ≤ h r s are the partition dual to the partition formed by the posi-tive long roots with respect to their special height in the dual root system R ˇ . The list { h i } r s i =1 in each of these types of R is given as B r : { r } , C r : { i } r − i =1 , F : { , } , G : { } . See Figure 1. Theorem 3.4. Let R be of type B, C or F . Let h , · · · , h r s be the special ex-ponents of ( R + s )ˇ . The graded multiplicity of the trivial module V in V V θ s isgiven by GM ( q ) = (1 + q h +1 ) · · · (1 + q h rs +1 ) . When R is of type G , GM ( q ) in V V θ s is GM ( q ) = (1 + q h )(1 + q h +1 ) . Remark 3.5. By Theorem 3.4 and [15, Table 1], when R is of type B, C or F there exists a graded subspace T = h T , · · · , T r s i ⊂ ( V V θ s ) g , such that14 r − r ...h R + s ˇ ( n ) n · · · ... . . . r − r − r − r − h R + s ˇ ( n ) n r − h R + s ˇ ( n ) n h R + s ˇ ( n ) n Figure 1: The partition formed by the coroots in R + s ˇ with respect to their special heights n when R is of type B r (top left), C r (top right), F (bottom left) and G (bottom right). each T i is a primitive generator of degree h i + 1 in V V θ s and ( V V θ s ) g is an exterior algebra over T. 4. The graded multiplicity of V θ s In this section we use the action of the operator Y θ ˇ from the doubleaffine Hecke algebra HH q, t s, l on a subset of the group algebra Q q, t s, l [ P ] , some properties of a subset of R associated with Y θ ˇ and the unitaryproperty of Y θ ˇ with respect to Cherednik’s inner product on Q q, t s, l [ P ] to find GM θ s ( q ) in V V θ s . We set t α = q − k ( α )2 (see [13, Sect. 4] and [2]) for α ∈ R. The following subsection is from [6, 7], but we maintain ( k s , k l ) = , for our unique integer label k ( α ) on the different W -orbits of R. HH q, t s, l Let ˆ W = Π ⋉ W a be the extended affine Weyl group, where W a is theaffine Weyl group and Π is the subgroup of ˆ W which leaves the affineDynkin diagram invariant. The group ˆ W is isomorphic to W ⋉ τ ( P ˇ) , where the subgroup τ ( P ˇ) are translations of the Euclidean space E asso-ciated with the root system R of the Lie algebra g by coweights. Let thedouble affine Hecke algebra HH q, t s, l be the quotient of the field Q q, t s, l byelements { T i } ri =0 , { e ω i } ri =1 and the group Π , modulo the relation ( T i − t i )( T i + t − i ) = 0 , ≤ i ≤ r, where T i , e ω i and Π generate and satisfy therelations on the double affine braid group B associated with HH q, t s, l . Let elements Y be pairwise commuting elements contained in the al-gebra HH q, t s, l such that Y λ = T ( τ ( λ )) , (24)where τ ( λ ) ∈ ˆ W is the translation of E by λ ∈ P ˇ . Then every element H ∈ HH q, t s, l can be uniquely written as H = X w ∈ W h w T w f w , where h w ∈ Q q, t s, l [ P ] and f w ∈ Q q, t s, l [ Y ] . Each h w ∈ Q q, t s, l [ P ] acts natu-rally on Q q, t s, l [ P ] while T w = Q pi =1 T j i when w has a reduced decompo-sition s j i · · · s j p , ≤ j i ≤ r, each T j acts on Q q, t s, l [ P ] via the Demazure-Lusztig operator as T j = t α j s j + ( t α j − t − α j ) (1 − s α j )1 − e α j , and Y acts on Q q, t s, l [ P ] via (24).Let θ be the highest root of R. In what follows we consider the re-duced decomposition of τ ( θ ˇ) ∈ ˆ W in order to determine the action of theoperator Y θ ˇ on Q q, t s, l [ P ] . .2. Operator Y θ ˇ Let ˆ α = α + nδ be an affine root in the affine root system ˆ R, where α ∈ R and δ is the constant function on E. Let G ˆ α be the operatordefined as G ˆ α = t α + ( t α − t − α ) ( s ˆ α − − q − n e α , (25)see [6, (2.17)] and [2, (2.3)]. Let ˆ w = πw ∈ ˆ W , where π ∈ Π and s j p · · · s j , ≤ j i ≤ r, is a reduced decomposition for w ∈ W a . Then by (2 . in [6] T ( ˆ w ) = ˆ wG α ( p ) · · · G α (1) , (26)where α (1) , · · · , α ( p ) is the chain of positive affine roots made negative by ˆ w and α ( i ) = s j · · · s j i − α j i . The following is from [2, Sect. 3]. We can write a reduced decomposi-tion for τ ( θ ˇ) ∈ ˆ W as τ ( θ ˇ) = s − θ + δ s θ = s s j p · · · s j s j s j − · · · s j − p , ≤ j i ≤ r, such that j i = j − i , − p ≤ i ≤ p, where α = − θ + δ is the zeroth-simpleroot of ˆ R. Let ˆ R ( ˆ w ) be the set of positive affine root made negative by ˆ w. Then ˆ R ( τ ( θ ˇ)) = R ( s θ ) ∪ { θ + δ } , where R ( s θ ) = { α ∈ R + | ( α , θ ˇ) > } . Let α ( − p ) , · · · , α (0) , · · · , α ( p ) , α ( p +1) be the chain of affine roots in ˆ R ( τ ( θ ˇ)) , where α ( i ) = s j − p · · · s j i − α j i , − p ≤ i ≤ p and α ( p +1) = θ + δ. Note that α ( − i ) = − s θ α ( i ) , − p ≤ i ≤ p. Thisimplies a symmetry in the lengths of roots about α (0) in the chain of roots α ( − p ) , · · · , α ( p ) in ˆ R ( τ ( θ ˇ)) . By (24) and (26) therefore, Y θ ˇ = τ ( θ ˇ) G θ + δ G α ( p ) · · · G α ( − p ) . (27)We will use the action of Y θ ˇ from (27) on e and e θ s ∈ Q [ q ± , t ± s, l ][ P ] to deduce GM θ s ( q ) in V V θ s from (9).17 .2.1. The action of Y θ ˇ Proposition 4.1. Let R be of type B, C or F. The following holds for the actionof Y θ ˇ on e and e θ s in terms of the double parameter t s, l .Y θ ˇ e = t Ll t Ss e ,Y θ ˇ e θ s = qt − l t − S +2 s e θ s − ( t s − t − s ) t L − l t − S +3 s e . (28) where L = ht l θ, S = ht s θ and t l (respectively t s ) = t α when α ∈ R l (respec-tively R s ) . Remark 4.2. The case of the double parameter t s, l for t in the action of Y θ ˇ in Proposition 4.1 provides a non-trivial and intricate extension of Ba-zlov’s result in [2, Sect. 3]. If we take t s = t l = t, then (28) reduces to theweaker results obtained in (11) and (12) in [2, Sect. 3].We will prove Proposition 4.1 using some properties of the roots in ˆ R ( τ ( θ ˇ)) to determine the actions of the operators G α in (27) on e and e θ s . Following [2], but using the double parameter t s, l instead of t, weintroduce another formula for G α . Let h α = t α − t − α , (29) ε ( α , β ) = ( − , ( α , β ) > , +1 , ( α , β ) ≤ , α, β ∈ E. We denote ε ( α ( i ) , α ( j )ˇ ) by ε i, j , α ( i ) , α ( j ) ∈ R ( s θ ) . Let α ∈ R, µ ∈ P . Then G α e µ = t εα e µ + ε · h α | ( µ , α ˇ) | + ε − X i =1 e µ + iεα , ε = ε ( α , β ) . (30)18n particular G α ( k ) e α ( i ) = t ε k,i α ( k ) e α ( i ) − δ k,i h α ( k ) e , α ( i ) ∈ R s ( τ ( θ ˇ)) ,G α ( k ) e = t α ( k ) e . (31)See [2].By Lemma in [2] and (31) G θ + δ G α ( p ) · · · G α ( − p ) e = t Ll t Ss e . We ap-ply τ ( θ ˇ) , which is identity on Q [ q ± , t ± s, l ] e , to this and obtain Y θ ˇ e in(28). To calculate Y θ ˇ e θ s we first compute G α ( p ) · · · G α ( − p ) e θ s and then ap-ply τ ( θ ˇ) G θ + δ to the result. By Lemma f ) in [2] there exists an index i such that θ s = α ( i ) ∈ R ( s θ ) . We fix this i. Using (31) we obtain G α ( p ) · · · G α ( − p ) e α ( i ) = p Y k = − p t ε i, k α ( k ) e α ( i ) − h s p Y k = i +1 t α ( k ) i − Y k = − p t ε i, k α ( k ) e . (32)We will simplify each of Q pk = − p t ε i, k α ( k ) , Q pk = i +1 t α ( k ) and Q i − k = − p t ε i, k α ( k ) to ob-tain G α ( p ) · · · G α ( − p ) e θ s , in terms of the double parameter t s, l , in Q q, t s, l [ P ] . By Lemma b ) , ( c ) in [2] we have ε i, k = − when α ( i ) , α ( k ) ∈ R s , k = − i,ε i, − i = 1 when α ( i ) ∈ R s ,t ε i, k α ( k ) t ε i, − k α ( k ) = 1 when α ( i ) ∈ R s , α ( k ) ∈ R l , k = 0 . (33)Let α ∈ R, then ( θ s , α ˇ) ∈ { , ± } , (34)see [4, Chap.VI, Sect. 1.3]. Hence, using Lemma a ) and Lemma in [2] p Y k = − p t ε i, k α ( k ) = t α ( − i ) t ε i, α (0) × Y α ( k ) ∈ R s \{ α ( i ) } t − α ( k ) = t − S +2 s t − l . (35)Since θ s is the highest short root in the sequence of short roots in thechain α ( − p ) , · · · , α ( p ) (Lemma e ) in [2]), it follows that the set of roots { α ( j ) | i < j } in the chain α ( − p ) , · · · , α ( p ) , where θ s = α ( i ) , are all long19oots. Therefore p Y k = i +1 t α ( k ) = t p − il . (36)By the symmetry in the lengths of roots about α (0) and since all the shortroots in R ( s θ ) lie between α ( − i ) and α ( i ) , combining (34) and (33) gives i − Y k = − p t ε i, k α ( k ) = t ε i, − i s t ε i, l − i − Y k = − p t ε i, k l × Y α ( k ) ∈ R + s ( θ ) \{ α ( ± i ) } t ε i, k s × i − Y k = − i +1 k =0 t ε i, k l = t − l t − S +3 s − i − Y k = − p t ε i, k l . (37)From (21) in [2] and (37) above (cid:16) i − X k = − p ε i, k (cid:17) = (cid:16) − i − X k = − p ε i, k (cid:17) − S + 2= p + i + 2 − ht α ( i ) . By Lemma f ) in [2] therefore, − i − X k = − p ε i, k = p + i − S − L + 1 . Hence, i − Y k = − p t ε i, k α ( k ) = t p + i − L − Sl t − S +3 s . (38)By Lemma in [2] R ( s θ ) = 2 p + 1 = 2 L + S + 1 . Combining (32), (35), (36)and (38) therefore gives G α ( p ) · · · G α ( − p ) e θ s = t − l t − S +2 s e θ s − h s t L − l t − S +3 s e . (39)Let wτ ( λ ) ∈ ˆ W , where w ∈ W and λ ∈ P ˇ . Then wτ ( λ )( e y ) = q ( λ ,y ) e w ( y ) , (40)20ee [2, Sect. 2.3]. By (25), (29) and (40) τ ( θ ˇ) G θ + δ ( e θ s ) = qt − l e θ s and τ ( θ ˇ) · G θ + δ ( e ) = t l e . Applying τ ( θ ˇ) G θ + δ to (39) then gives Y θ ˇ e θ s in (28). Y θ ˇ on the root system of G Let R be of type G with short and long simple root α and α re-spectively. We denote by β and θ s the short roots α + α and α + α respectively, by γ and θ the long roots α + α and α + 2 α respec-tively. The reflection s θ has the reduced decomposition s s s s s and Y θ ˇ = τ ( θ ˇ) G θ + δ G γ G θ s G θ G β G α . (41)See [2, Sect. 3.8]. Proposition 4.3. The action of Y θ ˇ on e and e θ s in terms of the double param-eter t s, l is given as Y θ ˇ e = t l t s e ,Y θ ˇ e θ s = qt − l t − s e θ s − ( t s − t − s ) t l t − s e . (42) Proof. Apply G α and τ ( θ ˇ) G θ + δ from (41) to e and e θ s using (25), (31)and (40).We will use Y θ ˇ e θ s , Y θ ˇ e and the unitary property of Y θ ˇ with respectto Cherednik’s inner product to find ( e α , for all α in R s . We will thenuse these to calculate (cid:0) χ ( V θ s ) , (cid:1) from which we will deduce GM θ s ( q ) in V V θ s . The next two lemmas deal respectively with the linear combination of α and ( e α , , α ∈ R + s , with respect to the dominant root θ s of R s . Lemma 4.4. Let R be of type B, C, F or G and let θ s be the dominant root of R s . If α ∈ R + s \{ θ s } , then there exists s i · · · s i k ∈ W such that α = s i · · · s i k θ s = θ s − ( α i + · · · + α i k ) , where s i j is the simple reflection along α i j ∈ Φ . roof. Recall the following. (1) The set of dominant weights P + of g is { ω ∈ P | ( ω , α i ˇ) ≥ ∀ α i ∈ Φ } , [4, Chap.VI, Sect. 1.10]. (2) P + ∩ R s = { θ s } , [17, Sect. 2]. (3) ( α , β ˇ) ∈ { , ± } for α ∈ R s , β ∈ R, [4, Chap.VI, Sect. 1.3]. Hence, if α ∈ R + s \{ θ s } , then there exists α i ∈ Φ such that s i ( α ) = α + α i . By Weyl conjugacy there exists s i k · · · s i ∈ W such that θ s = s i k · · · s i α = α + α i + · · · + α i k . The lemma holds since each s i j is an involution. Lemma 4.5. Let R be of type B, C, F or G. If there exists a constant X ∈ Q [ q ± , t ± s, l ] such that the formula ( e λ , 1) = t − ht l λl t − ht s λs X (43) holds when λ is the dominant weight θ s in R s , then the formula holds when λ = α for all α ∈ R + s . Proof. Let β, α ∈ R + s such that α = s i j β = β − α i j . By (26) and (30) T i j e β = s i j G α ij e β = t − α ij e α . Since T is unitary with respect to Cherednik’s inner product, t − α ij ( e α , 1) = t α ij ( e β , . Therefore, if (43) holds for β, then ( e α , 1) = t − ht l αl t − ht s αs X. Apply Lemma 4.4 and the lemma is proved.22 .3. Calculating GM θ s ( q ) Let R be of type B, C, F or G. Recall the unitary property of Y θ ˇ withrespect to Cherednik’s inner product ( , ) on the algebra Q q, t s, l [ P ] implies ( Y θ ˇ e θ s , 1) = ( e θ s , Y − θ ˇ . Using Y θ ˇ e θ s and Y θ ˇ e in (28) and (42) we have ( e θ s , 1) = t − ht l θ s l t − ht s θ s s qt − L − l t − S +2 s − t s − , , where L = ht l θ, S = ht s θ, ht l θ s = L +12 and ht s θ s = S − (Lemma d ) , ( e ) in [2]). Let α in R + s . By Lemma 4.5 ( e α , 1) = t − ht l αl t − ht s αs qt − L − l t − S +2 s − t s − , , for all α ∈ R + s . (44)By [2, Sect. 2.5] and (44) ( e − α , 1) = ( e α , ¯ ∗ = qt ht l α − ( L +1)) l t ht s α − S ) s qt − L − l t − S +2 s − t s − , , for all α ∈ R + s , where ¯ ∗ is the involution on Q q, t s, l [ P ] which acts on q, t s, l , and e α asfollows. ¯ ∗ : t s, l t − s, l , q q − , e α e α . Hence, ( P α ∈ R s e α , , 1) = t s − qt − L +1) l t − S − s − × X α ∈ R + s ( t − ht l αl t − ht s αs + qt ht l α − L +1) l t ht s α − Ss ) . (45)Recall the character of the little adjoint module χ θ s = r s + P α ∈ R s e α , where23 s = R s ∩ Φ) . Therefore, (cid:0) χ θ s , (cid:1) (1 , 1) = r s + ( P α ∈ R s e α , , . (46)Let λ ∈ P + , then h χ λ , ih , i = ( χ λ , , since χ λ and are W − invariant, see[12, (5.1.38)]. By (9) then, GM λ ( − q ) = GM ( − q ) ( χ λ , , (1 , , . We set t α = q − k ( α )2 , see [13, Sect. 4] and [2]. Recall from Sect.(3.1.3) that ( σ k , α ) = ht l α + 2 ht s α, where ht m α = P α i ∈ Φ m λ i , m ∈ { s, l } , when α = P ri =1 λ i α i . Using (45), (46) and ( k s , k l ) = (2 , , GM θ s ( − q ) becomes GM θ s ( − q ) = GM ( − q ) (cid:18) r s + (cid:0) − q − − q L +2 S (cid:1) X α ∈ R + s (cid:0) q ( σ k , α ) + q S + L +2 − ( σ k , α ) (cid:1)(cid:19) . (47)We will simplify this case by case for each R of type B, C, F or G. B r Recall the following in the root system of type B r .R + s = { X i ≤ m ≤ r α m , ≤ i ≤ r } , Φ l = { α , · · · , α r − } , Φ s = { α r } ,θ = α + 2 α + 2 α + · · · + 2 α r . See [4, Plate II]. If α = α i + · · · + α r , then ( σ k , α ) = r + 2 − i. Using GM ( q ) in (15), GM θ s ( q ) in (47) simplifies to GM θ s ( q ) = q + q r . (48)This result in (48) implies that when g is of type B r the only copies ofthe r + 1 -dimensional little adjoint module in its exterior algebra are theexterior powers V V θ s and V r V θ s , which are known to be isomorphic tothe little adjoint module as g -modules (recall the basis of V V θ s and its24oincar´e duality to V r V θ s , [14, 18]). C r Recall that in the root system of type C r R + s = J ∪ K = { X i ≤ m 1) = h K ˇ ( n ) . (50)Table 4 below is from (50) and Table 1.Observe that when r is odd h oK (4 p − 1) = h oK (4 p + 1) = h oK (4 r − p − 1) = h oK (4 r − p − 3) = p,p = 1 , · · · , r − . (51)25 h oK ( n ) h eK ( n )3 1 15 1 17 2 29 2 2 ... ... ... r − r − r − r − r − r − r − r − r r + 1 r − r − r + 3 r − r − ... ... ... r − r − Table 4: The multiplicity of the special heights of the roots of the subset K of R + s in type C r Whereas, when r is even h eK (4 p − 1) = h eK (4 p + 1) = h eK (4 r − p − 1) = h eK (4 r − p − 3) = p,p = 1 , · · · , r − 22 ; h eK (2 r − 1) = r . (52)Therefore, using (21), (49), (50) (51), (52), h J ˇ ( n ) in Table 1 and h K ( n ) inTable 4, GM θ s ( − q ) = GM ( − q ) (cid:18) r s + (cid:0) − q − − q L +2 S (cid:1) X α ∈ J ∪ K (cid:0) q ( σ k , α ) + q S + L +2 − ( σ k , α ) (cid:1)(cid:19) r is odd becomes GM θ s ( − q ) = r − Y p =1 (1 − q p +1 ) ( r − − q r − ) + (1 − q − ) (cid:18) r − X p =1 (cid:0) ( r − p ) × ( q p + q r − − p ) (cid:1) + (cid:0) r − X p =1 ( q p +1 + q r − − (4 p +1) + q p − + q r − − (4 p − + q r − p − + q r − − (4 r − p − + q r − p − + q r − − (4 r − p − ) (cid:1) + r − (cid:0) q r +1 + q r − − (2 r +1) + q r − + q r − − (2 r − + q r − + q r − − (2 r − (cid:1)(cid:19)! , and when r is even GM θ s ( − q ) becomes GM θ s ( − q ) = r − Y p =1 (1 − q p +1 ) ( r − − q r − ) + (1 − q − ) (cid:18) r − X p =1 (cid:0) ( r − p ) × ( q p + q r − − p ) (cid:1) + (cid:0) r − X p =1 ( q p +1 + q r − − (4 p +1) + q p − + q r − − (4 p − + q r − p − + q r − − (4 r − p − + q r − p − + q r − − (4 r − p − ) (cid:1) + r (cid:0) q r − + q r − − (2 r − (cid:1)(cid:19)! . In both cases GM θ s ( − q ) simplifies such that GM θ s ( q ) = (cid:18) r − Y p =1 (1 + q p +1 ) (cid:19) r − X p =1 ( q p − + q p ) . F Recall that in the root system of F , θ = 2 α + 3 α + 4 α + 2 α when Φ l = { α , α } and Φ s = { α , α } . See [4, Plate II]. We use GM ( q ) in (22)27nd ( σ k , α ) , α ∈ R + s , in Table 2 to simplify GM θ s ( q ) in (47) and obtain GM θ s ( q ) = (1 + q )( q + q + q + q ) . G Recall that in G , if Φ l = { α } and Φ s = { α } , then R + s = { α , α + α , α + α } , and θ = 3 α + 2 α . Using (23) GM θ s ( q ) in (47) simplifies to GM θ s ( q ) = (1 + q )( q + q + q ) . (53) Remark 4.6. Given the small sizes of R s in type B and G , the readercan verify GM ( q ) and GM θ s ( q ) in (15), (23), (48) and (53) by decompos-ing the exterior powers of V V θ s in both cases of R into their irreduciblemodules from their highest weight vectors. Theorem 4.7. Let R be of type B, C or F. The graded multiplicity of the littleadjoint module V θ s in its exterior algebra V V θ s is given by GM θ s ( q ) = r s − Y i =1 (1 + q h i +1 ) r s X i =1 ( q h i − (2 h − + q h i ); (54) while when R is of type G GM θ s ( q ) = (1 + q h )( q + q + q ) , where h , · · · , h r s are the special exponents of ( R + s )ˇ . Remark 4.8. Let R be of type B, C or F. Observe the following.1. The dimension of the isotypic component of V θ s in V V θ s , GM θ s (1) =2 r s r s . We note that in the case of the multiplicity of the adjoint module g in its exterior algebra, Kostant [10] proved GM [ V g : g ] (1) = 2 r r. 2. The formula for GM [ V V θs : V θs ] ( q ) in (54) suggests that the isotypic com-ponent of V θ s is a free module over a subring of the invariants in V V θ s . g GM [ V g : g ] ( q ) = r − Y i =1 (1 + q m i +1 ) r X i =1 ( q m i − + q m i ) , where m i , ≤ i ≤ r, are the exponents of R and ( V g ) g = V ( P , · · · ,P r ) , with each primitive invariant generator P i of degree m i + 1 in V g , [2]. DeConcini, Papi and Procesi proved that the isotypic compo-nent of g is a free module of rank r over V ( P , · · · , P r − ) , with basisvectors f i and u i , of degree m i and m i − respectively in V g , in terms of derivations of V g on primitive invariants. (See [8]). Moti-vated by this result of De Concini, Papi and Procesi in V g , we concludethis paper with a conjecture on the isotypic component of V θ s in V V θ s . Conjecture 4.8.1. Let V V θ s be the exterior algebra of the little adjointmodule of a simple Lie algebra g of type B, C or F. Let the skew invari-ants ( V V θ s ) g be the exterior algebra V ( T , · · · , T r s ) , { h j } r s j =1 the set of thespecial exponents of ( R + s )ˇ , where r s is the dimension of the zero weightspace of V θ s , let Der n V V θ s be the set of derivations of V V θ s such thatif ψ ∈ Der n V V θ s , ψ : V m V θ s → V n + m V θ s . The isotypic component of V θ s is a free module over the subring of invariants V ( T , · · · , T r s − ) withbasis vectors a j ∈ V h j V θ s and b j ∈ V h j − (2 h − V θ s , ≤ j ≤ r s , where if v ∈ V θ s , a j ( v ) = i θ s ( v ) T j , for some derivation i θ s ∈ Der − V V θ s . References [1] J. C. Baez, The Octonions , Bull. New. Ser. Am. Math. Soc. (2001)145–205.[2] Y. Bazlov, Graded multiplicities in the exterior algebra , Adv. Math. (2001) 129-153.[3] A. Borel, R. Friedman and J. W. Morgan, Almost commuting elements incompact Lie groups , Mem. Am. Math. Soc. No. 747 (2002).[4] N. Bourbaki, Lie groups and Lie algebras Chapters 4–6 , Springer-Verlag,Berlin, Heidelberg, New York, 2002.295] L. J. Boya, Geometric issues in quantum field theory and string theory ,in ”Geometric and Topological Methods for Quantum Field Theory,”Proceedings of the 2009 Villa de Leyva Summer School, CambridgeUniversity Press (2013) 241–273.[6] I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures ,Ann. of Math. (1995) 191 -216.[7] I. Cherednik, Double affine Hecke algebras , LMS Lecture Note SeriesCambridge University Press, Cambridge (2005) 305 -313.[8] C. De Concini, P. Papi and C. Procesi, The adjoint representation insidethe exterior algebra of a simple Lie algebra , Adv. Math. (2014) 21-46.[9] W. Fulton and J. Harris, Representation theory , Springer-Verlag, NewYork, 1991.[10] B. Kostant, Clifford algebra analogue of the Hopf–Koszul–Samelson theo-rem, the ρ -decomposition C ( g ) = End V ρ ⊗ C ( P ) , and the g -module struc-ture of V g , Adv. Math. (1997) 275–350.[11] G. Krolyi, A. Lascoux and S. O. Warnaar, Constant term identities andPoincar´e polynomials , Trans. Am. Math. Soc., (2015) 6809–6836.[12] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials , Cam-bridge Tracts in Mathematics (2003) Cambridge University Press,Cambridge.[13] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials ,S´´eminaire Bourbaki, 47eme ann´ee, 1994–95, no. 797, 4; Asterisque237 (1996), 189–207.[14] E. Meinrenken, Clifford algebras and Lie groups , Lecture Notes, Univer-sity of Toronto, 2009.[15] D. I. Panyushev, The exterior algebra and ‘spin’ of an orthogonal g-module ,Transform. Groups (2001) 371-396.[16] C. Parker and G. R ¨ohrle, Minuscule representations , in ”Combinatoricsof Minuscule Representations,” Cambridge Tracts in Mathematics,Cambridge University Press (2013) 89–117.3017] J. Stembridge, The partial order of dominant weights , Adv. Math. (1998) no. 2, 340–364.[18] J. Wilson,