ISOMORPHISMS OF FINITE INVARIANT FOR

ENVELOPING ALGEBRAS, SEMI-SIMPLE CASE

Philippe CALDERO

ABSTRACT. Let g be a finite dimensional semi-simple Lie algebra, U(g) its enveloping algebra and H a finite subgroup of Aut U(g). Let A be the invariant algebra U^{H}. In this article, we prove that the Lie algebra g is given (up to an isomorphism) by the algebra A. If we impose that H is a finite subgroup of the adjoint group of g acting on the enveloping algebra U(g), then the algebra A gives a unique group algebra C[H]. If g = sl_{2}, then the group H can be recovered from A.

0. Introduction.

A theorem of Alev and Polo, [1, Theorem 1], asserts that if g and g′ are two finite dimensional semi-simple Lie algebras, with enveloping algebras, respectively U(g) and U(g′), and if H is a finite subgroup of Aut U(g), then U(g)^{H} ≃ U(g′) ⟺ g ≃ g′ and H = {e}. The starting point of the article is the following question : can we recover in a unique way the semi-simple Lie algebra g and the finite subgroup H from the algebra A = U(g)^{H}? In this article, we prove the following rigidity theorems :

Theorem 1. Let g and g′ be two semi-simple Lie algebras, let H and H′ be finite subgroups of, respectively Aut U(g) and Aut U(g′). Then, we have U(g)^{H} ≃ U(g′)^{H′} ⟹ g ≃ g′.

Theorem 2. Let g and g′ be two semi-simple Lie algebras, let H and H′ be finite subgroups of the adjoint group of, respectively g and g′. Then, we have

U(g)^{H} ≃ U(g′)^{H′} ⟹ g ≃ g′ and C[H] ≃ C[H′].

Theorem 3. Let g be semi-simple Lie algebra, let H be finite subgroups of the adjoint group of g and H′ be a finite subgroup of P Sl_{2}. Then, we have

U(g)^{H} ≃ U(g′)^{H′} ⟹ g ≃ g′ et H ≃ H′.

The proof of Theorem 1 is organized as follows. Let A be an invariant algebra for the action of a finite subgroup of Aut U(g). We know [loc. cit., Proposition 1.4], [15] that the center of A is the invariant algebra of the center of U(g) for the action of a subgroup of the group of diagram automorphisms. As for the enveloping algebra, we define the integral regular central characters χ_{λ} as the central characters of simple finite dimensional A-modules. According to an idea of Patrick Polo, we define a function h̃t^{+} on the set of these characters. By lemma 1.5, this function is an analog of the height function on the integral dominant weights lattice P^{+} of an enveloping algebra. With the help of this reference function, we can determine, by a growth argument, the Casimir element corresponding to each component, c.f. (1.2.1), of g. In the appendix, we show that a component of g can be recovered by the corresponding Casimir element (except for one case, that we treat separately).

For the proof of Theorem 2, we proceed as follows. Let H be a finite subgroup of the adjoint group of g. Then, H acts on the algebra S = C[G/N], in which G is the simply connected semi-simple group and N a maximal unipotent subgroup of G. S is a finite integral algebra with a P^{+}-graduation and H acts on S as a graded algebra. More precisely, S = ⨁_{λ∈P^{+}} V (λ), in which V (λ) denotes the (finite dimensional) U-module with highest weight λ. In [10], [4], the dimension of a U^{H}-submodule of V (λ) is seen as the multiplicity of an irreducible representation of H. With the help of a multiparametered version of a theorem of Howe, [6], settled in 3.1, we obtain the following fact : the quotients of the dimensions of the finite dimensional U^{H}-modules corresponding to an integral regular central character χ_{λ}, tend, when λ tends toward infinity ( inside a cone), to the quotients of the dimensions of the irreducible representations of H. So, U^{H} “sees” these dimensions and we can recover the algebra C[H]. In the case SL_{2}, the classification of the finite subgroups as well as the constant τ, [4], associated to each one of these subgroups, enable us to recover the group H.

We are in debt to M. Brion, T. Levasseur, G. Perets, P. Polo and M. Van den Bergh for the precious help they gave.

1. Notations et preliminaries.

1.1. Let g be the semi-simple complex Lie algebra of the simply connected semi-simple Lie group G and U := U(g) its enveloping algebra. Let n be the rank of g. We fix a Cartan subalgebra of g and a root system R. We note Δ = {α_{i}} the set of simple roots of R, P the associated weights lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} Nϖ_{i} the semigroup of dominant weights. Let W be Weyl group generated by ref lections corresponding to the simple roots s_{α}, α ∈ Δ. We note ( , ) the W -invariant form on P. We have (α_{j}, ϖ_{i}) = δ_{ij} \frac{(α_{i},α_{i})}{2}.

Let Z be the center of U. We know that Z embeds in the symetric algebra of the Cartan subalgebra. The center Z will inherit a natural N-graduation from this embedding. To each simple component of g corresponds a Casimir element in Z and the space of elements of degree 2 in Z is generated by the set C of such elements. So, any diagram automorphism of g acts by permutation on C and fixes the sum C of elements of C.

For all λ in P^{+}, we note V (λ) the simple U-module with highest weight λ. We know that this module is finite dimensional and any finite dimensional irreducible U-module is a V (λ). To each λ corresponds a central character χ_{λ} such that z | V (λ) = χ_{λ}(z).Id for all z in Z. We know that

(1.1.1),  χ_{λ}(C) = (λ, λ) + 2(λ, ρ)

in which ρ stands for the sum of fundamental weights. Finally, the U-module V (λ) is the unique simple quotient of the Verma module M(λ) whose annihilator is the primitive, completely prime ideal U.Kerχ_{λ}, [5; 8.4.3, 8.4.4].

1.2. Let H be a finite subgroup of the group of algebra automorphisms of U, and U^{H} be the invariant algebra of U for the action of H. The center Z is stable by H and we know , [15], that the action of H on Z restricts to the action of a subgroup of the group of diagram automorphisms of g. Let D be this subgroup. So, the group H acts on the minimal primitive ideals U.Kerχ_{λ}, λ ∈ P^{+}, by diagram automorphisms. Moreover, the group D acts by permutation on the set of connected components of the Dynkin diagram of g. Now, let’s remark that an element of D maps a simple component of the Dynkin diagram of g to a component of the same type. So, we can decompose g into a direct sum of D-orbits :

(1.2.1)  g = ⨁_{i=1}^{m} g_{i},

in which g_{i} is a multiple of a simple component of g.

By [1], the center of U^{H} is Z^{H} = Z^{D}. It results from 1.1 that the element C defined above is in Z^{H}, and that the subspace of degree 2 in Z^{H} is generated by the C_{i}, 1 ≤ i ≤ m, in which C_{i} denotes the sum of the elements of a D-orbit in C.

1.3. In this section, we shortly present the Montgomery isomorphism in the context of enveloping algebras. The references for these results are [13] and [8, par. 2].

We define in Spec U the equivalence relation ∼ as follows : P ∼ P′ ⇔ hP = P′, for a h in H. We have a projection (of algebras) φ from the crossed product U ∗ H onto U^{H}, cf. [12, Lemma 2.1]. We define in Spec U^{H} the equivalence relation, still noted ∼ by : p ∼ p′ ⇔ φ^{−1}(p) ∩ U = φ^{−1}(p′) ∩ U.

Let O, resp. o, in Spec U/ ∼, resp. in Spec U^{H}/ ∼. We note I_{O}, resp. I_{o}, the intersection of all (prime) ideals of O, resp. o. Such an ideal will be called the ideal associated to the orbit O, resp. o. Let’s remark that I_{O} is a semi-prime ideal preserved by H and that P ∩ U^{H} = I_{O}^{H}, for all P in O. We know that Spec U/ ∼ has a partial order noted ≤ : O ≤ O′ ⇔ I_{O} ⊂ I_{O′}. We define a partial order on Spec U^{H}/ ∼ : o ≤ o′ ⇔ φ^{−1}(p) ∩ U ⊂ φ^{−1}(p′) ∩ U, for p in o and one p′ in o′. The following lemma is proved in [8, par. 2]

Lemma. Let O, O′ in Spec U/ ∼ , o and o′ in Spec U^{H}/ ∼. We have

(i) O ≤ O′ ⇔ for all P in O, there exists P′ in O′ such that P ⊂ P′. (ii) O ≤ O′ ⇔ for all P′ in O′, there exists P in O such that P ⊂ P′.

(iii) o ≤ o′ ⇔ for all p′ in o′, there exists p in o such that p ⊂ p′. ♢

The following theorem describes the Montgomery order isomorphism.

Theorem. There exists an order bijection b from Spec U/ ∼ in Spec U^{H}/ ∼ such that for all O in Spec U/ ∼, we have I_{O}^{H} = I_{b(O)}. Moreover, the set of elements of an orbit is exactly the set of minimal prime ideals over the ideal associated to this orbit. ♢

Remark 1. The bijection b maps the set of classes of finite codimensional primitive ideals of Spec U onto the set of classes of finite codimensional primitive ideals of Spec U^{H}.

Remark 2. If P is completely prime in U, then P ∩ U^{H} is completely prime in U^{H} and it is the unique element of the orbit b(O(P)), O(P) being the orbit of P.

1.4. As in [1], we will use some of the knowledge on primitive ideals over U.Kerχ_{λ}, λ ∈ P^{+}. Let d be the Gelfand-Kirillov dimension, and rk the Goldie rank. We know that we can index the minimal prime ideals over U.Kerχ_{λ} by the set Δ of simple roots of g. Let us call P_{α}, α ∈ Δ such ideals. For all subset S of Δ, we set P_{S} = ∑_{α∈S} P_{α}. We have

Theorem. Let us fix λ in P^{+}. Then, with the previous notations, we have :

(i) For all subset S of Δ, the P_{S} are primitive ideals of U. Moreover, P_{S} ⊂ P_{S′} ⇒ S ⊂ S′.

(ii) d(U/P_{S}) = # R − #(R_{S}), in which R_{S} stands for the roots system generated by S.

(iii) rk(U/P_{S}) = ∏_{α∈R_{S}^{+}} \frac{(λ + ρ, αˇ )}{(ρ, αˇ )}, in which α_{i}ˇ = 2\frac{α_{i}}{(α_{i}, α_{i})}.

Proof. The second assertion of (i) comes from [8, 4.6 (*)]. The other assertions can be found in [1, Proposition 1.12] and [7, 12.6]. ♢

Remark 1. Let λ = ∑_{i} λ_{i}ϖ_{i}. We have rk(U/P_{α_{i}}) = λ_{i} + 1. Hence, the sum of the rk(U/P) when P runs on the set of minimal ideals over U.Ker χ_{λ} is equal to ht^{+}(λ) := ht(λ + ρ) = ∑_{i} (λ_{i} + 1).

Remark 2. If α and β are two simple roots, then d(U/P_{α} + P_{β}) = d(U) − 2m_{αβ}, in which m_{αβ} stands for the order of the element s_{α}.s_{β} in W, [1].

1.5. We set A = U^{H}. What is interesting in the following definitions is that they only depend on A and not on the enveloping algebra U. The definition of h̃t^{+} is due to P. Polo.

Definition. A central character χ on A is called regular integral if the ideal A.Kerχ is contained in a primitive finite codimensional ideal. We will note Ξ the set of regular integral central character. For a χ in Ξ, let P_{χ} be the set of minimal prime ideals over A.Kerχ. We set

h̃t^{+}(χ) = ∑_{P ∈P_{χ}} rk(A/P).

With the notations of 1.1 and 1.2, we have the lemma :

Lemma. Let λ in P^{+}. Let χ̄_{λ} be the restriction of χ_{λ} to Z^{D}. Any regular integral central character of A is a χ̄_{λ}. Moreover,

\frac{1}{#H} h̃t^{+}(χ̄_{λ}) ≤ ht^{+}(χ_{λ}) ≤ #Hh̃t^{+}(χ̄_{λ}).

Proof. The first assertion of the lemma results from [1.3, remark 1]. Moreover,

h̃t^{+}(χ̄_{λ}) = ∑_{p∈Pχ̄_{λ}} rk(A/p) = ∑_{o∈Oχ̄_{λ}} rk(A/I_{o}),

in which Oχ̄_{λ} is the set of orbits of minimal prime ideals over A.Kerχ̄_{λ}. By [12, 5.3], and 1.3 we have :

h̃t^{+}(χ̄_{λ}) ≤ ∑_{O∈b^{−1}(Oχ̄_{λ})} rk(U/I_{O}) ≤ #Hh̃t^{+}(χ̄_{λ}).

As Kerχ_{λ} is a central maximal ideal, an ideal of U can contain at most only one minimal primitive ideal U.Kerχ_{λ}. By 1.3, we then obtain that b^{−1}(Oχ̄_{λ}) is a disjoint union of orbits of minimal prime ideals over U.Kerχ′_{λ}, in which λ′ runs on the D-orbit of λ. By [1.4, remark 1], we have

∑_{O∈b^{−1}(Oχ̄_{λ})} rk(U/I_{O}) = ∑_{λ′∈D.λ} ht^{+}(λ′) = #(D.λ).ht^{+}(λ) ≤ #H.ht^{+}(λ).

The lemma follows from this. ♢

1.6. In this section, we focus on prime ideals of U^{H} over the ideal generated by the kernel of the trivial character χ̄_{0} := χ_{0} | Z^{D}.

Proposition. Let P_{α}, α ∈ Δ, the prime ideals of U over U.Kerχ_{0}. Then, for all subset S ⊂ Δ, P_{S} is completely prime. The ideal P_{S}^{H} := P_{S} ∩ U^{H} is completely prime. Let α and β be two simple roots of Δ. Then, for all P_{α′}, resp. P_{β′}, in the H-orbit of P_{α}, resp. P_{β}, (P_{α′} +P_{β′})^{H} is a minimal prime ideal over P_{α}^{H} +P_{β}^{H}. Moreover, any minimal prime ideal over P_{α}^{H} + P_{β}^{H} has such a form.

Proof. The first part of the proposition is exactly [1, proposition 1.12 (b)].

Let’s show that (P_{α′} + P_{β′})^{H} minimal prime ideal over P_{α}^{H} + P_{β}^{H}. With the hypothesis of the proposition, (P_{α′} + P_{β′})^{H} is completely prime and contains P_{α′}^{H} = P_{α}^{H} as well as P_{β′}^{H} = P_{β}^{H}. Then, P_{α}^{H} + P_{β}^{H} ⊂ (P_{α′} + P_{β′})^{H}. Let’s suppose that P_{α}^{H} + P_{β}^{H} ⊂ q ⊂ (P_{α′} + P_{β′})^{H}, with q prime. Then, o(P_{α}^{H}), o(P_{β}^{H}) ≤ o(q) ≤ o((P_{α′} + P_{β′})^{H}). As a consequence of lemma 1.3 and theorem 1.4 (i), we obtains that O(P_{α′} + P_{β′}) is a minimal orbit among the orbits greater than O(P_{α}) and O(P_{β}). By Theorem 1.3, we obtain our assertion.

Conversely, let p be a prime ideal of U^{H}, minimal over P_{α}^{H} + P_{β}^{H}. It comes that o(P_{α}^{H}), o(P_{β}^{H}) ≤ o(p). So, b^{−1}(o(p)) is a minimal H-orbit containing the orbits of P_{α} and P_{β}. By Lemma 1.3, we deduce b^{−1}(o(p)) = O(P_{α′} + P_{β′}), for a P_{α′}, resp. P_{β′}, in the H-orbit of P_{α}, resp. P_{β}. So, o(p) = o((P_{α′} +P_{β′})^{H}) = (P_{α′} +P_{β′})^{H}, by [1.3, remark 2]. Then, p = (P_{α′} + P_{β′})^{H}. ♢

1.7. In this section, we focus on the case when H is a subgroup of G. In this case, H has a natural action on the finite dimensional representations V (λ), λ ∈ P^{+}. Moreover, this action is compatible with the action of U on V (λ), cf. [4]. Let ϕ_{λ} the projection from U onto U/Ann(V (λ)). As in [10] and [4], we show that for any λ in P^{+}, the algebras ϕ_{λ}(U^{H}) and ϕ_{λ}(C[H]) are in duality. This duality is a consequence of the following lemma :

Lemma. Let A be a simple finite dimensional C-algebra, B a semi-simple subalgebra of A, C the commutant of B in A and M a simple finite dimensional A-module. Then, we have the decomposition into B ⊗ C-modules : M = ⨁_{i} N_{i} ⊗ P_{i}, in which the N_{i}, resp. P_{i}, are distinct simple B-modules, resp. C-modules.

Proof. Let M = ⨁_{i} n_{i}N_{i} be the decomposition of M into distinct simple B-modules. By hypothesis, we have A = End M and B = ⨁_{i} n_{i}End N_{i}. Let’s set P_{i} := End_{B} (N_{i}, M)- = End_{B} (N_{i}, n_{i}N_{i}).

Then, P_{i} is a C-module with dimension n_{i} and C = ⨁_{i}(dimN_{i})EndP_{i}. So, the P_{i} are simple C-modules. Moreover, as N_{i} = End_{C} (P_{i}, M), we obtain that the P_{i} are distincts. ♢

We can prove the theorem :

Theorem. For all λ in P^{+}, we have the decomposition : V (λ) = ⨁_{i} N_{i} ⊗ P_{i}, in which the N_{i}, resp. P_{i}, are distincts simple C[H]-modules, resp. U^{H}-modules. In particular, dimP_{i} = [V (λ) : N_{i}] and dimN_{i} = [V (λ) : P_{i}].

Proof. As C[H] is semi-simple, the surjection ϕ_{λ} restricts into a surjection from U^{H} onto End(U)^{H} = End_{C[H]} U. So, the theorem is a consequence of the previous lemma with A = End V (λ), B = ϕ_{λ}(V (λ)), and M = V (λ). ♢

Remark 1. This theorem due to Kraft and Small, [10], is precised in [4], where the description of the U^{H}-module P_{i} and its annihilator is given from the data of N_{i}. Let’s note that the method in [4] can be generalized, by the results of [14], in the case of the action of a semi-simple finite dimensional Hopf algebra H, on an algebra whose finite dimensional representations are completely reducible, and on which H acts in a compatible way.

Remark 2. The theorem is easily generalized in the case when H is a torus of G or in the case when H is a reductive subgroup of G.

2. The general case.

2.1. In this section, we prove that we an recover, up to a multiplicative constant, the elements C_{i} defined in 1.2, from the algebra A = U^{H}. We have the proposition

Proposition. The monoid E = ∑_{i} Q^{+}C_{i} is exactly the set of elements z in the center of A such that (i) χ(z) ∈ Q^{+} for all χ in Ξ. (ii) There exists χ in Ξ such that χ(z) = 0. (iii) There exists a constant K of Q^{+} such that for all χ in Ξ, we have χ(z) ≤ Kh̃t^{+}(χ)^{2}.

Proof. Let’s first prove that any element z in E verifies the three assertions. (i) is provided by (1.1.1) applied to each component g_{i} of g. The trivial character (the zero of P is indeed D-invariant), vanishes on z. As z has degree 2 for the graduation defined in 1.2, the assertion (iii) comes easily from lemma 1.5.

Conversely, let’s suppose that z verify the three assertions of the proposition. Let’s decompose z into graduated components : z = ∑_{j=0}^{d} z_{j} in which z_{j} has degree j. As (P^{+})^{D} defines a Zariski dense set in Spec(Z^{D}), we can find a χ_{λ} in Ξ such that χ_{λ}(z_{d}) is non zero. So, χ_{qλ}(z_{d}) is a function with degree d of the variable q ∈ N. By (iii), we can prove by lemma 1.5 that d ≤ 2. As d is the degree of a central element, there comes d = 2 or d = 0. By 1.2, z is a linear combination of the C_{i} and of unity. We can complete the proposition by assertions (i) and (ii) applied to z. ♢

We have shown how to recover the semigroup E from the invariant algebra A. Now, we can easily obtain the lines generated by the C_{i} as follows. Let ≺ be the partial order defined on E by

z ≺ z′ ⇔ ∀χ, χ ∈ Ξ, (z′(χ) = 0 ⇒ z(χ) = 0).

So, the union of the Q^{+∗}C_{i} is the set of non zero elements, which are minimal for this order.

2.2. Let’s decompose P^{+} = ⨁_{j=1}^{m} P_{j}^{+}, in which P_{j}^{+} is the semigroup of integral dominant weights corresponding to the component g_{j} of g defined in 1.2. by the theorem of Harish-Chandra, we have

(2.2.1)  U/ ⋂_{λ∈P_{i}^{+}} Ann V (λ) = U(g_{i}).

By hypothesis, each P_{j}^{+} is stable by D. So, by (2.2.1) we have an action of H on U(g_{i}). For all i, 1 ≤ i ≤ m, let Ξ_{i} be the set of integral regular central characters (of A) χ which verify χ(C_{j}) = 0 for all j not equal to i. Let’s note I_{i} the ideal of A which is the intersection of the finite codimensional primitive ideals over some A.Kerχ, χ ∈ Ξ_{i}. This definition is intrinsic, i.e. only depends on the algebra A and not on the embedding of A into U, because the lines generated by the C_{i} are so, c.f. 2.1. Let’s prove the lemma

Lemma. We have

(i) For all i, 1 ≤ i ≤ m, each character of Ξ_{i} is defined in a unique way by a weight in (P_{i}^{+} | D).

(ii) The algebra A/I_{i} is isomorphic to U(g_{i})^{H}.

Proof. (i) comes easily from (1.1.1). (ii) is provided by (i), (2.2.1) and [1.3, remark 1]. ♢

2.3. We can now prove the theorem Theorem. Let g and g′ be two semi-simple Lie algebras, H and H′ subgroups of, respectively Aut U(g) and Aut U(g′). We have U(g)^{H} ≃ U(g′)^{H′} ⟹ g ≃ g′.

Proof. It is enough to recover the Lie algebra g from the algebra A = U(g)^{H}. By 2.2, we can suppose that g has only one type of simple component, up to isomorphism, i.e. g = ⨁_{i=1}^{k} c_{i}, in which the c_{i}, 1 ≤ i ≤ k, are simple Lie algebras isomorphic to the the same Lie algebra c. Let’s decompose P^{+} = ⨁_{i=1}^{k} P_{i}^{+}, P_{i}^{+} being the semigroup of dominant integral weights corresponding to the component c_{i} de g. We have

(2.3.1)  d(A) = d(U(g)) = kd(U(c)) = k#R_{c},  d(Z(A)) = k#Δ_{c},

in which R_{c}, resp. Δ_{c} denotes the root system, resp. the set of simple roots, of c. So,

(2.3.2)  \frac{d(A)}{d(Z(A))} = \frac{#R_{c}}{#Δ_{c}} .

By 2.1, we can choose in Z(A) an element z such that z = rC, r ∈ Q^{+∗}, C being the Casimir element of g. The element C decompose into a sum of Casimir elements corresponding to each simple component c_{i}, 1 ≤ i ≤ k, of g : C = ∑_{i=1}^{k} C_{i}. Let χ_{λ} be in Ξ, corresponding to the integral dominant weight λ, with λ = ∑_{i=1}^{k} λ_{i}, λ_{i} ∈ P_{i}^{+}. We have

(2.3.3)  χ_{λ}(z) = r ∑_{i=1}^{k} χ_{λ_{i}}(C_{i}).

Let ζ_{1} < ζ_{2} be the two smallest non zero values of χ_{λ}(z) for λ in P^{+} and let κ := \frac{ζ_{2}}{ζ_{1}} .

From [Appendice, Assertion 1] and (2.3.3), we obtain that κ is a constant which does only depend on the simple Lie algebra c. If we can recover the Lie algebra c, then by (2.3.1), we can deduce k, then g and the theorem is proved. With the help of (2.3.2) and [Appendice, Assertion 2], we see that it is enough to distinguish the types C_{n} and D_{n+1}. This distinction can be realized in the following way. The Casimir element of U(g) allows us to define the trivial central character by χ̄_{0}(C) = 0. Let’s note Π(P, Q) a minimal prime ideal over P + Q, P and Q distincts in the set M_{0} of minimal prime ideals over A.Kerχ̄_{0}. By the proposition 1.6, and [1.4, remark 2] we have

d(A/Π(P, Q)) ≥ d(A) − sup_{α,β∈Δ}2m_{α,β},

this bound being reached for a Π(P, Q), P, Q ∈ M_{0}. So, we can complete the proof. ♢

3. Case when H is a finite subgroup of the adjoint group.

In this section, we suppose that H is a finite subgroup of the adjoint group of g. The adjoint action of H on g extends naturally on the enveloping algebra and we can embed H into Aut U. We want to show that, in this case, we can recover the group algebra C[H] from the invariant algebra U^{H}. 3.1. Let N be a maximal unipotent subgroup of G and S = C[G/N]. So, S finite integral algebra and S = ⨁_{λ∈P^{+}} V (λ) is a decomposition of S as G-module (or Umodule). We know that this decomposition defines a P^{+}-graduation for S. Moreover, this graduation is H-stable. In the following, P^{+} is embedded in a natural way into Spec Z. Then, the monoid P^{+} is equipped with the induced Zariski topology. The following theorem is an adaptation to the multi-graduate case of a theorem of Howe, [6], [3, 4.1].

Proposition. Let ψ_{0} be the trivial representation of H. There exists a dense open set T of P^{+} such that lim_{||λ||→+∞,λ∈T}[V (λ) : ψ]/[V (λ) : ψ_{0}] = dim ψ, for all irreducible

representation ψ of H.

Proof. For all h non trivial in H, we can choose a λ_{h} in P^{+} such that h | V (λ_{h}) is not a constant. Let λ be a weight in P^{+} we have V (λ).V (λ_{h}) ⊂ V (λ + λ_{h}). As h acts in a semi-simple way on each graduate component of S and as S is integral, we can deduce that h is non constant on all V (λ + λ_{h}), λ ∈ P^{+}. We set μ = ∑_{h∈H} λ_{h}. Let B be the following subalgebra of S B := ∑_{n≥0} V (nμ). B is a finite integral algebra with N-graduation and, by construction, the subgroup of elements of H which acts by scalar on each graduate component of B is trivial. We fix h in H and et let o(h) be its order. As in [6], we can find a integer m such that the action of h on V (mμ) is spectraly complete, i.e. the spectrum of h | V (mμ) contains all o(h)-th of unity. Then, for λ in P^{+}, the action of h on V (λ + mμ) is spectraly complete. For all λ in P^{+} and for all o(h)-th root of unity w, let’s note V_{h}(λ, w) the eigenspace V (λ) corresponding to the eigenvalue w for the action of h. As in [6, (12)], we have for all couple of o(h)-th roots of unity (w_{1}, w_{2}) : dim V_{h}(λ+2mμ, w_{1}) ≥ dim V_{h}(λ+mμ, w_{2}). Adding this inequalities for all w_{2}, we obtain dim V_{h}(λ + 2mμ, w_{1}) ≥ o(h)^{−1}dim V (λ + mμ). Now, by the Weyl formula,

lim_{||λ||→+∞} \frac{dim V (λ + 2mμ)}{dim V (λ + mμ)} = 1.

So, we obtain

lim inf_{||λ||→+∞} \frac{dim V_{h}(λ + 2mμ, w_{1})}{dim V (λ + 2mμ)} ≥ o(h)^{−1}.

As in the proof of Howe’s theorem, we have the equality and this implies our assertion by setting T = {λ + 2mμ, λ ∈ P^{+}}. ♢

3.2. We can prove the theorem :

Theorem. Let g and g′ be two semi-simple Lie algebras, H and H′ finite subgroups of respectively the adjoint group of g and of g′. We have

U(g)^{H} ≃ U(g′)^{H′} ⟹ g ≃ g′ and C[H] ≃ C[H′].

Proof. By Theorem 2.3, it is enough to recover the group algebra C[H] from the algebra A = U(g)^{H}. As H is a subgroup of the adjoint group, the group D associated to H is the trivial group and the center of A is equal to the center Z of the enveloping algebra. Let χ be in Ξ. Let V_{i}^{χ}, 1 ≤ i ≤ k be the simple finite dimensional A-modules whose annihilator contains A.Kerχ. Let’s suppose V_{1}^{χ} be of minimal dimension and

let’s set d_{i}^{χ} = \frac{dim V_{i}^{χ}}{dim V_{1}^{χ}}. Then, by Theorem 1.7, Proposition 3.1 and Lemma 1.5, there

exists a dense open set of Ξ for the Zariski topology of Spec Z induced on Ξ, such that lim_{_{ht}^{+}(χ)→+∞,χ∈T} d_{i}^{χ} is equal to the dimension of a irreducible representation of H. By

Proposition 3.1, we obtain the dimensions of all irreducible representations of H. The theorem follows from this. ♢

3.3. In the case when g = sl_{2}, we can strengthen the previous theorem and recover the group H from the invariant algebra A. The proof of the next theorem uses the classification of the finite subgroups of SL_{2}, [16], each finite subgroup of SL_{2} being associated to the type of a semi-simple Lie algebra by the McKay correspondence, [11], [9].

Theorem. Let g be a semi-simple Lie algebra, H un finite subgroup of the adjoint group of g and H′ a finite subgroup of P SL_{2}. We have

U(g)^{H} ≃ U(sl_{2})^{H′} ⟺ g ≃ sl_{2} and H ≃ H′.

Proof. By Theorem 2.3, the hypothesis U(g)^{H} ≃ U(sl_{2})^{H′} implies g ≃ sl_{2}. For each finite subgroup H of SL_{2}, we can associate a constant τ equal to the number of given dimensional representations of U^{H}. Let’s recall, [4] that τ(C_{n}) = n^{2}, τ(D_{n}) = n(n + 7)/2, τ(E_{6}) = 29, τ(E_{7}) = 53,τ(E_{8}) = 106. With the help of these constants, we can prove that if H and H′ are distincts, then H has type, say, C_{n}, and H′ has type D_{m}. Now, by Theorem 3.2, we have C[H] ≃ C[H′]. This is only possible for possible for the cyclic group of order 4, C_{4}, and the dihedral group of order 4, D_{2}. As τ(C_{4}) = 16 is different than τ(D_{2}) = 9, we have our theorem.

APPENDICE

In this section, c is a simple Lie algebra and C is the Casimir of c. We note ζ_{1} the minimal non zero value of C on the (discrete) set P^{+}. We note ζ_{2} the minimal value

strictly greater than ζ_{1} of C on P^{+}. Let κ = \frac{ζ_{2}}{ζ_{1}} . We give in the following tables the

values of the constants ζ_{1},ζ_{2}, κ, and \frac{#R}{#Δ} for all simple algebras c. The calculus have

been made with the help of [2, Planches I à IV]. Let’s note that the minimas ζ_{1} and ζ_{2} are always obtained on fundamental weights.

\begin{matrix}Type & A_{n} & B_{n} & C_{n} & D_{n} \\ ζ_{1} & \frac{n(n + 2)}{n + 1} & 4n & 2n + 1 & 2n − 1 \\ ζ_{2} & \frac{2(n − 1)(n + 2)}{n + 1} & 8n − 4 & 4n & 4n − 4 \\ κ & \frac{2(n − 1)}{n} & \frac{2n − 1}{n} & \frac{4n}{2n + 1} & \frac{4n − 4}{2n − 1} \\ \frac{#R}{#Δ} & n + 1 & 2n & 2n & 2(n − 1) \end{matrix}

\begin{matrix}Type & E_{6} & E_{7} & E_{8} & F_{4} & G_{2} \\ ζ_{1} & 52/3 & 57/2 & 60 & 24 & 12 \\ ζ_{2} & 24 & 36 & 96 & 36 & 24 \\ κ & 18/3 & 72/57 & 8/5 & 3/2 & 2 \\ \frac{#R}{#Δ} & 12 & 18 & 60 & 12 & 6 \end{matrix}

The following assertions are deduced from the tables above.

Assertion 1. Let c be a simple Lie algebra. With the previous notations, we have ζ_{2} ≤ 2ζ_{1}. ♢

Assertion 2. Let c and c′ be two non isomorphic simple Lie algebras. Then,

(κ(c), \frac{#R}{#Δ}(c)) = (κ(c′), \frac{#R}{#Δ}(c′)) if and only if c has type C_{n} and c′ has type D_{n+1}

(or inversely). ♢

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Institut Girard Desargues, UPRES-A-5028 Université Claude Bernard Lyon I, Bat 101 69622 Villeurbanne Cedex, France e-mail : caldero@desargues.univ-lyon1.fr