Primitive ideals of C_{q}[SL(n)]

Timothy J. Hodges^{∗} University of Cincinnati Cincinnati, OH 45221-0025, U.S.A.

Thierry Levasseur Université de Bretagne Occidentale 29287 Brest Cedex, France

Abstract

The primitive ideals of the quantum group C_{q}[SL(n)] are classified in the case where q is a non-zero complex number which is not a root of unity. It is shown that the orbits in Prim C_{q}[SL(n)] under the action of the character group H ≅ (C^{∗})^{n−1} are parameterized naturally by W × W where W is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals of C_{q}[SL(n)] and symplectic leaves of the associated Poisson algebraic group SL(n, C).

Let q be a non-zero complex number which is not a root of unity. In [3] the authors classified the primitive ideals of the quantum group C_{q}[SL(3)], showing that there is a natural bijection between the primitive ideals of C_{q}[SL(3)] and the symplectic leaves of SL(3, C) for the associated Poisson group structure. Here we generalize this result to the quantum group C_{q}[SL(n)]. Denote by W the associated Weyl group and let l(w) be the length of the element w ∈ W. Let H be the usual maximal torus of SL(n). Then SL(n) has a natural H-invariant Poisson structure and H acts by left translation on the set Symp SL(n) of symplectic leaves of SL(n). The H-orbits in Symp SL(n) are parameterized by the double Weyl group W × W. For more details the reader is referred to the appendix of [3] where a complete description of the symplectic leaves is given using results of Semenov-Tian-Shansky and Lu and Weinstein (see section four for a definition of s(w)).

Theorem 1) Symp SL(n) ≅ ⨆_{w∈W ×W} Symp_{w}SL(n).

∗Partially supported by a grant from the N.S.A.

Key words and phrases: Quantum group, primitive ideal, Poisson Lie group, symplectic leaves.

AMS Mathematics Subject Classification: 17B37, 16W30, 16S80, 16S30, 58F06, 81R50.

2) For each w ∈ W ×W, Symp_{w}SL(n) is a non-empty H-orbit. If Aẇ ∈ Symp_{w}SL(n), then H/Stab_{H}Aẇ is a torus of rank equal to n − 1 − s(w).

3) The dimension of Aẇ is l(w) + s(w).

The group H occurs again in the quantum case as the character group which acts naturally as automorphisms on C_{q}[G]n. The primitive spectrum P rim C_{q}[SL(n)] therefore decomposes into a union of H-orbits. Following ideas of Soibelman [7, 8], we define for each w ∈ W × W a locally closed H-invariant subset P rim_{w} C_{q}[SL(n)] of P rim C_{q}[SL(n)]. The main result of this paper is the following theorem which was conjectured in [3].

Theorem 4.2 1) Prim C_{q}[SL(n)] ≅ ⨆_{w∈W ×W} Prim_{w}C_{q}[SL(n)].

2) For each w ∈ W ×W, Prim_{w}C_{q}[SL(n)] is a non-empty H-orbit. If Pẇ ∈ Prim_{w}C_{q}[SL(n)], then H/Stab_{H}Pẇ is a torus of rank equal to n − 1 − s(w).

3) The Gelfand-Kirillov dimension of C_{q}[SL(n)]/Pẇ is l(w) + s(w).

Since the proof follows the geometry closely, it is useful for the reader to understand in a little more detail the d escription of the symplectic leaves of G = SL(n, C). Let D = G × G, identify G with the diagonal subgroup of D and let G_{r} be the dual group. Denote by p the natural projection G → D/G_{r}. The symplectic leaves of G are precisely the connected components of the inverse images of the left G_{r}-orbits in D/G_{r}. Set Γ = ker p and Ḡ = p(G). Then Γ is a finite subgroup of H and Ḡ = G/Γ is an open subset of D/G_{r}. For each w ∈ W × W, let C_{w} be the image of the corresponding Bruhat cell of D in D/G_{r}. Let B_{w} = C_{w} ∩ Ḡ and let A_{w} = p^{−1}(C_{w}). Since the G_{r}-orbits in C_{w} form a single H-orbit, it follows that the symplectic leaves in A_{w} also form a single H-orbit. The algebras A_{w}, B_{w} and C_{w} defined below may be considered as quantizations of the algebras of functions on A_{w}, B_{w} and C_{w} respectively. The role of the hamiltonian vector fields in the quantized situation is played by the adjoint action of C_{q}[SL(n)] on itself. For this reason, the key result is the description of the adjoint action given in section three.

Much of the inspiration for this work came from work of Soibelman [7, 8]. When q is real, q ≠ 1, the quantum group C_{q}[G] together wit h a natural involution * can be viewed as a deformation of C[K], the algebra of functions on a maximal compact subgroup K of G. Soibelman showed that the irreducible unitary representations of C_{q}[K] correspond to the symplectic leaves of K. A number of key definitions and results are taken from these papers.

These results were first announced at the Symposium on Noncommutative Rings in Durham in July 1992. More recently Joseph has generalized the main result to the case of an arbitrary simply connected semi-simple group.

1 Preliminaries

Denote by g and G the Lie algebra sl(n, C) and the Lie group SL(n, C) respectively. Let q be a non-zero complex number which is not a root of unity and let Q be the subgroup of C^{∗} generated by q. Denote by C_{q}[G] the usual quantization of C[G] and by U_{q}(g) the quantized enveloping algebra. The notation used for the weights and roots of g is as in Bourbaki [1]. All other undefined notation will be as in [3]. The Weyl group W can be identified in a standard way with the symmetric group S_{n} and we shall often make this identification.

Let L(ϖ_{i}) be the finite dimensional simple U_{q}(g)-module corresponding to the fundamental weight ϖ_{i}. Then L(ϖ_{i}) has a basis of the form {v_{wϖ_{i}}}_{w∈W} where each v_{wϖ_{i}} has weight wϖ_{i}. Let l_{−wϖ_{i}} be a dual basis of L(ϖ_{i})^{∗}. For y, t ∈ W, the elements c_{−yϖ_{i},tϖ_{i}}^{ϖ_{i}} of C_{q}[G] are defined by:

∀u ∈ U_{q}(g), c_{−yϖ_{i},tϖ_{i}}^{ϖ_{i}}(u) = l_{−yϖ_{i}}(uv_{tϖ_{i}}).

We set c_{i,y}^{+} = c_{−yϖ_{i},ϖ_{i}}^{ϖ_{i}} and c_{i,y}^{−} = c_{−yw_{0}ϖ_{n−i},w_{0}ϖ_{n−i}}^{ϖ_{n−i}} where w_{0} is the longest element of W.

For each i ∈ {1, …, n − 1}, we define a relation on W by y ≤_{i} w if and only if yϖ_{i} ≥ wϖ_{i}. We say y =_{i} w if and only if y^{−1}wϖ_{i} = ϖ_{i}. If we identify W with S_{n} then this partial order has the following interpretation. If I = {1, …, i}, then y ≤_{i} w if and only if yI ≤ wI (where yI ≤ wI means that if yI = {y_{1} < … < y_{i}} and wI = {w_{1} < … < w_{i}}, then y_{i} ≤ w_{i} for all i). Set W_{i} = Stab_{W} (ϖ_{i}).

The following proposition is proved as in [8, Proposition 3.2]. It is a consequence of the isomorphism L(ϖ_{i}) ⊗ L(ϖ_{j}) ≅ L(ϖ_{j}) ⊗ L(ϖ_{i}) given by the universal R-matrix.

Proposition 1.1 Let ϖ_{i}, ϖ_{j} be fundamental weights and x, y, t ∈ W.Then

c_{−yϖ_{i},tϖ_{i}}^{ϖ_{i}}c_{−xϖ_{j},ϖ_{j}}^{ϖ_{j}} = (q^{2})^{(tϖ_{i},ϖ_{j})−(xϖ_{j},yϖ_{i})}c_{−xϖ_{j},ϖ_{j}}^{ϖ_{j}}c_{−yϖ_{i},tϖ_{i}}^{ϖ_{i}}+ ∑_{{u∈W \atop x<_{j}ux,uy<_{i}y}} g_{u}(q)c_{−uxϖ_{j},ϖ_{j}}^{ϖ_{j}}c_{−uyϖ_{i},tϖ_{i}}^{ϖ_{i}}

for some g_{u}(q) ∈ C.

Definition Let w = (w_{+}, w_{−}) ∈ W × W. Set I_{w} = ⟨c_{i,y}^{ε} | y ≰_{i} w_{ε}⟩ and let E_{w} = {c_{i,w_{+}}^{+}, c_{i,w_{−}}^{−} | i = 1, …, n − 1}.

Theorem 1.2 Let P ∈ Spec C_{q}[G]. Then there exists a unique w ∈ W × W such that P ⊇ I_{w} and P ∩ E_{w} = ∅.

Proof. First observe that for all i = 1, …, n − 1, there exists a w_{i} ∈ W such that c_{i,w_{i}}^{+} ∉ P but c_{i,y}^{+} ∈ P for all y ≰_{i} w_{i}. To see this let w and w′ be two distinct elements of {y ∈ W | c_{i,y}^{+} ∉ P } which are maximal for ≤_{i}. Suppose that w ≠_{i} w′. It follows from Proposition 1.1 that c_{i,w}^{+} and c_{i,w′}^{+} are normal and regular modulo P. Apply Proposition 1.1 to these elements and notice that the exponent of q^{2} is symmetric in w and w′. This implies that (ϖ_{i}, ϖ_{i}) = (wϖ_{i}, w′ϖ_{i}). Hence w =_{i} w′. Thus we may take w_{i} to be any maximal element of {y ∈ W | c_{i,y}^{+} ∉ P }.

The same remark as above concerning the symmetry in w_{i} and w_{j} in Proposition 1.1 shows that

(ϖ_{i}, ϖ_{j}) = (w_{i}ϖ_{i}, w_{j}ϖ_{j}), for all 1 ≤ i, j ≤ n − 1.

It is easy to see that this condition is equivalent to the existence of a unique element w_{+} ∈ W such that w_{+} =_{i} w_{i} for all i.

A similar argument produces an analogous element w_{−}. The element w = (w_{+}, w_{−}) is then the unique element such that P ⊇ I_{w} and P ∩ E_{w} = ∅. □

Corollary 1.3 Let Spec_{w}C_{q}[G] = {P ∈ Spec C_{q}[G] | P ⊇ I_{w} and P ∩ E_{w} = ∅}. Then Spec C_{q}[G] = ⨆_{w∈W ×W} Spec_{w}C_{q}[G].

Let E_{w} be the multiplicatively closed set generated by the images of the elements of E_{w} in C_{q}[G]/I_{w}. Since the elements of E_{w} are normal we may localise with respect to these elements. Denote by A_{w} the localised algebra (C_{q}[G]/I_{w})_{E_{w}}. The analysis of Spec C_{q}[G] reduces to the analysis of Spec A_{w}. Recall that A_{w} ≠ 0 for all w ∈ W × W [3, Theorem 2.2.2].

Another important relation between the c_{i,y}^{+}’s is the following.

Lemma 1.4 Let y, y′ ∈ W and i, j ∈ {1, …, n − 1} such that yW_{i} ∩ y′W_{j} = ∅. Then there exist scalars a_{u,u′} such that

c_{i,y}^{+}c_{j,y′}^{+} = ∑_{u,u′∈W} a_{u,u′}c_{i,u}^{+}c_{j,u′}^{+}

and a_{u,u′} ≠ 0 only if uϖ_{i} + u′ϖ_{j} = yϖ_{i} + y′ϖ_{j} and either u >_{i} y or u′ >_{j} y′.

Proof. This is a consequence of the Plucker relations [5, 1.2]. Suppose that i ≤ j. Then the hypothesis yW_{i} ∩ y′W_{j} = ∅ is equivalent to yI ⊈ y′J where I = {1, …, i} and J = {1, …, j}. Let t ∈ yI\y′J. Apply [5, 1.2] in the following situation:

I_{1} = I, I_{2} = J, J_{1} = yI\{t}, J_{2} = ∅, K = y′J ∪ {t}.

In the notation of that paper, we conclude that:

∑_{K′∪K′′=K} sgn_{q}(J_{1}, K′)sgn_{q}(K′, K′′)sgn_{q}(K′′, J_{2})ξ_{I}^{J_{1}∪K′}ξ_{J}^{K′′} = 0

where #(K′) = 1 and #(K′′) = j. We may pick u, u′ ∈ W such that J_{1} ∪ K′ = uI and K′′ = u′J. In this case ξ_{I}^{J_{1}∪K′} = c_{i,u}^{+} and ξ_{J}^{K′′} = c_{j,u′}^{+}. It is easily verified that if K′ = {t′}, then t′ > t ⇒ u >_{i} y, t = t′ ⇒ u =_{i} y, u′ =_{i} y′ and t′ < t ⇒ u′ >_{j} y′. □

2 The Structure of C_{w}^{H}

Recall some of the notation and basic results from section 2 of [3]. Denote by H the group of one dimensional representations of C_{q}[G]. Notice that H identifies naturally with the usual maximal torus of SL(n, C) via χ ↦ (χ(X_{ij})). The group H acts naturally on C_{q}[G] by a^{χ} = ∑ a_{(1)}χ(a_{(2)}). The elements c_{i,y}^{ε} are eigenvectors for this action and hence there is an induced action of H on A_{w} for all w. Denote by Γ the subgroup of H consisting of representations χ such that χ(X_{ii}) = ±1. Set B_{w} = A_{w}^{Γ}. Then B_{w} contains the elements:

z_{i,y}^{ε} = c_{i,y}^{ε}(c_{i,w_{ε}}^{ε})^{−1}, t_{i} = c_{i,w_{−}}^{−}(c_{i,w_{+}}^{+})^{−1}.

We define C_{w} to be the algebra

C_{w} = C[t_{i}^{±1}, z_{i,y}^{ε} | ε = ±, y <_{i} w_{ε}, i = 1, …, n − 1].

It is shown in [3, 2.6] that there exists a normal element d ∈ C_{w} such that B_{w} = C_{w}[d^{−1}]. The action of H restricts to an action on C_{w} and the fixed ring is the subalgebra generated by the z’s. That is

C_{w}^{H} = C[z_{i,y}^{ε} | ε = ±, y <_{i} w_{ε}, i = 1, …, n − 1].

The monomials in the t_{i}^{±1} form a basis for C_{w} over C_{w}^{H}. Hence C_{w} is an iterated skew Laurent extension of C_{w}^{H}.

Definition Set R^{+}(w) = {α ∈ R | α > 0, w_{+}(α) < 0}, R^{−}(w) = {α ∈ R | α < 0, w_{−}(α) > 0} and set R(w) = R^{+}(w) ∪ R^{−}(w). For α ∈ R define j_{α} = min{i | (α, ϖ_{i}) ≠ 0} and l_{α} = 1 + max{i | (α, ϖ_{i}) ≠ 0}. For α ∈ R^{ε}(w) set z_{α} = z_{j_{α},w_{ε}s_{α}}^{ε}. Notice that α is completely determined by ε, j_{α} and l_{α} and that #R(w) = l(w).

Theorem 2.1 Assume that 1 ≤ j ≤ n − 1, y ≤_{j} w_{ε}. Then for ε = ±,

z_{j,y}^{ε} ∈ C[z_{α} | α ∈ R^{ε}(w) and j_{α} < j or j_{α} = j and w_{ε}s_{α} ≥_{j} y].

Hence C_{w}^{H} = C[z_{α} | α ∈ R(w)].

Proof. We prove the assertion for z_{j,y}^{+}, the proof in the other case being similar. The result is trivially true in the case z_{1,w_{+}}^{+} = 1. By induction we may assume that the result is true for all z_{i,y′}^{+} where i < j or i = j and y <_{j} y′ ≤_{j} w_{+}. First suppose that there exists an i < j such that w_{+}W_{i} ∩ yW_{j} = ∅. Then Lemma 1.4 implies that

z_{j,y}^{+} = ∑_{u,u′} a_{u,u′}z_{i,u}^{+}z_{j,u′}^{+}

where y <_{j} u′ ≤_{j} w_{+} and a_{u,u′} ∈ C. The result then follows by induction. On the other hand suppose that w_{+}W_{i}∩yW_{j} ≠ ∅ for all i < j. In this case we may assume that y =_{i} w_{+} for all i < j. But then it is easily verified that

w_{+}ϖ_{j} − yϖ_{j} = ε_{w_{+}(j)} − ε_{y(j)} = w_{+}(γ)

where γ = ε_{j} − ε_{w_{+}^{−1}y(j)} ∈ R^{+}(w). Thus y =_{j} w_{+}s_{γ} and j_{γ} = j. Hence z_{j,y}^{+} = z_{γ}. □

Definition Define a total ordering ≺ on R(w) by the following rule: if α ∈ R^{ε}(w) and β ∈ R^{η}(w) then

β ≺ α if \{ \begin{matrix}j_{β} < j_{α}, & or \\ j_{β} = j_{α}, η = +, ε = −, & or \\ j_{β} = j_{α}, ε = η, w_{ε}s_{β} >_{j_{α}} w_{ε}s_{α}. \end{matrix}

Notice that when j_{β} = j_{α}, the condition w_{ε}s_{β} >_{j_{α}} w_{ε}s_{α} is equivalent to w_{ε}(l_{β}) > w_{ε}(l_{α}). (This is not quite the same order as that defined in [3, Definition 3.3].)

Theorem 2.2 Let α ≻ β ∈ R(w). Then there exists an a ∈ Q such that z_{α}z_{β} − az_{β}z_{α} ∈ C[z_{γ} | γ ≺ α].

Proof. Let j = j_{α}, i = j_{β}. First consider the case when α, β ∈ R^{+}(w). If i = j, then it follows from the Plucker relations that z_{α} and z_{β} quasi-commute. Suppose on the other hand that i < j. Set x = w_{+}s_{α} and y = w_{+}s_{β}. It follows from Proposition 1.1 that there exist a ∈ Q, and b_{u} ∈ C such that

z_{α}z_{β} − az_{β}z_{α} = ∑_{{x<_{j} ux≤_{j} w \atop uy<_{i}y}} b_{u}z_{j,ux}^{+}z_{i,uy}^{+}.

From the previous theorem we have that

z_{j,ux}^{+} ∈ C[z_{γ} | γ ∈ R^{+}(w) and j_{γ} < j or j_{γ} = j and ws_{γ} ≥_{j} ux],

and z_{i,uy}^{+} ∈ C[z_{γ} | j_{γ} ≤ i]. Since j_{γ} = j and ws_{γ} ≥_{j} ux >_{j} x = ws_{α} implies that γ ≺ α, the result follows in this case. The proof in the case when α and β belong to R^{−}(w) is similar. Now consider the case α ∈ R^{−}(w) and β ∈ R^{+}(w). Set x = w_{−}s_{β} and y = w_{+}s_{α}. From Proposition 1.1, we have that :

z_{α}z_{β} − az_{β}z_{α} = ∑_{x<_{i}ux,uyw_{0}<_{n−j}yw_{0}} b_{u}z_{i,ux}^{+}z_{j,uy}^{−}.

From the previous theorem, we deduce that

z_{i,ux}^{+} ∈ C[z_{γ} | γ ∈ R^{+}(w), γ < β], z_{j,ux}^{−} ∈ C[z_{γ} | γ ∈ R^{−}(w), γ < α].

Hence the result follows easily when α and β belong to distinct R^{ε}(w). □

3 The Adjoint Action

Denote by X_{ij} the usual generators for C_{q}[G]; that is, X_{ij} = c_{−(1,i)ϖ_{1},(1,j)ϖ_{1}}^{ϖ_{1}}. For J ⊂ Z

and k ∈ Z, define δ_{k,J} = \{ \begin{matrix}1 & if k ∈ J, \\ 0 & if k ∉ J; \end{matrix} and set q̂ = q^{2} − q^{−2}.

Lemma 3.1 Let k, l ∈ {1, …, n}, j ∈ {1, …, n − 1} and y ∈ W. Set J = {1, …, j}.

X_{kl}c_{j,y}^{+} = (q^{2})^{δ_{l,J}} [(q^{2})^{−δ_{k,yJ}} c_{j,y}^{+}X_{kl} + (δ_{k,yJ} − 1) ∑_{a<k,a∈yJ} q̂(−q^{2})^{d(a,y,k)}c_{j,(a,k)y}^{+}X_{al}]

where d(a, y, k) = #{x ∈ yJ | a < x < k}.

Proof. This follows from the identities [6, 4.5.1,5.1.2]. Let C_{q}[M(m)] be the usual quantized algebra of functions on m × m matrices and let D_{ij} be the quantum minor obtained by deleting the i-th row and the j-th column. Then the formulae in [6, 4.5.1,5.1.2] may be combined to give:

X_{kl}D_{rm} = (q^{2})^{1−δ_{l,m}}[(q^{2})^{δ_{k,r}−1}D_{rm}X_{kl} − δ_{k,r} ∑_{a=1}^{k−1} q̂(−q^{2})^{k−a−1}D_{al}X_{al}].

Interpreting this formula for the c_{j,y}^{+} yields the desired result. □

The adjoint action of an element a ∈ C_{q}[G] on b ∈ A_{w} is given by (ad a)b = ∑ a_{(1)}bS(a_{(2)}) where S is the antipode. We denote ad X_{kl} by ad_{k,l}. It follows from the description of the action of Γ that B_{w} is invariant under the adjoint action. The following results imply in particular that C_{w} and C_{w}^{H} are also invariant and that the adjoint action on C_{w} is locally finite.

Theorem 3.2 a) Let α ∈ R^{+}(w), j = j_{α}, l = l_{α}, and y = w_{+}s_{α}. Then

ad_{k,m}z_{α} = \{ \begin{matrix}(q^{2})^{δ_{k,w_{+}J} −δ_{k,yJ}} z_{α} & if k = m \\ a_{k,m,y}z_{j,(m,k)y}^{+} & if w_{+}(l) ≤ m < k ≤ w_{+}(j), k ∉ yJ, m ∈ yJ \\ 0 & otherwise \end{matrix}

where a_{k,m,y} ∈ C^{∗}. In particular ad_{w_{+}(j),w_{+}(l)} z_{α} ∈ C^{∗}.

b) Let β ∈ R^{−}(w), j = j_{β}, l = l_{β}, and y = w_{−}s_{β}. Then

ad_{k,m}z_{β} = \{ \begin{matrix}(q^{2})^{δ_{m,w_{−}J} −δ_{m,yJ}} z_{β} & if k = m \\ b_{k,m,y}z_{j,(m,k)y}^{−} & if w_{−}(l) ≤ k < m ≤ w_{−}(j), k ∈ yJ, m ∉ yJ \\ 0 & otherwise \end{matrix}

where b_{k,m,y} ∈ C^{∗}. In particular ad_{w_{−}(l),w_{−}(j)} z_{β} ∈ C^{∗}.

Proof. a) Lemma 3.1 implies that

c_{j,w_{+}}^{+}X_{kl} = (q^{2})^{−δ_{l,J} +δ_{k,w_{+}J}} X_{kl}c_{j,w_{+}}^{+}

modulo I_{w}. Therefore we deduce from the lemma that

X_{kl}z_{j,y}^{+} = (q^{2})^{δ_{k,w_{+}J}} [(q^{2})^{−δ_{k,yJ}} z_{j,y}^{+}X_{kl} + (δ_{k,yJ} − 1) ∑_{a<k,a∈yJ} q̂(−q^{2})^{d(a,y,k)}z_{j,(a,k)y}^{+}X_{al}]

Since ad_{k,m}z_{j,y}^{+} = ∑_{l} X_{kl}z_{j,y}^{+}S(X_{lm}) and ∑_{b} X_{ab}S(X_{bd}) = δ_{a,d} we deduce that

ad_{k,m}z_{j,y}^{+} = \{ \begin{matrix}(q^{2})^{δ_{k,w_{+}J} −δ_{k,yJ}} z_{j,y}^{+} & if k = m \\ −q̂(q^{2})^{δ_{k,w_{+}J}} (−q^{2})^{d(m,y,k)}z_{j,(m,k)y}^{+} & if m < k, m ∈ yJ, k ∉ yJ, (m, k)y ≤_{j} w_{+} \\ 0 & otherwise. \end{matrix}

In the case where y = ws_{α} and j = j_{α}, it is easily seen that the condition w_{+}(l) ≤ m < k ≤ w_{+}(j), k ∉ yJ, m ∈ yJ is equivalent to m < k, m ∈ yJ, k ∉ yJ, (m, k)y ≤_{j} w_{+}. A similar argument proves part (b). □

Corollary 3.3 Let r ∈ N.

1. If α ∈ R^{+}(w) and y = w_{+}s_{α}, then

ad_{k,m}z_{α}^{r} = \{ \begin{matrix}az_{α}^{r} & if k = m \\ a′z_{α}^{r−1} & if k = w_{+}(j_{α}) and m = w_{+}(l_{α}) \\ 0 & unless w_{+}(l_{α}) ≤ m < k ≤ w_{+}(j_{α}), k ∉ yJ, m ∈ yJ \\ & or k = m \end{matrix}

where a ∈ Q and a′ ∈ C^{∗}.

2. If β ∈ R^{−}(w) and y = w_{−}s_{β}, then

ad_{k,m}z_{β}^{r} = \{ \begin{matrix}bz_{β}^{r} & if k = m \\ b′z_{β}^{r−1} & if k = w_{−}(l_{β}) and m = w_{−}(j_{β}) \\ 0 & unless w_{−}(l_{β}) ≤ k < m ≤ w_{−}(j_{β}), k ∈ yJ, m ∉ yJ \\ & or k = m \end{matrix}

where b ∈ Q and b′ ∈ C^{∗}.

Proof. The proof is analogous to that of [3, 3.9.2]. □

It follows from Theorem 2.2 that the algebra C_{w}^{H} is generated by monomials of the form

z = ∏_{γ∈R^{+}(w)} z_{γ}^{m_{γ}} ∏_{β∈R^{−}(w)} z_{β}^{m_{β}}

where the product is taken over the γ’s according to the inverse ordering given by ≺ on R^{+}(w) and the second product is taken over the β’s according to the ordering ≺ on R^{−}(w). Such monomials will be called standard monomials. In order to define an ordering on such expressions we define a new ordering on R(w).

Definition. Define a new total ordering ≺′ in the following way: if α ∈ R^{ε}(w) and β ∈ R^{η}(w), then

α ≻′ β if \{ \begin{matrix}ε = − η = + & or \\ ε = η, w_{ε}(l_{α}) < w_{ε}(l_{β}), & or \\ ε = η, w_{ε}(l_{α}) = w_{ε}(l_{β}), & j_{α} < j_{β}. \end{matrix}

The following proposition follows easily from the definition.

Proposition 3.4 Suppose α, β ∈ R^{ε}(w).

1. If α ≻′ β and α ≻ β, then j_{α} ≥ j_{β} and w_{ε}(l_{α}) < w_{ε}(l_{β}).

2. If α ≻′ β and α ≺ β, then j_{α} < j_{β} and w_{ε}(l_{α}) ≤ w_{ε}(l_{β}).

For a standard monomial z = ∏_{α∈R(w)} z_{α}^{m_{α}} of the form given above, we define

Supp(z) = {α ∈ R(w) | m_{α} ≠ 0}

and MSupp(z) to be the largest element of Supp(z) under the ordering ≺′. Define the degree of a non-zero standard monomial, deg(z) to be the element (m_{α})_{α} of N^{R(w)} and define e_{α} by (e_{α})_{β} = δ_{α,β}.

Proposition 3.5 Let α ∈ R(w). Set (k, m) = (w_{+}(j_{α}), w_{+}(l_{α})) if α ∈ R^{+}(w), (k, m) = (w_{−}(l_{α}), w_{−}(j_{α})) if α ∈ R^{−}(w).

1. If MSupp(z) = α, then ad_{k,m}z is a scalar multiple of a standard monomial and deg(ad_{k,m}z) = deg(z) − e_{α}.

2. If MSupp(z) ≺′ α, then ad_{k,m}z = 0.

Proof. Let z be a monomial such that MSupp(z) ⪯′ α. Then we may write

z = (∏_{γ} z_{γ}^{m_{γ}} ) z_{α}^{m_{α}} (∏_{β} z_{β}^{m_{β}} )

where the products are ordered as described above. We prove the result by induction on e(z) = #(Supp(z)\{α}). We first consider the case where α ∈ R^{−}(w). We prove that if k′ ≤ k then

ad_{k′,m}z = δ_{k′,k} (∏_{γ} ad_{k,k}z_{γ}^{m_{γ}} ) ad_{k,m}z_{α}^{m_{α}} (∏_{β} ad_{m,m}z_{β}^{m_{β}} )

Suppose that e(z) = 0. Then z = z_{α}^{m_{α}} and the result is true by Corollary 3.3. Now assume that the proposition is true for all monomials y such that MSupp(y) ⪯′ α and e(y) < e(z). We consider three cases.

(1) Suppose that z = ξy, where ξ = z_{γ}^{m_{γ}} and γ ∈ R^{+}(w). Write ad_{k′,m}z = ∑_{s} ad_{k′,s}ξad_{s,m}y. Suppose that ad_{k′,s}ξ and ad_{s,m}y are non-zero for some s. From ad_{k′,s}ξ ≠ 0 it follows that s ≤ k′ ≤ k. By induction we conclude from ad_{s,m}y ≠ 0 that s = k = k′.

(2) Suppose that z = ξy, ξ = z_{γ}^{m_{γ}} where γ ∈ R^{−}(w) and γ ≺ α. Proposition 3.4 implies that j_{γ} ≤ j_{α} and k = w_{−}(l_{α}) < w_{−}(l_{γ}). Write ad_{k′,m}z = ∑_{s} ad_{k′,s}ξad_{s,m}y. Suppose that ad_{k′,s}ξ and ad_{s,m}y are nonzero for some s. Then k′ ≤ s. If s ≠ k′, we have w_{−}(l_{γ}) ≤ k′ < s ≤ w_{−}(j_{γ}). Hence w_{−}(l_{γ}) ≤ k′ ≤ k, a contradiction. Thus s = k′ ≤ k and by induction k ≤ k′ = s ≤ m. Hence k′ = s = k.

(3) Suppose that z = yξ where ξ = z_{β}^{m_{β}} where β ∈ R^{−}(w) and β ≻ α. Then Proposition 3.4 implies that j_{β} > j_{α} and k = w_{−}(l_{α}) ≤ w_{−}(l_{β}). Write ad_{k′,m}z = ad_{k′,s}yad_{s,m}ξ and suppose that ad_{k′,s}y ≠ 0 and ad_{s,m}ξ ≠ 0. Then s ≤ m. If s < m, we must have

w_{−}(j_{α}) = m ∉ yJ = {w_{−}(1), …, w_{−}(j_{β} − 1), w_{−}(l_{β}}.

But this contradicts j_{α} < j_{β}. Hence s = m. Then ad_{k′,m}y ≠ 0 and induction yields the result. The case when α ∈ R^{+}(w) is similar. □

Using induction on the degree of a standard monomial we then obtain:

Theorem 3.6 Let z be a standard monomial. Then there exists a monomial f in the X_{ij} such that

1. (adf)z ∈ C^{∗};

2. (adf)z′ = 0 if z′ is a standard monomial with deg z′ ≺′ deg z.

The theorem implies in particular that the standard monomials are linearly independent. This fact together with the commutation relations of Theorem 2.2 implies the following.

Corollary 3.7 The algebra C_{w}^{H} is an iterated skew polynomial ring in l(w) variables.

As we noted above, C_{w} is a skew Laurent extension of C_{w}^{H} in rk G variables. Thus C_{w} is a deformation of the algebra of functions on the variety C_{w} ≅ C^{l(w)} × (C^{∗})^{rk G} described in [3, A.2]. Similarly the algebra B_{w} = C_{w}[d^{−1}] is a deformation of the algebra of functions on the open subset B_{w} of C_{w}.

Lemma 3.8 Let k, l ∈ {1, …, n} and let j ∈ {1, …, n − 1}. Then

ad_{k,l}t_{j} = δ_{kl}q^{2ψ_{kj}}t_{j}

where ψ_{kj} = δ_{k,w_{+}(J)} − δ_{k,w_{−}(J)}.

Proof. It follows from Lemma 3.1 that X_{kl}c_{j,w_{+}}^{+} = (q^{2})^{δ_{l,J} −δ_{k,w_{+}(J)}}c_{j,w_{+}}^{+}X_{kl}. Similarly X_{kl}c_{j,w_{−}}^{−} = (q^{2})^{δ_{l,J} −δ_{k,w_{−}(J)}}c_{j,w_{−}}^{−}X_{kl}. The result is then clear. □

Denote by Ψ the n × (n − 1) matrix (ψ_{kj}). It induces a map Ψ : Z^{n−1} → Z^{n}.

Theorem 3.9 The subalgebra C[t_{i}^{±1} | i = 1, … n − 1] is invariant and diagonalisable under the adjoint action. The subalgebra of ad-invariant elements is

C[t_{i}^{±1} | i = 1, … n − 1]^{ad} = C[t^{m} | Ψm = 0]

which is the algebra of functions on a (commutative) torus with rank dim ker_{h^{∗}}(w_{+}w_{−}^{−1} − Id).

Proof. It is clear from the lemma that ad_{i,i}t^{m} = q^{2(Ψm)_{i}}t^{m}. Thus the subalgebra of ad-invariants is the linear span of the monomials t^{m} for m ∈ ker Ψ. Identify h^{∗} with the usual subspace of V = ⟨ε_{1}, …, ε_{n}⟩C. Let Φ be the linear map on V given by the matrix [Ψ : 0] with respect to this basis. Then Φ(ε_{i}) = (w_{+} − w_{−})(ε_{1} + ⋯ + ε_{i}). Hence rank(ker Ψ) = dim(ker Φ) − 1 = dim(ker_{h^{∗}}(w_{+} − w_{−})) = dim(ker_{h^{∗}}(w_{+}w_{−}^{−1} − Id)). □

4 Primitive Spectrum of C_{q}[G]

We are now able to deduce the main result.

Definition. Let K be a subgroup of Z^{n−1} such that Z^{n−1} = ker Ψ ⊕ K. Set

Cẇ = C_{w}^{H}[t^{m} | m ∈ K]

and let Bẇ = Cẇ[d^{−1}].

It follows from [3, 2.7] and Corollary 3.7 that Cẇ is a localization of an iterated skew polynomial ring. Hence Cẇ and Bẇ are integral domains. Furthermore C_{w} ≅ Cẇ ⊗ C_{w}^{ad}, B_{w} ≅ Bẇ⊗C_{w}^{ad} as C_{q}[G]-module algebras and C_{w}^{ad} = C[t_{i}^{±1} | i = 1, … n−1]^{ad} as described in Theorem 3.9.

Set s(w) = codim(ker_{h^{∗}}(w_{+}w_{−}^{−1} − Id)), (c.f. [3, A.2]).

Theorem 4.1 B_{w} ≅ Bẇ ⊗ C_{w}^{ad} and Bẇ is a simple algebra. The center of B_{w} is C_{w}^{ad} and the ideals of B_{w} are generated by their intersection with the center. Thus Spec B_{w} ≅ Spec C_{w}^{ad} and Prim B_{w} ≅ Prim C_{w}^{ad}. The primitive ideals of B_{w} are maximal and form an H-orbit. If P ∈ Prim B_{w}, then H/Stab_{H}P is a torus of rank equal to rank G − s(w) and the Gelfand-Kirillov dimension of B_{w}/P is l(w) + s(w).

Proof. Let P_{e} be the ideal of B_{w} generated by {t^{m} − 1 | m ∈ ker Ψ}. Then P_{e} is an adinvariant ideal and Bẇ ≅ B_{w}/P_{e} as C_{q}[G]-module algebras. Suppose that Bẇ is not simple. Then by ‘going up’ [4, 10.5.15] Bẇ contains a proper non-zero ad-invariant ideal, I. In this case I ∩Cẇ is a proper non-zero ad-invariant ideal of Cẇ. Let CK = C[t^{m} | m ∈ K]. Since CK is diagonalizable under the adjoint action and Cẇ ≅ C_{w}^{H} ⊗ CK as C_{q}[G]-modules, it follows that the socle of Cẇ under the adjoint action is given by

Soc Cẇ ≅ Soc C_{w}^{H} ⊗ CK ≅ C ⊗ CK.

Thus Soc Cẇ is a direct sum of distinct one-dimensional submodules each generated by a unit. Since I ∩ Cẇ must intersect nontrivially at least one summand of the socle, it must contain a unit, a contradiction. Thus Bẇ is simple.

Let σ : H → H/Γ be the map σ(h) = h^{2}Γ. Then the action of H on C[t_{i}^{±1} | i = 1, … n − 1] factors through σ and the induced action of H/Γ is the natural action of a torus on its algebra of functions. This, together with Theorem 3.9 implies the assertions concerning the H-action. The assertion concerning the Gelfand-Kirillov dimension follows from a slight generalization of [4, 8.2.10]. □

Notice that Theorem 4.1 implies that all prime ideals of C_{q}[G]^{Γ} are completely prime. This result may also be deduced from [2] where it is proved that all prime ideals of C_{q}[G] are completely prime.

Recall that a Noetherian C-algebra A is said to satisfy the Dixmier-Moeglin condition if the following are equivalent for P ∈ Spec A: a) P is primitive; b) P is rational (the center of the ring of fractions of A/P is C); c) P is locally closed in Spec A.

Theorem 4.2 1) Prim C_{q}[G] ≅ ⨆_{w∈W ×W} Prim_{w}C_{q}[G].

2) For each w ∈ W × W, Prim_{w}C_{q}[G] is a non-empty H-orbit. If Pẇ ∈ Prim_{w}C_{q}[G], then H/Stab_{H}Pẇ is a torus of rank equal to rank G − s(w).

3) The Gelfand-Kirillov dimension of C_{q}[G]/Pẇ is l(w) + s(w).

4) C_{q}[G] satisfies the Dixmier-Moeglin condition.

Proof. Notice first that C_{q}[G] satisies the nullstellensatz, [4, 9.1.8]. Therefore a) ⇒ b) and c) ⇒ a) for any P ∈ Spec C_{q}[G]. Let P ∈ Spec_{w}C_{q}[G] be rational and let Q = P ∩C_{q}[G]^{Γ}.

Since C_{w}^{ad} is central in A_{w}, P A_{w} ∩ C_{w}^{ad} must be a maximal ideal of C_{w}^{ad}. Thus QB_{w} is maximal by Theorem 4.1. It then follows from ‘going up’ [4, 10.5.15] that any prime ideal strictly containing P must contain the product of the elements of E_{w}. Hence P is locally closed and primitive. Thus C_{q}[G] satisfies the Dixmier-Moeglin condition. Since Γ acts transitively on the primitive ideals of C_{q}[G] lying over Q and H acts transitively on the maximal ideals of B_{w}, it follows easily that H acts transitively on P rim_{w} C_{q}[G]. Since the action of H is algebraic H/Stab_{H}P must be a torus whose rank is rk G − s(w) by Theorem 4.1. The assertion concerning the Gelfand-Kirillov dimension follows from [4, 8.2.9]. □

Comparing this result with Theorem A.3.2 of [3], we see that there exists an Hequivariant bijection β : Prim C_{q}[G] ⟶ Symp G such that β(Prim_{w}C_{q}[G]) = Symp_{w}G for all w ∈ W × W and such that dim β(P) = GKdim C_{q}[G]/P for all P ∈ Prim C_{q}[G]. Thus Conjecture 1 of [3] is true.

References

[1] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris 1981.

[2] K. R. Goodearl and E. S. Letzter, Prime factors of the coordinate ring of quantum matrices, preprint.

[3] T. J. Hodges and T. Levasseur, Primitive ideals of C_{q}[SL(3)], Comm. Math. Phys., to appear.

[4] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley Interscience, New York 1987.

[5] M. Nuomi, H. Yamada and K. Mimachi, Finite dimensional representations of the quantum group GL_{q}(n + 1, C) and the zonal spherical functions on U_{q}(n)\U_{q}(n + 1), preprint.

[6] B. Parshall and J. Wang, Quantum Linear Groups, Memoirs Amer. Math. Soc., No. 439 , 1991.

[7] Ya. S. Soibelman, Irreducible representations of the function algebra on the quantum group SU(n) and Schubert cells, Soviet Math. Dokl., 40 (1990), 34-38.

[8] Ya. S. Soibelman, The algebra of functions on a compact quantum group, and its representations, Leningrad Math. J. 2 (1991), 161-178.