QHWM of the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators

In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators. We also realize them in terms of the irreducible quasifinite highest weight modules of the Lie algebras of infinite matrices with finitely many nonzero diagonals and its classical Lie subalgebras of types B, C and D.

1 Introduction 2 Lie algebras gℓ [m] ∞ and its classical Lie subalgebras In this section we will give a description of the Lie algebra of infinite matrices with finitely many nonzero diagonals gℓ [m] ∞ and its classical Lie subalgebras of B, C and D types. We will follow the notation in Sect. 1 of [KWY].
Denote R m = C[u]/(u m+1 ) the quotient algebra of the polynomial algebra C[u] by the ideal generated by u m+1 (m ∈ Z ≥0 ). Let 1 be the identity element in R m . Denote by gℓ [m] ∞ the complex Lie algebra of all infinite matrices (a i,j ) i,j∈Z with only finitely many nonzero diagonals with entries in R m . Denote E i,j the infinite matrix with 1 at (i, j)-entry and 0 elsewhere. There is a natural automorphism ν of gℓ [m] ∞ given by (2.1) Let the weight of E i,j be j − i. This defines the principal Z-gradation gℓ where J = i≤0 E i,i . The Z-gradation of the Lie algebra gℓ [m] ∞ extends to gℓ [m] ∞ by putting the weight of R m to be 0. In particular, we have the triangular decomposition, ∞ ) * 0 , we let where j ∈ Z and 0 ≤ i ≤ m. Let L( gℓ [m] ∞ , λ) be the irreducible highest weight gℓ [m] ∞ -module with highest weight λ. The a λ (i) j are called labels and c i are the central charges of L( gℓ [m] ∞ , λ). Consider the vector space R m [t, t −1 ] and take the basis v i = t −i , i ∈ Z over R m . Now consider the following C-bilinear form on R m [t, t −1 ]: (2.6) Denote byb ∞ ) the Lie subalgebra of gℓ [m] ∞ which preserves the bilinear form B − (·, ·) (resp. B + (·, ·)). We havē ∞ ) given by the restriction of the 2-cocycle (2.2), defined in gℓ [m] ∞ . The subalgebra b [m] ∞ (resp. b [m] ∞ ) inherits from gℓ [m] ∞ the principal Z-gradation and the triangular decomposition (see [KR2] and [FKRW] for notation), Note that the Lie algebrab [m] ∞ is isomorphic to b [m] ∞ via the isomorphism that sends the elements u k E i,j − (−u) k E −j,−i to u k E i,j + (−1) 1+i+j (−u) k E −j,−i , i, j ∈ Z, k ∈ Z + . Their Cartan subalgebra coincides. In particular, when m = 0, we have the usual Lie subalgebra of gℓ ∞ , denoted b ∞ (see [K]) (resp.b ∞ , see [W]). Given λ ∈ (b ∞ , λ) the irreducible highest weight module over b [m] ∞ with highest weight λ.
where i ∈ N and 0 ≤ j ≤ m. The b λ ∞ , λ). Now consider the following C-bilinear form on R m [t, t −1 ]: (2.8) Denote byc [m] ∞ the Lie subalgebra of gℓ [m] ∞ which preserves the bilinear form C( , ). We havē Denote by c [m] ∞ =c [m] ∞ ⊕ R m the central extension ofc [m] ∞ given by the restriction of the 2-cocycle (2.2), defined in gℓ [m] ∞ . This subalgebra inherits from gℓ [m] ∞ the principal Z-gradation and the triangular decomposition, (see [KWY] and [K] for notation) In particular when m = 0, we have the usual Lie subalgebra of gℓ ∞ , denoted by c ∞ . (2.10) where j ∈ N and i = 0, · · · , m. For later use, it is convenient to put c h The c λ ∞ , λ).

The Lie algebra S q,N
Consider C[z, z −1 ] the Laurent polynomial algebra in one variable. We denote S a q the associative algebra of quantum pseudo-differential operators. Explicitly, let T q denote the operator on C[z, z −1 ] given by where q ∈ C × = C\{0}. An element of S a q can be written as a linear combination of operators of the form z k f (T q ), where f is a Laurent polynomial in T q . The product in S a q is given by Denote S q the Lie algebra obtained from S a q by taking the usual commutator. Take S ′ q := [S q , S q ]. It follows: Let N be a positive integer. As of this point, we shall denote by M at N A the associative algebra of all N × N matrices over an algebra A and E ij the standard basis of M at N C.
Let S a q,N = S a q ⊗ M at N C be the associative algebra of all quantum matrix pseudodifferential operators, namely the operators on C N [z, z −1 ] of the form In a more useful notation, we write the matrix of pseudodifferential operators as linear combinations of elements of the form z k f (T q )A, where f is a Laurent polynomial, k ∈ Z and A ∈ M at N C. The product in S a q,N is given by Let S q,N denote the Lie algebra obtained from S a q,N with the bracket given by the conmutator, namely: (3.4) Taking the trace form tr 0 ( j c j w j ) = c 0 , and denoting by tr the usual trace in M at M C, we obtain, by a general construction (cf. Sec. 1.3 in [KR1]), the following 2-cocylce in S q,N denote the central extension of S ′ q,N by a one-dimensional center CC corresponding to the twococycle ψ. The bracket in S q,N is given by The elements z k T m q E ij (k ∈ Z, m ∈ Z, i, j ∈ {1, · · · , N }) form a basis of S q,N . We define the weight on S q,N by (3.8) This gives the principal Z -gradation of S a q,N , S q,N and S q,N , An anti-involution σ of S a q,N is an involutive anti-automorphism of S a q,N , ie, σ 2 = Id, σ(ax+by) = aσ(x) + bσ(y) and σ(xy) = σ(y)σ(x), for all a, b ∈ C and x, y ∈ S a q,N . From now on we will assume that |q| = 1.
The following Corolary was proved in [BB].
Corollary 3.1. Let σ = σ A,B,c,r,N be given by (3.10b) Then σ = σ A,B,c,r,N extends to an anti-involution on S a q,N which preserves the principal Zgradation.
Remark 3.2. For each n < N , a Z-gradation preserving anti-involution can be constructed in a similar way. In [BB] all anti-involutions of S a q,N preserving the Z-gradation were classified. Let where σ A,B,c,r,N is the anti-involution given by Corolary 3.1. , where ǫ is 1 or −1, and 1 is the matrix c with c i = 1 except for the fixed points that are 1 or −1, which keep their sign.
Thus, the anti-involution is of the following form: where ǫ = ±1, r ∈ C × . For simplicity, denote S σ,N q,N the Lie subalgebras of S q,N fixed by minus σ ǫ,r,N .
We will denote δ m,even = 1 if m is even 0 otherwise .
S σ,N q,N inherits a Z-gradation from S q,N since σ preserves the principal Z−gradation of S a q,N . Thus S σ,N q,N = ⊕ j ∈ Z (S σ,N q,N ) j . We can now give a description of (S σ,N q,N ) j . By the division algorithm, let We denote again ψ the restriction of the 2-cocycle in (3.5) to S σ,N q,N . Denote by S σ,N q,N the central extension of S σ,N q,N by CC corresponding to the 2-cocycle ψ. S σ,N q,N is a Lie subalgebra of S q,N by definition.
4 Parabolic subalgebras of S σ,N q,N In order to characterize the quasifiniteness of the highest weight modules (HWMs) of S σ,N q,N we will study the structure of its parabolic subalgebras and apply general results for quasifinite representations of Z-graded Lie algebras obtained in [KL]. We refer to [KL] for proofs and details. Let g be a Z-graded Lie algebra over C, where g i is not necessarily of finite dimension. Let g ± = ⊕ j>0 g ±j . A subalgebra p of g is called parabolic if it contains g 0 ⊕ g + as a proper subalgebra, that is where p j = g j for j ≥ 0, and p j = 0 for some j < 0.
Following [KL], we assume the following properties of g: Cf. [KL] Lemmas 2.1 and 2.2.
In [BKLY], for the case of the central extension of the Lie algebra of matrix differential operators on the circle, the existance of some parabolic subalgebras p such that p j = 0 for j ≫ 0 was observed. Having in mind that example, they give the following definition.
We will also require the following condition on g.
(P3) If p is a nondegenerate parabolic subalgebra of g, then there exists an nondegenerate element a such that p a ⊆ p.
Now take a parabolic subalgebra p of S σ,N q,N . Observe that for each j ∈ N, j = kN + p with 0 ≤ p ≤ N − 1, we have Let us check conditions (P1),(P2) and (P3) for S σ,N q,N . Observe that (P1) is immediate from the definition of ( S σ,N q,N ) 0 . (P2) follows from computing the following bracket and the particular case To prove (P3), let f (w), g(w) be Laurent polynomials in the variable w with f ∈ I i −j , and let p −j with j = kN + p as in (4.1). Let us first consider 1 ≤ i ≤ N − p. If p = 0, suppose i = (N + 1)/2. We compute the following bracket and if p is odd for I Thus, since C[w, w −1 ] is a principal ideal domain, we have proven the following Let [k] denote the integer part of a number k. Now we have the following important Proposition.
, part (a) follows. Let p be any parabolic subalgebra of S σ,N q,N , using Lemma 2.1 and 2.2 in [KL], we get p −1 = 0. Then using (a) and p d ⊆ p (for any nonzero d ∈ p −1 ) we obtain (b). Finally, part (c) follows by computing the commutators Summarizing, we have proven that the following properties are satisfied by S σ,N q,N : (P3) if p is a nondegenerate parabolic subalgebra of S σ,N q,N , then there exists a nondegenerate element a, such that p a ⊆ p.
5 Characterization of quasifinite highest weight modules of S σ,N q,N Now, we begin our study of quasifinite representations over the lie algebras S σ,N q,N . Let g be a Lie algebra. For a Lie algebra g, The Verma module over g is defined as usual: where C λ is the one-dimensional (g 0 ⊕ g + )-module given by h → λ(h) if h ∈ g 0 , g + → 0, and under the action of g is induced by the left multiplication in U (g). Here and further U (g) stands for the universal enveloping algebra of the Lie algebra g. Any highest-weight module V (g, λ) is a quotient module of M (g, λ). The irreducible module L(g, λ) is the quotient of M (g, λ) by the maximal proper graded module. We shall write M (λ) and L(λ) in place of M (g, λ) and L(g, λ) if no ambiguity may arise. Consider a parabolic subalgebra p = ⊕ j ∈ Z p j of g and let λ ∈ g * 0 be such that λ| g 0 ∩[p,p] = 0. Then the (g 0 ⊕ g + )-module C λ extends to a p-module by letting p j act as 0 for j < 0, and we may construct the highest-weight module called the generalized Verma module. Clearly all these heighest weight modules are graded.
From now on we will consider λ ∈ g * 0 . By Theorem 2.5 in [K], we have the following.
Consider g = S σ,N q,N . A functional λ ∈ ( S σ,N q,N ) * 0 is described by its labels, with l ∈ Z ≥0 , 1 < i ≤ [N/2] + δ N,even and the central charge c = λ(C). We shall consider the generating series Recall that a quasipolynomial is a linear combination of functions of the form p(x)q αx , where p(x) is a polynomial and α ∈ C. That is, it satisfies a nontrivial linear differential equation with constant coefficients. We also have the following well-known proposition.
Proposition 5.2. Given a quasipolynomial P , and a polynomial B( If the polynomial B is even we call P an even quasipolynomial. As a result, one has the following characterization of quasifinite heighest weight modules over g. 2. There exist quasipolynomials P i and even quasipolynomials P ǫ N such that (n ∈ N) These conditions can be rewritten as follows: Let us first analyze (5.8). Multiplying both sides by x −l and adding over l ∈ Z, we get . The equivalence of (1) and (2) for this case follows from the fact that (5.2) holds since it also holds multiplying both sides of this formula by x m i with m i ≥ 0. Due to Proposition 5.2, the existence of the qualispolynomials P i (x) for Once again, (5.3) holds since it also holds multiplying both sides of this formula by x p with it is easy to see that if α = 0 is a root of b ǫ (x), then 1/α is also a root of b ǫ (x). Now we can apply Proposition 5.2 and due to the relationship between the roots of B and b in this proposition it follows that the B ǫ (x) corresponding to our b ǫ (x) is an even polynomial. This implies that the quasipolynomial P ǫ N (x) such that P ǫ N (n) = △ N,n for n = 0 and P (0) = 2c is even, finishing the proof for this case. Finally, let us analyze 5.10 for the case N even. Proceding similarly as with the previous equation, we multiply by (x n − x −n ) and add over n ∈ Z ≥0 . Using the fact that △ 1+N/2,l = −△ N/2,−l we obtain Making use once again of Proposition 5.2 we prove that P N/2 (x) such that P N/2 (n) = △ N/2,n − △ N/2+1,n for n ∈ Z is an even quasipolynomial.
Given a quasifinite irreducible highest weight S σ,N q,N -module V by Theorem 5.3, we have that there either exist a quasipolynomials P i (x) (for 1 ≤ i ≤ [N/2] − δ N,even ) satisfying (5.5), an even quasipolynomials P ǫ N (x) verifying (5.6), and if N is even, P N/2 (x) satisfying (5.7). We will write with p j,N (x) and p j,N/2 (x) (respectively, q j,N (x) and q j,N/2 (x)) even (respectively, odd) polynomials, p e,i (x) a polynomial, e, e + j and e − j distinct complex numbers. Also, cosh q (x) = q x +q −x 2 and sinh q = q x −q −x 2 . The last two expresions in (5.12) are unique up to a sign of e + j or a simultaneous change of signs of e − j and the respective q j (x). We call e + j (respectively, e − j ), even type (respectively odd type) exponents of V with multiplicities p j (x) (respectively, q j (x)). As in [KWY], we denote e + the set of even type exponents e + j with multiplicity p j (x) and by e − the set of odd type exponents e − j with multiplicity q j (x). Therefore, the pair (e + ; e − ) uniquely determines V . Analogously for the first formula, we call e i the exponents of V with multiplicities p e,i (x), and we denote e the set of exponents e i with multiplicity p e,i (x). We will denote this module by L( S σ,N q,N ; e; e + ; e − ).

Interplay between S σ,N q,N and the infinite rank classical Lie algebras
In this section we will discuss the connection between S σ,N q,N and the Lie algebra of infinite matrices with finitely many nonzero diagonals over the algebra of truncated polynomials and its classical Lie subalgebras. Let O be the algebra of all holomorphic functions on C × with the topology of uniform convergence on compact sets, and denote Let R be an associative algebra over C and denote R ∞ a free R-module with a fixed basis {v j } j∈Z and denote R m = C[t]/(t m+1 ) where m ∈ Z + .
We consider the vector space S O a q,N spanned by the quantum pseudo differential operators (of infinite order) of the form z k f (T q )E i,j , where f ∈ O. The bracket in S q,N extends to (S q,N ) O . In a similar fashion, we define a completion (S σ,N q,N ) O of S σ,N q,N consisting of all pseudo differential operators of the form and the opposite diagonal which are Lie algebra homomorphisms. A restriction of these homomorphisms of Lie algebras to S σ,N q,N gives a family of homomorphisms of Lie algebras ϕ For each s ∈ C and k ∈ Z, set s,k,ǫ }. Choose a branch of log q. Let τ = log q/2πi. Then any s ∈ C × is uniquely written as s = q a , with a ∈ C/τ −1 Z. Fix s = (s 1 , · · · , s M ) ∈ C M such that if each s i = q a i , we have Proposition 6.1. Given s and m as above, we have the exact sequence of Z−graded Lie algebras, provided that |q| = 1 : Proof. The injectivity part is clear from (6.1). For the sake of simplicity, we will prove the surjectivity of ϕ [ m] s for the case M = 1, m = m and s = s = q a . We will make use of the well-known fact that for every discrete sequence of points of C and a non-negative integer m there exists f (w) ∈ O having prescribed values of its first m derivatives at these points. By conditions (6.2) and |q| = 1 and since a / ∈ Z/2 we have that {q (n−1)/2+j+a } and {q −(n−1)/2−j−a } are discrete and disjoint sequences of points in C. Therefore we can find f ∈ O such that every element t j E a,b is in the image, finishing the proof.
Proof. It is a straightforward computation restricting the formulaφ is defined for any s ∈ C. However, for a ∈ Z/2, it is no longer surjective. These cases are described by the following propositions. Proposition 6.3. For a = 1, we have the following exact sequence of Lie algebras: Proof. We will first prove the case ǫ = 1. The homomorphism ϕ ∞ introduced in [?] is surjective. The anti-involution of S q,N defined in (3.12) transfers, via ϕ [m] s , to an antiinvolution ω : gℓ Therefore, the Lie algebra of −σ fixed points in S q,N , explicitly, S σ,N q,N , maps surjectively to the Lie algebra of −ω fixed points in gℓ where i = q i N + r i and j = q j N + r j , with 1 ≤ r i ≤ N and 1 ≤ r j ≤ N . As a result of the surjectivity described, it is enough to show that ω is conjugated by an automorphism T ′ of gℓ ∞ . To that end, we define Proposition 6.4. If a = 1/2 and N is odd, we have the following exact sequence of Lie algebras: where g ≃b Proof. If ǫ = 1, replace in the proof of the last proposition ω by Therefore, the Lie algebra of −σ fixed points in S q,N , explicitly, S σ,N q,N , maps surjectively to the Lie algebra of −ω fixed points in gℓ [m] ∞ . Consequently, it is enough to see that ω is conjugated by an automorphism T of gℓ [m] ∞ to the anti-involution definingb It is easy to check that this extends to an automorphism of the algebra gℓ [m] ∞ that conjugates ω to the anti-involution definingb (6.11) where i = q i N + r i and j = q j N + r j , with 1 ≤ r i ≤ N y 1 ≤ r j ≤ N . The automorphism of gℓ [m] ∞ for this case is D = T • T ′ , where T is the same that in the previous case and we have (6.12) with a = q a N + r a and b = q b N + r b , for 0 ≤ r a ≤ N − 1 and 0 ≤ r b ≤ N − 1. It is easy to see that ω is conjugated by D to the anti-involution definingb Proposition 6.5. If a = 1/2 and N is even, we have the following exact sequence of Lie algebras: Proof. This proof follows the same steps as last proposition. If ǫ = 1, because ω is the same as before, it is enough to replace T by The rest of the proof is the same for this case. If ǫ = −1, ω is the same formula as in the last proposition, so it is enough to replace T ′ by 14) where a = q a N + r a and b = q b N + r b , with 0 ≤ r a ≤ N − 1 and 0 ≤ r b ≤ N − 1.
Remark 6.6. (a) By an abuse of notation, for a = 1 and a = 1/2, in view of Propositions 6.3 to 6.5, we will denote again ϕ (b) Recall that ν was defined in (2.1). If ǫ = 1, for arbitrary a ∈ Z, the image of S σ,N q,N under the homomorphism ϕ ∞ ) if N is odd. As a consequence, it is enough to study the cases a = 1 and a = 1/2. The same conclusions can be obtained for ǫ = −1. Therefore, we will only consider a = 1 and a = 1/2 throughout this paper.
Given vectors s = (s 1 , · · · , s M ) = (q a 1 , · · · , q a M ) ∈ C M and m = (m 1 , · · · , m M ) ∈ Z M such that if a i ∈ Z, then a i = 1; if a i ∈ Z + 1/2 then a i = 1/2; and a i − a j / ∈ Z + τ −1 Z for i = j. Combining this with Propositions 6.1 to 6.5, we obtain a surjective Lie algebra homomorphism if a i = 1/2 and N is odd, if a i = 1/2 and N is even or a i = 1.
if a i = 1/2 and N is odd, if a i = 1/2 and N is even, 7 Realization of quasifinite highest weight modules of S σ,N q,N In this section g [m] will be gℓ [m] ∞ or one of its classical subalgebras. The proof of the following Proposition is standard (cf. [K]) Proposition 7.1. The g [m] −module L(g [m] , λ) is quasifinite if and only if all but finitely many Given m = (m 1 , · · · , m M ) ∈ Z M ≥0 , take a quasifinite λ i ∈ (g [m i ] ) * 0 for each 1 ≤ i ≤ M , and let L(g [m i ] , λ i ) be the corresponding g [m i ] -module. Let λ = (λ 1 , · · · , λ M ). Then the tensor product , λ) can be regarded as a S σ,N q,N −-module via the homomorphism ϕ [ m] s and will be denoted by L [ m] s ( λ). We shall need the following results.
Proposition 7.2. Let V be a quasifinite S σ,N q,N -module. Then the action of S σ,N q,N on V naturally extends to the action of ( S σ,N q,N ) O u on V , for any u = 0.
Proof. The proof is similar to the proof of Proposition (4.3) of [KL], replacing B = adD 2 − k 2 by the following: Theorem 7.3. Let V be a quasifinite g [ m] -module, which is regarded as a S σ,N q,N -module via the homomorphism ϕ [ m] s . Then any S σ,N q,N -submodule of V is also a g [ m] -submodule. In particular, the S σ,N q,N -module L Proof. Let W be a S σ,N q,N -submodule of V . Due to the fact that W is a quasifinite S σ,N q,N -module as well, by Proposition 7.2 it can be extended to ( S σ,N q,N ) O u for u = 0. As a result of (6.15), the map ϕ ) u is surjective for any u = 0. Therefore, W is invariant with respect to all members of the principal gradation of (g [ m] ) u with u = 0. Since g [ m] coincides with its derived algebra, this proves the theorem. Now, we will proceed to show that all the irreducible quasifinite S σ,N q,N -modules can be realized as some L [ m],k,ǫ s ( λ), for some m ∈ Z M ≥0 and s ∈ C M , with s i = q a i such that a i − a j / ∈ Z + τ −1 Z for i = j. For simplicity, we will consider the case M = 1 to calculate the generating series We will introduce the following notation Making use of Theorem (5.3), take an irreducible quasifinite weight S σ,N q,N -module V with central charge c and generating series △ i (x), P ǫ N (x) an even quasipolynomial such that P ǫ N (n) = △ N,n for n = 0 and P ǫ N (0) = −2c, (7.3) P i (x) a quasipolynomial such that P i (n) = △ i,n − △ i+1,n (7.4) for 1 < i ≤ [N/2]−δ N,even are quasipolynomials, and when N is even, P N/2 (x) an even quasipolynomial such that P N/2 (n) = △ N/2,n − △ N/2+1,n . (7.5) We write where p j,N (x) and p j,N/2 (x) (respectively, q j,N (x) and q j,N/2 (x)) are even ( ∞ or one of its classical subalgebras. Then with 1 < i ≤ [N/2] + δ N,even , and ∞ , λ) regarded as a S σ,N q,N -module is isomorphic to L( S σ,N q,N ; e; e + ; e − ), where (a) The exponents e are −1/2+a−l and 1/2−a+l with l ∈ Z and their respective mutiplicities are Proof. If 1 < i ≤ [N/2] − δ N,even , combining the formulas of proposition 6.2 with (2.5) and (7.7), we have that Then, (l−1)N +i q −n(−1/2+a−l) Making use of the definitions of multiplicities and exponents for the quasipolynomial P i (x) in (7.6), we complete the proof for (a).
If i = N , as before, considering (7.8) and (2.5), we obtain Shifting the index l to l − 1 in the first sum, we get Since 2 sinh q (n/2)η u (a − l, n) = (n log q) u u! (q n(a−l) + (−1) u q −n(a−l) ) and making use of the definitions of multiplicities and exponents for the quasipolynomials P ǫ N (x) in (7.6), we finish the proof for (b).
If N even, following the same steps as in the proof of (a), we have Then, splitting the sums according to the parity of u, we get the multiplicities and exponents expected.
(c) Moreover, if N is even, for i = N/2 the exponents e + and e − are l ≥ 0 and their multiplicities are, if l ≥ 1, and (7.14) Proof. Consider first the case ǫ = 1. By Remark 6.6, part (a), we have that the embeddinĝ ∞ is in fact the embedding given by proposition 6.2 composed by T −1 , where T is the automorphism of gℓ [m] ∞ defined in (6.13). If 1 < i ≤ [N/2] − δ N,even , using (7.7) for the embedding in this case, we get (−n log q) u u! q nl t r E (l−1/2)N +i,(l−1/2)N +i + m u=1 η u (−1/2, n)c u .
In order to complete the proof, we study the parity of u and split the sums accordingly. As a result of the definitions of multiplicities and exponents for the quasipolynomials P ǫ N (x) in (7.6), we find the exponents and multiplicities expected for (b).
For i = N/2, we follow the same steps as in (a), but with an adequate change of variables in l. This way, we get Lastly, by splitting this sums according to the parity of u, we find the exponents and multiplicities expected, finishing the proof for this case.
∞ is in this case the embedding given by proposition 6.2 composed by D = T • T ′ , where T ′ is the automorphism of gℓ [m] ∞ defined in (6.14). The results for this embedding are the same as for ǫ = 1. and for l = 0, and P ǫ l,N (0) = −2c 0 for i = N .
To complete the proof, we study the parity of u and split the sums accordingly. As a result of the definitions of multiplicities and exponents for the quasipolynomials P ǫ N (x) in (7.6), we find the exponents and multiplicities expected.
Consider now ǫ = −1. The embeddingφ ∞ is in this case the embedding given by proposition 6.2 composed by D = T • T ′ , where T ′ is the automorphism of gℓ [m] ∞ defined in (6.12). Proceeding in an analogous way as for the case ǫ = 1, we get the expected results.
where † represents c or d depending on whether g [m] is c Proof. By Remark 6.6, part (a), we have that the embeddingφ [m] s : S σ,N q,N −→ d [m] ∞ is in fact the embedding given by proposition 6.2 composed by T −1 , where T is the automorphism of gℓ [m] ∞ defined in (6.13).
In order to finish the proof, we study the parity of u and split the sums accordingly. As a result of the definitions of multiplicities and exponents for the quasipolynomials P ǫ N (x) in (7.6), we find the exponents and multiplicities expected for (b).
Studying the parity of u and splitting the sums accordingly, we find the exponents and multiplicities expected for this case, finishing the proof.
Consider now ǫ = −1. The embeddingφ ∞ is in this case the embedding given by proposition 6.2 composed by D = T • T ′ , where T ′ is the automorphism of gℓ [m] ∞ defined in (6.8). Proceeding in an analogous way as for case ǫ = 1, we obtained the expected results.
Pick a representative s in an equivalence class S such that s = q if the equivalence class lies in Z and s = q 1/2 if the equivalence class lies in Z + 1/2. Let S = {q a , q a+t 1 , q a+t 2 , . . . } be such an equivalence class. Take t 0 = 0 and let m = max s∈S {deg p s,i , deg p ǫ s,N , deg q ǫ s,N , deg p s,N/2 , deg q s,N/2 }. It is easy to see that if a = 1 or a = 1/2, then t i ∈ Z. Now, we will associate S to a g [m] -module L [m] s (λ S ) in one of the following ways.
• If a /