Extremal projectors in the semi-classical case

Sophie Chemla

September 16, 1997

Abstract Using extremal projectors, Zhelobenko solved extremal equations in a generic Verma module of a complex semi-simple Lie algebra. We will solve similar equations in the semi-classical case. Our proof will be geometric. In the appendix, we give a factorization for the extremal projector of the Virasoro algebra in the semi-classical case.

Résumé En utilisant les projecteurs extrémaux, Zhelobenko a résolu des équations extrémales dans le cas d’un module de Verma générique d’une algèbre de Lie semi-simple complexe. Nous résolvons des équations similaires dans le cas semi-classique. Notre preuve sera géométrique. Dans l’appendice, nous donnons une factorisation du projecteur extrémal pour l’algèbre de Virasoro dans le cas semi-classique.

1 Introduction

Let 𝔤 be a complex semi-simple finite dimensional Lie algebra, 𝔥 a Cartan subalgebra of 𝔤 and Δ the root system associated to 𝔥. We will write Δ^{+} (respectively Δ^{−}) for the set of positive (respectively negative) roots of Δ

and put ρ = \frac{1}{2} ∑_{γ∈Δ^{+}}γ. We will denote by B = (α_{1}, …, α_{l}) the set of simple roots. Let 𝔤_{γ} be the root space associated to the root γ. We put

𝔫 = ⊕_{γ∈Δ^{+}}𝔤_{γ}, 𝔟 = 𝔥 ⊕ 𝔫 , 𝔫_{−} = ⊕_{γ∈Δ^{+}}𝔤_{−γ}.

Let R(𝔥) be the field of rational functions on 𝔥^{∗}. One introduces the algebra U′(𝔤) = U(𝔤) ⊗_{S(𝔥)}R(𝔥). Let us consider the generic Verma module

V = \frac{U′(𝔤)}{U′(𝔤)𝔫}. Zhelobenko ([Z1]) showed that V^{𝔫} = R(𝔥)1_{+} (where 1_{+} =

1 + U′(𝔤)𝔫). The decomposition V = 𝔫^{−}V ⊕ R(𝔥)1_{+} defines a projector p onto R(𝔥)1_{+} called the extremal projector. Inspired by a work of Asherova, Smirnov and Tolstoy ([A-S-T]), Zhelobenko ([Z1]) showed that p factorizes into elementary projectors. Let (γ_{1}, …, γ_{m}) be a normal ordering on the positive roots. Introduce the following notations:

p_{α} = ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!f_{α,k}} e_{−α}^{k}e_{α}^{k}

f_{α,0} = 1,

if k > 0, f_{α,k} = (h_{α} + ρ(h_{α}) + 1) … (h_{α} + ρ(h_{α}) + k)

(e_{δ} being the root vector associated to the root δ and h_{δ} the coroot). We have p = p_{γ_{1}} … p_{γ_{m}} ([Z1]). Let w = s_{1} … s_{j} be a reduced decomposition of w ∈ W (with s_{k} = s_{β_{k}}, β_{k} a simple root ). Put w_{i} = s_{1} … s_{i}. The roots γ_{i} = w_{i−1}(β_{i}) (w_{0} = 1) are pairwise distinct and

Δ_{w} = {α ∈ Δ_{+} | w^{−1}(α) < 0} = {γ_{1}, …, γ_{j}}.

Put 𝔫_{w} = ⊕_{α∈Δ_{w}} 𝔤_{α}. In [Z2], Zhelobenko gives an explicit description of V^{𝔫_{w}}.

We will establish similar results for the symmetric algebra (the so-called semi-classical case).

Let us consider the analytic manifold (𝔤/𝔫)^{∗}. We will endow it with the following coordinate system ((e_{−α})_{α∈Δ_{+}}, (h_{α_{i}})_{i∈[1,l]}). We will call U_{δ} the open subset of (𝔤/𝔫)^{∗} defined by the equation h_{δ} ≠ 0. We define Φ_{δ} to be the following rational map of U_{δ}:

∀λ ∈ U_{δ}, Φ_{δ}(λ) = exp (\frac{e_{−δ}(λ)}{h_{δ}(λ)}e_{δ}) · λ

where the dot denotes natural action of 𝔫 on (𝔤/𝔫)^{∗}. By composition, Φ_{δ} defines an algebra morphism of A(U_{δ}) which we call π_{δ}. We put

U_{w} = U_{γ_{1}} ∩ … ∩ U_{γ_{j}}.

We will denote by P(U_{w}) (respectively A(U_{w})) the set of regular functions (respectively analytic functions) on U_{w} and we will write P(U_{w})^{𝔫_{w}} (respectively A(U_{w})^{𝔫_{w}}) the set of invariant functions of P(U_{w}) (respectively A(U_{w})) under the action of 𝔫_{w}. We prove the following result:

Theorem The algebra morphism π_{w} = π_{γ_{1}} ∘ … ∘ π_{γ_{j}} does not depend on the reduced expression of w. It establishes an isomorphism between

C_{w} = \{f ∈ A(U_{w}) | \frac{∂f}{∂e_{−γ_{1}}} = … \frac{∂f}{∂e_{−γ_{j}}} = 0\}

and A(U_{w})^{𝔫_{w}}. Moreover π_{w} sends C_{w} ∩ P(U_{w}) onto P(U_{w})^{𝔫_{w}}.

Let N_{w} be the connected simply connected group whose Lie algebra is 𝔫_{w}. The main ingredient of the proof will be the choice of a point in each N_{w}-orbit lying in U_{w} in accordance with the following proposition :

Proposition Let λ be in U_{w}. The point Φ_{γ_{j}}Φ_{γ_{j−1}} … Φ_{γ_{1}}(λ) is the unique point of the orbit N_{w} · λ whose coordinates e_{−γ_{1}}, …, e_{−γ_{j}} vanish.

In the appendix, we shall give a factorization for the extremal projector of the Virasoro algebra in the semi-classical case. Note that the non commutative case is still open. It is very different from the semi-simple case because the Virasoro algebra does not admit any normal ordering.

Notations

Along all this article 𝔤 will denote a complex semi-simple finite dimensional Lie algebra and 𝔥, Δ, Δ_{+}, Δ_{−}, 𝔫, 𝔫_{−}, B = (α_{1}, …, α_{l}) will be as above. Denote by W the Weyl group associated to these choices and \overline{w} its longest element. Let γ be an element of Δ^{+} and let h_{γ} be the unique element of [𝔤_{γ}, 𝔤_{−γ}] such that γ(h_{γ}) = 2. If e_{γ} is in 𝔤_{γ}, then there exists a unique e_{−γ} such that (h_{γ}, e_{γ}, e_{−γ}) is a sl(2)-triple. If α and β are two roots, we set [e_{α}, e_{β}] = C_{α,β}e_{α+β} with the convention that C_{α,β} is zero if α + β is not a root.

The ordering (γ_{1}, …, γ_{m}) on the positive roots is normal if any composite root is located between its components. Thus for all positive roots γ_{i}, γ_{j}, γ_{k}, the equality γ_{k} = γ_{i}+γ_{j} implies i ≤ k ≤ j or j ≤ k ≤ i. There is a one to one correspondence between normal orderings and reduced expression of \overline{w} ([Z2]). Let us recall it. Denote by s_{i} the reflexion with respect to a simple root β_{i}. If \overline{w} = s_{1} … s_{m}, then (β_{1}, s_{1}(β_{2}), …, s_{1} … s_{i−1}(β_{i}), …, s_{1} … s_{m−1}(β_{m})) are in normal ordering.

If V is a vector space, S(V) will be the symmetric algebra of V. Lastly, if P is in S(V), S(V)_{P} will be the localization of S(V) with respect to {P^{n} | n ∈ ℕ}.

Acknowledgments

I would like to thank M. Duflo for suggesting to me to study the semiclassical case and for helpful discussions. I am grateful to the referee for indicating to me that the main theorem could be proved in a geometric way.

2 Extremal equations in (𝔤/𝔫)^{∗}

We consider (𝔤/𝔫)^{∗} as an analytic manifold. We endow it with the following coordinate system ((e_{−α})_{α∈Δ_{+}} , (h_{α_{i}})_{i∈[1,l]}). If δ is a positive root, we will denote by U_{δ} the open subset of (𝔤/𝔫)^{∗} defined by the equation h_{δ} ≠ 0. If U is an open subset for the Zariski topology, we will write A(U) for the algebra of analytic functions on U and P(U) for the algebra of regular functions on U. We will define Φ_{δ} to be the following rational map of U_{δ}

∀λ ∈ U_{δ}, Φ_{δ}(λ) = exp (\frac{e_{−δ}(λ)}{h_{δ}(λ)}e_{δ}) · λ.

By composition, Φ_{δ} defines an algebra morphism of A(U_{δ}) which we call π_{δ}. We will denote by X_{δ} the natural action of e_{δ} on P(U_{δ}). Remark that X_{δ} is a derivation. If f is in P(U_{δ}), we have

(∗) π_{δ}(f) = ∑_{k=0}^{∞}(−1)^{k} \frac{e_{−δ}^{k}}{k!h_{δ}^{k}} X_{δ}^{k} · f

where e_{−δ} denotes the multiplication by e_{−δ}. The operator π_{δ} is the commutative analog of the Zhelobenko’s elementary projector.

Let w = s_{1} … s_{j} be a reduced decomposition of w ∈ W (with s_{k} = s_{β_{k}}, β_{k} ∈ B). Put w_{i} = s_{1} … s_{i}. The roots γ_{i} = w_{i−1}(β_{i}) (w_{0} = 1) are pairwise distinct and

Δ_{w} = {α ∈ Δ_{+} | w^{−1}(α) < 0} = {γ_{1}, …, γ_{j}}.

An ordering in Δ_{w} is called normal if it coincides with the initial segment of some normal ordering in Δ^{+} (that is compatible with one of the reduced expression of \overline{w}). Note that (γ_{1}, …, γ_{j}) is a normal ordering of Δ_{w}. Put

U_{w} = ⋂_{δ∈Δ_{w}}U_{δ}.

We have

P(U_{w}) = (\frac{S(𝔤)}{S(𝔤)𝔫})_{h_{γ_{1}}...h_{γ_{j}}} = S (\frac{𝔤}{𝔫})_{h_{γ_{1}}...h_{γ_{j}}} .

We will denote by N_{w} the connected and simply connected group whose Lie algebra is 𝔫_{w} = ⊕_{α∈Δ_{w}}𝔤_{α}. We will start by proving the following proposition.

Proposition 2.1 Let λ be in U_{w}. The point Φ_{γ_{j}}Φ_{γ_{j−1}} … Φ_{γ_{1}}(λ) is the unique point of the orbit N_{w} ·λ whose coordinates e_{−γ_{1}}, …, e_{−γ_{j}} vanish. In particular Φ_{γ_{j}}Φ_{γ_{j−1}} … Φ_{γ_{1}} does not depend on the normal ordering on Δ_{w}.

Proof of the proposition 2.1: Complete (γ_{1}, …, γ_{j}) into a normal ordering on the positive roots (γ_{1}, …, γ_{m}). 𝔤/𝔫 is endowed with the basis (e_{−γ_{1}}, …, e_{−γ_{m}}, h_{α_{1}}, …, h_{α_{l}}). Let (e_{−γ_{1}}^{∗}, …, e_{−γ_{m}}^{∗}, h_{α_{1}}^{∗}, …, h_{α_{l}}^{∗}) be the dual basis. We will often identify the point a_{γ_{1}}e_{−γ_{1}}^{∗} + … + a_{γ_{m}}e_{−γ_{m}}^{∗} + b_{1}h_{α_{1}}^{∗} + … + b_{l}h_{α_{l}}^{∗} with its coordinates (a_{γ_{1}}, …, a_{γ_{m}}, b_{1}, …, b_{l}). Let us see that there is a unique point in N_{w}·λ whose coordinates e_{−γ_{1}}, …, e_{−γ_{j}} vanish. Assume that there are two such points f = (0, …, 0, a_{γ_{j+1}}, …, a_{γ_{m}}, b_{1}, …, b_{l}) and f′ = (0, …, 0, a′_{γ_{j+1}}, …, a′_{γ_{m}}, b′_{1}, …, b′_{l}). Then there exist complex numbers (t_{1}, …, t_{j}) such that exp(t_{1}e_{γ_{1}} + … + t_{j}e_{γ_{j}}) · f = f′. One can show easily the following equalities

\begin{matrix}e_{γ_{l}} · e_{−γ_{k}}^{∗} = −C_{γ_{l},−γ_{k}}e_{−γ_{l}−γ_{k}}^{∗} \\ e_{γ_{l}} · h_{−α_{i}}^{∗} = −h_{−α_{i}}^{∗}(h_{γ_{l}})e_{−γ_{l}}^{∗}. \end{matrix}

From these equalities, one deduces easily that the term in e_{−γ_{1}}^{∗} of exp(t_{1}e_{γ_{1}} + …+t_{j}e_{γ_{j}})·(0, …, 0, a_{γ_{j+1}}, …, a_{γ_{m}}, b_{1}, …, b_{l}) is −t_{1}f(h_{γ_{1}}). As f is in U_{w}, we get t_{1} = 0. We reproduce the same reasoning to show that t_{2}, t_{3}, …, t_{j} are zero. So that we have proved that the two points f and f′ coincide. It is not difficult to deduce from the normal ordering property that Φ_{γ_{i}} sends the point (x_{γ_{1}}, …, x_{γ_{m}}, y_{1}, …, y_{l}) to a point (x′_{γ_{1}}, …, x′_{γ_{i−1}}, 0, x′_{γ_{i+1}}, …, x′_{γ_{m}}, y_{1}, …, y_{l}) and that it sends the point (0, …, 0, x_{γ_{i}}, …, x_{γ_{m}}, y_{1}, …, y_{l}) to a point (0, …, 0, x′_{γ_{i+1}}, …, x′_{γ_{m}}, y_{1}, …, y_{l}). So that Φ_{γ_{j}}Φ_{γ_{j−1}} … Φ_{γ_{1}}(λ) is the unique point of N_{w} · λ whose coordinates e_{−γ_{1}}, …, e_{−γ_{j}} vanish. This finishes the proof of the proposition 2.1.

As a consequence of the previous proposition, we may write Φ_{w} for the operator Φ_{γ_{j}}Φ_{γ_{j−1}} … Φ_{γ_{1}}. The algebra homomorphism defined by Φ_{w} on A(U_{w}) will be denoted by π_{w}. Using the proposition 2.1, we will give a geometric proof of the following result.

Theorem 2.2 1) If \overline{𝔫}_{w} denotes the linear hull of (e_{−α})_{α∈Δ_{w}}, one has Kerπ_{w} = \overline{𝔫}_{w}A(U_{w}).

2) The operator π_{w} is the projector onto A(U_{w})^{𝔫_{w}} with kernel \overline{𝔫}_{w}A(U_{w}) and its restriction to P(U_{w}) is the projector onto P(U_{w})^{𝔫_{w}} with kernel \overline{𝔫}_{w}P(U_{w}).

3) The operator π_{w} establishes an isomorphism Π_{w} between

C_{w} = \{f ∈ A(U_{w}) | \frac{∂f}{∂e_{−γ_{1}}} = … \frac{∂f}{∂e_{−γ_{j}}} = 0\}

and A(U_{w})^{𝔫_{w}}. Moreover Π_{w} sends C_{w} ∩ P(U_{w}) onto P(U_{w})^{𝔫_{w}}. If f is in A(U_{w})^{𝔫_{w}}, Π_{w}^{−1}(f) is the restriction of f to the subvariety of equations e_{−γ_{1}} = … = e_{−γ_{j}} = 0.

Proof of the theorem 2.2: From the previous proposition, the inclusion \overline{𝔫}_{w}A(U_{w}) ⊂ Kerπ_{w} is clear. Moreover, a standard reasoning shows that

A(U_{w}) = C_{w} ⊕ \overline{𝔫}_{w}A(U_{w}).

Then one sees easily that Kerπ_{w} ∩ C_{w} = \{0\}. So that we have \overline{𝔫}_{w}A(U_{w}) = Kerπ_{w}.

Let us now show that Imπ_{w} = A(U_{w})^{𝔫_{w}} and that π_{w} is a projector. Let α be in Δ_{w}. For any f in A(U_{w}) and any λ in U_{w}, we have

(e_{α} ∘ π_{w}) (f)(λ) = \frac{d}{dt}f (Φ_{γ_{j}} … Φ_{γ_{1}}exp(−te_{α})λ)_{|t=0} .

But for any t, Φ_{γ_{j}} … Φ_{γ_{1}}exp(−te_{α})λ is the unique point of N_{w} · λ whose coordinates e_{−γ_{1}}, …, e_{−γ_{j}} vanish. So that e_{α} ∘ π_{w} = 0. We have thus proved the inclusion Imπ_{w} ⊂ A(U_{w})^{𝔫_{w}}. Now it is clear that π_{w} is a projector : check that π_{w} ∘π_{w} = π_{w} on coordinates using the formula (*). The reverse inclusion A(U_{w})^{𝔫_{w}} ⊂ Imπ_{w} will be a consequence of the following lemma.

Lemma 2.3 Let k be in [1, j] and let f be in A(U_{w}). If X_{γ_{k}}f = 0, then π_{γ_{k}}f = f.

Proof of the lemma 2.3 : We first remark that (π_{γ_{k}}(e_{−γ_{1}}), …, π_{γ_{k}}(e_{−γ_{k−1}}), e_{−γ_{k}}, π_{γ_{k}}(e_{−γ_{k+1}}), …, π_{γ_{k}}(e_{−γ_{m}}), h_{α_{1}}, …, h_{α_{l}}) is a coordinate system in U_{w}. Indeed, one may see by induction that for any i ≤ k − 1 (respectively i ≥ k + 1), e_{−γ_{i}} may be expressed as a regular function of (π_{γ_{k}}(e_{−γ_{1}}), …, π_{γ_{k}}(e_{−γ_{i}}), h_{α_{1}}, …, h_{α_{l}}) (respectively (π_{γ_{k}}(e_{−γ_{i}}), …, π_{γ_{k}}(e_{−γ_{m}}), h_{α_{1}}, …, h_{α_{l}}) ). We put (ϵ_{1}, …, ϵ_{m+l}) = (π_{γ_{k}}(e_{−γ_{1}}), …, π_{γ_{k}}(e_{−γ_{k−1}}), e_{−γ_{k}}, π_{γ_{k}}(e_{−γ_{k+1}}), …, π_{γ_{k}}(e_{−γ_{m}}), h_{α_{1}}, …, h_{α_{l}}). In

these coordinates, we have X_{γ_{k}} = h_{γ_{k}} \frac{∂}{∂ϵ_{k}}. So that if X_{γ_{k}}f = 0, then f does

not depend on ϵ_{k} and it becomes clear that there exists g such that f = π_{γ_{k}}g. As π_{γ_{k}} is a projector, we have π_{γ_{k}}f = π_{γ_{k}}π_{γ_{k}}g = π_{γ_{k}}g, which finishes the proof of the lemma. It is clear from the proof that π_{w} sends C_{w} ∩ P(U_{w}) onto P(U_{w})^{𝔫_{w}}.

In particular π_{\overline{w}|P(U\overline{w})} is the projector onto S(𝔥)_{h_{γ_{1}}...h_{γm}} with kernel 𝔫_{−}P(U\overline{w}). By analogy to Asherova, Tolstoy, Smirnov and Zhelobenko’s work, we will call it the extremal projector.

The proposition 2.1 gives a geometric interpretation of the projector π_{w}.

3 Appendix : Extremal projector for the Virasoro algebra in the semi-classical case

In this section, we shall give a factorization of the Virasoro algebra extremal projector in the semi-classical case. Note that the non commutative case is still open. It is very different from the semi-simple case because the Virasoro algebra does not admit any normal ordering. Recall that the Virasoro algebra V ir is the infinite dimensional Lie algebra generated by {e_{i} | i ∈ ℤ} ∪ {c} with commutation rules

[e_{i}, e_{j}] = (j − i) e_{i+j} + \frac{(j^{3} − j)}{12}δ_{i+j,0}c, [e_{i}, c] = 0.

V ir admits the following triangular decomposition

V ir = V ir_{+} ⊕ V ir_{0} ⊕ V ir_{−}

where

V ir_{+} = ⊕_{i≥1}ℂe_{i}, V ir_{0} = ℂe_{0} ⊕ ℂc, V ir_{−} = ⊕_{i≤−1}ℂe_{i}.

We will also use the notation

V ir_{r,+} = ⊕_{i≥r}ℂe_{i} and V ir_{r,−} = ⊕_{i≤−r}ℂe_{i}.

V ir_{r,+} and V ir_{r,−} are Lie subalgebras of V ir.

Let R(V ir_{0}) be the field of fractions of S(V ir_{0}). We introduce the algebra

S′(V ir) = S(V ir) ⊗_{S(V ir_{0})} R(V ir_{0}) = S′ (V ir/V ir_{−}) .

There is a natural action of V ir_{−} on S′ (V ir/V ir_{−}). Through this action, for

any negative i, e_{i} defines a derivation X_{i} of S′ (\frac{V ir}{V ir_{−}} ). Set

T_{r} = (\frac{S′(V ir)}{S′(V ir)V ir_{−}} )^{V ir_{r,−}}

The result and the proof of the following lemma is left to the reader.

Lemma 3.1

T_{r} = ⊕_{k_{1},...,k_{r−1}∈ℕ}R(V ir_{0})e_{1}^{k_{1}} … e_{r−1}^{k_{r−1}}

As a consequence of the lemma 3.1, we have the following decomposition

S′ (V ir/V ir_{−}) = T_{r} ⊕ V ir_{r,+}S′ (V ir/V ir_{−}) .

The proof of the next lemma is an easy computation.

Lemma 3.2 For any i > 1, the operator π_{i} = ∑_{k=0}^{∞} \frac{(−1)^{k}}{k! (2ie_{0} + \frac{(i^{3}−i)c}{12})^{k}} e_{i}^{k}X_{−i}^{k}

is an algebra morphism and satisfies the relations

X_{−i} ∘ π_{i} = 0 and π_{i} ∘ e_{i} = 0

(where e_{i} denotes multiplication by e_{i}).

It is not hard to see that the operator Π_{r} = ∏_{i=r}^{∞} π_{i} is well defined. Actually

∏_{i=r}^{∞} π_{i}(e_{1}^{a_{1}} … e_{k}^{a_{k}}) = ∏_{r≤i≤k}π_{i}(e_{1}^{a_{1}} … e_{k}^{a_{k}}) = 0 (by the lemma 3.2).

Theorem 3.3 The operator Π_{r} satisfies the relations

∀i ≥ r, X_{−i} ∘ Π_{k} = 0, Π_{k} ∘ e_{i} = 0.

It is the projector onto T_{r} with kernel V ir_{r,+}S′ (\frac{V ir}{V ir_{−}} ) .

In particular, Π_{1} is the extremal projector.

Proof of the theorem 3.3 The relations of a) are easy to check and they prove that Π_{r} is a projector. To prove that the kernel of Π_{r} is V ir_{r,+}, we proceed as in the semi-simple case. The inclusion ImΠ_{r} ⊂ T_{r} is a consequence of a). To prove the reverse inclusion, remark that if x is in T_{r}, then Π_{r}x = x, so that x is in ImΠ_{r}.

4 Bibliography

[A-S-T] Asherova R. M.-Smirnov Y. F.-Tolstoi V. N., Description of a class of projection operators for semi-simple complex Lie algebras, Matem. Zametki 26 No. 1, 15-25, (1979).

[Z1] Zhelobenko D. P. , An introduction to the theory of S-algebras over reductive Lie algebras, Representations of infinite Lie groups and algebras, Gordon and Breach, New York (1986).

[Z2] Zhelobenko D. P, Extremal cocycles of Weyl groups, Functional analysis and its applications, 21, No 3, (1987), p. 11-21.

Sophie Chemla Institut de Mathématiques Université Paris VI, Case 82 4, place Jussieu, 75252 Paris Cedex 05 (FRANCE) schemla@math.jussieu.fr Fax: +33 01 44 27 26 87