Journal of Lie Theory Volume 10 (2000) 323–330 C ◯2000 Heldermann Verlag

On the Chevalley restriction theorem

E. A. Tevelev

Communicated by E. B. Vinberg

Abstract. Let 𝔤 be a complex semisimple Lie algebra with adjoint group G. Suppose that σ is an involutive automorphism of 𝔤. Then σ induces an involution of G also denoted by σ. Let K=G^{σ} be the subgroup of σ -fixed points. Consider the direct decomposition 𝔤=𝔨⊕𝔭 of 𝔤 into the eigenspaces

for σ, so that 𝔨={x∈𝔤 | σ(x)=x}, 𝔭={x∈𝔤 | σ(x)=−x}. Then 𝔭 is a K -

module. Denote by 𝔞⊂𝔭 a maximal abelian ad-diagonalizable subalgebra. Consider the “baby Weyl group” W_{0}=N_{K}(𝔞)/Z_{K}(𝔞). The aim of this paper is to prove that the restriction map ψ: ℂ[𝔭×𝔭]^{K}→ℂ[𝔞×𝔞]^{W0} is surjective.

§0. Introduction

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G. Suppose that σ ≠ id is an involutive automorphism of 𝔤. Then σ induces an involution of G also denoted by σ. Let K = G^{σ} be the subgroup of σ -fixed points. Consider the direct decomposition 𝔤 = 𝔨 ⊕ 𝔭 of 𝔤 into the eigenspaces for σ, so that

𝔨 = {x ∈ 𝔤 | σ(x) = x}, 𝔭 = {x ∈ 𝔤 | σ(x) = −x}.

Then, clearly, 𝔨 is the Lie algebra of K and 𝔭 is a K -module, the representation of K in 𝔤 is certainly nothing else but the isotropy representation of the symmetric space G/K (see e.g. [KR]). Denote by 𝔞 ⊂ 𝔭 a maximal abelian ad-diagonalizable subalgebra. Consider the “baby Weyl group” W_{0} = N_{K}(𝔞)/Z_{K}(𝔞). It is well-known that W_{0} is a finite group generated by reflections as a linear group operating on 𝔞 (with respect to some real form of 𝔞). Let ψ : ℂ[𝔭]^{K} → ℂ[𝔞]^{W_{0}} be the restriction map of algebras of invariants. Then the famous Chevalley restriction theorem states that ψ is an isomorphism (see, eg., [He1]; in [Vi] a more general result is obtained in the context of so-called θ-groups attached to any periodic automorphism of 𝔤).

Now consider the following special case. The adjoint representation G : 𝔤 could be identified with the isotropy representation of the symmetric space induced by the involution σ of G × G given by σ(x, y) = (y, x). Then the cited

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag Chevalley restriction theorem is equivalent to the “usual” Chevalley restriction theorem ℂ[𝔤]^{G} ≃ ℂ[𝔥]^{W} , where 𝔥 ⊂ 𝔤 is a Cartan subalgebra and W is the usual Weyl group. In the paper [J] A. Joseph obtained a remarkable “multivariable” analogue of this theorem. Namely, he proved that the restriction map

ℂ[𝔤 × 𝔤]^{G} → ℂ[𝔥 × 𝔥]^{W}

is surjective. This theorem has important applications to the representation theory and the geometry of commuting varieties (cf. [Ri]), because it is essentially equivalent to the fact that the ring of G-invariant functions on the commuting variety is integrally closed (see [Hu, J]). The aim of this paper is to extend Joseph’s arguments in order to prove the following generalization: Theorem. The restriction map ψ : ℂ[𝔭 × 𝔭]^{K} → ℂ[𝔞 × 𝔞]^{W_{0}} is surjective.

Actually, this (and Joseph’s) theorem also holds for any number of summands (a generalization of the proof is immediate), but we will prove it in this form in order to make the notation easier.

Certainly, one could try to deduce this Theorem directly from Joseph’s result arguing as follows: take f ∈ ℂ[𝔞 × 𝔞]^{W_{0}} and lift it to an element of ℂ[𝔥×𝔥]^{W} ; then by Joseph’s theorem the latter element can be lifted to an element of ℂ[𝔤 × 𝔤]^{G}, which can be resricted to an element of ℂ[𝔭 × 𝔭]^{K}. Unfortunately, the first step of this construction fails even in the case of one summand (see examples in [He2]). Moreover, I don’t know how to prove the natural analogue of the theorem in the context of arbitrary θ-groups (see [Vi]) and this seems to be an interesting problem. I should mention also that in some particular cases these results could be proved using “generalized polarizations” (see, e.g., [Hu]), but these results do not cover our theorem (as well as Joseph’s one). In [Pa1] the Theorem was proved in the case of an involution of maximal rank (and in this case the argument in the beginning of this paragraph is evidently valid).

The paper is organized as follows. In §1 we look for a proof of Chevalley restriction theorem in the case of one summand such that this proof could be generalized to the case of two summands. Another reason for doing that separately is that all the ideas distinct from the ones used by Joseph already appear in this case. In §2 we prove the theorem in full generality showing that our main tool (class 1 representations) is compatible with Joseph’s idea to use the PRV Conjecture (or, better said, the Kumar-Mathieu Theorem) to “separate layers”.

I would like to thank D. Saltman for useful discussions and his warm hospitality during my stay in the University of Texas in Austin. The research was supported by CRDF grant RM1-206, and by INTAS grant INTAS-OPEN-97-1570.

§1. Surjectivity of ψ : ℂ[𝔭]^{K} → ℂ[𝔞]^{W_{0}}

Recall, that there is 1-1 correspondence between involutions of 𝔤 and real forms of 𝔤 (see, e.g., [W, §1.1]). Namely, suppose that 𝔤^{ℝ} is a real form of 𝔤 (that is, 𝔤 = 𝔤^{ℝ} + i𝔤^{ℝ}). Let 𝔤^{ℝ} = 𝔨^{ℝ} ⊕ 𝔭^{ℝ} be a Cartan decomposition and 𝔤 = 𝔨 ⊕ 𝔭 its complexification. Then the map σ defined by σ(x) = x for x ∈ 𝔨 and σ(x) = −x for x ∈ 𝔭 is an involution of 𝔤 and any involution can be obtained in this way. Now, a maximal ad-diagonalizable abelian subalgebra 𝔞 in 𝔭 (all of them are K -conjugate) can be chosen to be a complexification of some 𝔞^{ℝ} ⊂ 𝔭^{ℝ}. ¿From now on by complex conjugation we will understand the complex conjugation with respect to a given real form 𝔤^{ℝ}.

Let B be a Killing form on 𝔤. Then the restriction of B on 𝔨^{ℝ} is negative-definite and on 𝔭^{ℝ} is positive definite. In particular we can identify 𝔤 ≃ 𝔤^{∗} and 𝔞 ≃ 𝔞^{∗} by means of this scalar product. Then the restriction map ψ : ℂ[𝔭]^{K} → ℂ[𝔞]^{W_{0}} can be viewed as a homomorphism of symmetric algebras ψ : (S𝔭)^{K} → (S𝔞)^{W_{0}} . Now, it is well-known that K = G^{σ} is a reductive group, so we have a K -module decomposition S𝔭 = (S𝔭)^{K} ⊕ (S𝔭)_{K}, where (S𝔭)_{K} is a sum of all non-trivial K -submodules. Consider the Reynolds operator (K -equivariant projection on the first factor of this decomposition). Let θ : (S𝔞)^{W_{0}} → (S𝔭)^{K} be its restriction to (S𝔞)^{W_{0}} . It is clear that we have a commutative diagram

S𝔞 ⟶^{θ_{1}} (S𝔭)^{K}
θ_{2} ↘  ↗_{θ}
(S𝔞)^{W_{0}}

where θ_{1} is a restriction of the Reynolds operator to S𝔞 and θ_{2} is a W_{0} - equivariant projection. Now we are ready to prove the following lemma:

Lemma 1.1. If θ is injective then ψ is surjective.

Proof. It is clear that both maps ψ and θ are homogeneous (with respect to natural grading of S𝔞 and S𝔭). So the composition ψθ is a homogeneous map of degree 0 from (S𝔞)^{W_{0}} to (S𝔞)^{W_{0}} . Now it is clear that if θ is injective and Ker ψ ∩ Im θ = 0 then ψθ is injective, hence surjective, so ψ is surjective as well. Certainly the second assertion follows from the fact that ψ is injective (and this is an easy part of Chevalley restriction theorem), but we are looking for a proof that could be applicable in the case of two summands.

First we could extend B to the scalar product in S𝔭 in the usual way. Since the restriction of B on 𝔭^{ℝ} is positive definite, it is clear that the restriction of B on S𝔭^{ℝ} is also positive definite (more precisely, the restriction of B on any finite-dimensional subspace of S𝔭^{ℝ} is positive definite). In particular, if V ⊂ S𝔭 is any subspace stable under complex conjugation, then V ∩ V^{⊥} = 0. Take V to be the minimal K -submodule of S𝔭 generated by S𝔞. Since S𝔞 is stable under complex conjugation, so is V , because V is spanned by the vectors of the form k · f, k ∈ K, f ∈ S𝔞, and since σ commutes with complex conjugation on 𝔤, K is stable under complex conjugation (we are in the adjoint group!).

By definition of Reynolds operator, Im θ = V ∩ (S𝔭)^{K}. By the construction of B on S𝔭, it is clear that Ker ψ = (S𝔞)^{⊥} ∩ (S𝔭)^{K} and by K -invariance of B we have Ker ψ = V^{⊥} ∩ (S𝔭)^{K}. Since V ∩ V^{⊥} = 0, we finally obtain Ker ψ ∩ Im θ = 0.

So to prove the surjectivity of ψ it remains to prove that θ is injective. This idea (as well as many others in this paper) belongs to Joseph, the only exception is that Joseph didn’t use Reynolds operator directly. Instead of that he defined the “Letzer map” purely in infinitesimal terms, but it is easy to see that in the adjoint situation Joseph’s “Letzer map” actually coincides with good old Reynolds operator! One could mention that in the proof of Lemma 1.1 we didn’t use the fact our representation is an isotropy representation of a symmetric space. So the argument is applicable in a little bit more general setup.

Now it is time to slightly change our notations and to introduce new ones. First of all, since S𝔭 is a K -submodule of S𝔤, injectivity of θ will follow from the injectivity of the restriction on (S𝔞)^{W_{0}} of the Reynolds operator S𝔤 → (S𝔤)^{K}. Since the definition of the Reynolds operator depends only on the K -module structure of S𝔤, we can substitute for S𝔤 the universal enveloping algebra U𝔤, since U𝔤 has the same K -module structure by the Poincare-Birkhoff-Witt theorem (this K -module isomorphism could be written down explicitly as the symmetrization map S𝔤 → U𝔤). Moreover, we have an isomorphism S𝔞 ≃ U𝔞. Notice that the obvious embedding U𝔞 ⊂ U𝔤 corresponds to an embedding S𝔞 ⊂ S𝔤 via the symmetrization map. So from now on θ will denote the map (U𝔞)^{W_{0}} → (U𝔤)^{K} induced by the Reynolds operator. We need to show that it is injective.

Take a torus S ⊂ G such that 𝔞 = Lie S. Then S is a maximal σ - anisotropic torus in the terminology of [Vu]. Fix a Borel subgroup B and maximal torus T ⊂ B such that KB is dense in G (so K is spherical in G and if 𝔟 is the Lie algebra of B then 𝔨 + 𝔟 = 𝔤) and T contains S (such B, T exist by [Vu]). Then automatically σ(T) = T. Then 𝔥 = Lie T is a Cartan subalgebra of 𝔤. Let 𝔥_{0} = 𝔥 ∩ 𝔨, so 𝔥 = 𝔥_{0} ⊕ 𝔞. Let 𝔛^{∗}(T) (resp. 𝔛_{∗}(T)) be the character group (resp. the set of one-parameter subgroups) of T. We use the additive notation for the operation in 𝔛^{∗}(T) and sometimes identify a character with its differential. Let P ⊂ 𝔛^{∗}(T) be the weight lattice and P_{+} ⊂ P the semigroup of dominant weights. Set

Q = {λ ∈ P : σ(λ) = −λ, λ|_{S} ∈ 2𝔛^{∗}(S)}.

It is clear that for any λ ∈ Q we have λ|_{𝔥_{0}} = 0, so we may view such λ as an element λ̃ of 𝔞^{∗}. Let w ∈ W_{0} , μ̃ = wλ̃. Then we can extend μ̃ to an element μ of 𝔛^{∗}(T) ⊗ ℚ. But it is clear that actually μ ∈ Q. Moreover by the structure theory of the baby Weyl group (see e.g. [W, §1.1]), there exists w ∈ W such that μ = wλ. Another useful tip is that if two elements of 𝔞^{∗} are conjugate with respect to W, then they are also conjugate w.r.t. W_{0} (this follows from the fact that any W_{0} -fundamental chamber in 𝔞^{∗} entirely lies in some W -fundamental chamber, see [loc. cit.]). Set Q_{+} = Q ∩ P_{+} .

Lemma 1.2. In the identifications above, Q = W_{0}Q_{+} .

Proof. This follows from the results of T. Vust (see [Vu]). Namely, passing to dual lattices, it is sufficient to prove that if R ∈ 𝔛_{∗}(S) is any one-parameter subgroup then there exists w ∈ W_{0} such that R^{w} = wRw^{−1} is dominant, that is, for any b ∈ B the limit lim_{t→0} R^{w}(t)bR^{w}(t)^{−1} exists. Consider a subgroup

P (R) = {g ∈ G | lim_{t→0} R(t)gR(t)^{−1} exists}.

Then P (R) is parabolic, moreover, P (R) and σP (R) are opposite parabolics, so that P (R) ∩ σP (R) = Z_{G}(R), the centralizer of R in G. Such parabolics are called σ -anisotropic. Since Z_{G}(R) is a σ -invariant reductive subgroup and S ⊂ Z_{G}(R), there exists minimal σ -anisotropic parabolic subgroup P′ of Z_{G}(R) such that P′ ∩ σP′ = Z_{Z_{G}(R)}(S). Then P′′ = P′R_{u}(P (R)) is a minimal σ -anisotropic parabolic subgroup in G and P′′ ∩ σP′′ = Z_{G}(S). Since Z_{G}(R) ⊂ P (R), we have P′′ ⊂ P (R). Now, it is known that K^{0}SR_{u}(P′′) is open in G, where K^{0} is an identity component of K (algebraic analogue of Iwasawa decomposition), so if B′ is a Borel subgroup of Z_{G}(S) then B′′ = B′R_{u}(P′′) is a Borel subgroup in G that contains S, is opposite to K, and is contained in P (R). It remains to notice that all pairs (B, T) such that B is a Borel subgroup opposite to K, T ⊂ B is a maximal torus that contains a maximal σ -anisotropic torus, are conjugate with respect to K (see [loc. cit.]). In particular, there exists w̃ ∈ N_{K}(S) such that B = w̃B′′. Since B′′ ⊂ P (R), B ⊂ P (w̃Rw̃^{−1}) and hence R^{w} = w̃Rw̃^{−1} is dominant.

Returning to proof of Theorem, for any spherical subgroup H of G and any irreducible representation V_{λ} of G with highest weight λ, it is known that dim V_{λ}^{H} ≤ 1 and the set

Γ_{H} = {λ ∈ P_{+} | dim V_{λ}^{H} = 1}

is a submonoid in P_{+} (see eg. [Pa2]). In our case by [Vu], it is known that Γ_{K} = Q_{+} . Irreducible representations V_{λ} such that λ ∈ Q_{+} are known as representations of class 1. Fix λ ∈ Q_{+} . For any v ∈ V_{λ} and μ ∈ P, let (v)_{μ} denote the μ-weight component of v, and set Supp v = {μ ∈ P | (v)_{μ} ≠ 0}. Let v_{λ} be a highest weight vector in V_{λ}. Fix a K -invariant vector v_{K} ≠ 0 in V_{λ}.

Lemma 1.3. (i) Supp v_{K} ⊂ Q; (ii) λ ∈ Supp v_{K};

(iii) Supp v_{K} ∩ W λ = W_{0}λ.

Proof. Fix μ ∈ Supp v_{K}. Since 𝔥_{0}v_{K} = 0 we have μ(𝔥_{0}) = 0, and so σμ = −μ. Moreover, if τ ∈ S has order 2, then τ ∈ K, so τv_{K} = v_{K} and μ ∈ 2𝔛^{∗}(S). This proves (i). To prove (ii) let us notice that if B^{0} is a Borel subgroup, opposite to B (so that B ∩ B^{0} = T), then KB^{0} is dense in G (this follows from the description of B as in proof of Lemma 1.2). Then 𝔤 = 𝔟^{0} + 𝔨, so U𝔤 = U𝔟^{0}U𝔨. Then

V_{λ} = (U𝔤)v_{K} = (U𝔟^{0})v_{K}.

But for any x ∈ U𝔟^{0} , Supp xv_{K} < Supp v_{K} (with respect to a usual order relation on P), so λ ∈ Supp v_{K}. The last assertion follows from (i), (ii), N_{K}(S)- invariance of v_{K} and the discussion before Lemma 1.2.

Now we can finish the proof of the theorem in the case of one summand. Suppose that f ∈ Ker θ, f ≠ 0. To obtain a contradiction, it is sufficient to prove that f (viewed as an element of S𝔞) vanishes at λ̃ for all λ ∈ Q. Since f is W_{0} -invariant we need to prove that f(λ̃) = 0 for all λ ∈ Q_{+} (by Lemma 1.2). Arguing by induction on the standard order relation in P_{+} , we may assume that f(μ̃) = 0 for all μ ∈ Q_{+} , μ < λ. Suppose that f(λ̃) ≠ 0. Let π denote the K -invariant projection V_{λ} → V_{λ}^{K}. Let F denote the operator on V_{λ} of left multiplication by f (viewed as an element of U𝔤). Then, clearly, F (v_{λ}) = f(λ̃)v_{λ}. Since θf = 0 and since the Reynolds operator commutes with K -module homomorphisms, we have πF π = 0 (because πF π is clearly K -invariant, so Reynolds operator maps it onto itself, on the other hand, the Reynolds operator commutes with left (and right) multiplication on (K -invariant!) element π). Hence πF v_{K} = 0. By Lemma 1.3 and by induction we have

F v_{K} = ρf(λ̃) ∑_{w∈W_{0}} w̃v_{λ},

where w̃ is some representative of w in N_{K}(S) and ρ ≠ 0. Since π is K - invariant, we get π(v_{λ}) = 0. Therefore (U𝔨)v_{λ} ≠ V_{λ}. But

V_{λ} = (U𝔤)v_{λ} = (U(𝔨 + 𝔟))v_{λ} = (U𝔨)(U𝔟)v_{λ} = (U𝔨)v_{λ}.

Contradiction.

§2. Surjectivity of ψ : ℂ[𝔭 × 𝔭]^{K} → ℂ[𝔞 × 𝔞]^{W_{0}}

First, we can identify ℂ[𝔭 × 𝔭] with S(𝔭 × 𝔭), and ℂ[𝔞 × 𝔞] with S(𝔞 × 𝔞) by means of the Killing form. Then we can identify S(𝔞 × 𝔞) with U(𝔞 × 𝔞). Let θ denote the restriction of the Reynolds operator U(𝔤 × 𝔤) → U(𝔤 × 𝔤)^{K} on U(𝔞 × 𝔞)^{W_{0}} (with respect to the diagonal action).

Lemma 2.1. If θ is injective then ψ is surjective.

Proof. The proof is the same as that of Lemma 1.1 taking into account the discussion after it.

Now suppose that f ∈ Ker θ. In order to show that f = 0 it is sufficient to show that f (viewed as an element of S(𝔞×𝔞)) vanishes at all (λ, μ) ∈ Q×Q. We will proceed by the induction on the standard order relation. Take (λ, μ) ∈ Q_{+} × Q_{+} and suppose that f vanishes at all (λ′, μ′) ∈ Q × Q such that there exist w_{1}, w_{2} ∈ W_{0} with (w_{1}λ′, w_{2}μ′) ∈ Q_{+} × Q_{+} and (w_{1}λ′, w_{2}μ′) < (λ, μ). We need to show that f(w_{1}λ, w_{2}μ) = 0 for any w_{1}, w_{2} ∈ W_{0} .

Consider the G×G module W = V_{λ}⊗V_{μ}. Let π denote the K -invariant projection W → W^{K} (with respect to the diagonal action). Let F denote the operator of left multiplication by f (viewed as an element of U(𝔤 × 𝔤)). Since θf = 0, we get πF π = 0 as in the proof for a case of one summand. Let v_{K}^{λ} (resp. v_{K}^{μ}) denote a non-zero K -invariant vector in V_{λ} (resp. V_{μ}). Then πF (v_{K}^{λ} ⊗ v_{K}^{μ}) = 0. Let v_{λ} (resp. v_{μ}) denote the highest weight vector of V_{λ} (resp. V_{μ}). By Lemma 1.3

v_{K}^{λ} ⊗ v_{K}^{μ} = ∑_{{w′∈W_{0}/(W_{0})_{λ} \atop w′′∈W_{0}/(W_{0})μ}} (w̃′v_{λ}) ⊗ (w̃′′v_{μ}) + U,

where Supp U lies in Q × Q and for any (λ′, μ′) ∈ Supp U by induction (and Lemma 1.2) we have f(λ′, μ′) = 0. So we obtain that

∑_{{w′∈W_{0}/(W_{0})_{λ} \atop w′′∈W_{0}/(W_{0})μ}} f(w′λ, w′′μ)(w̃′v_{λ}) ⊗ (w̃′′v_{μ})

lies in the kernel of π. Now for any w′ ∈ W_{0}/(W_{0})_{λ}, w′′ ∈ W_{0}/(W_{0})_{μ} we can find an element (w_{1}(w′, w′′), w_{2}(w′, w′′)) ∈ W_{0} × W_{0} such that w_{1} = yw′, w_{2} = yw′′ for some y ∈ W_{0} and such that w_{1}λ + w_{2}μ ∈ Q_{+} . Denote by w_{1}^{i}, w_{2}^{i}, i = 1, …, r the complete set of such pairs obtained for all w′ ∈ W_{0}/(W_{0})_{λ}, w′′ ∈ W_{0}/(W_{0})_{μ}. Now by W_{0} -invariance of f and by K -invariance of π we proved that an element

∑_{i=1,...r} a_{i}f(w_{1}^{i}λ, w_{2}^{i}μ)(w̃_{1}^{i}v_{λ}) ⊗ (w̃_{2}^{i}v_{μ})

lies in the kernel of π for some non-zero integers a_{i}. It is sufficient to prove that all f(w_{1}^{i}λ, w_{2}^{i}μ) are equal to zero. So we reduced our proof to the following claim: Lemma. Suppose that an element

A = ∑_{i=1,...r} b_{i}(w̃_{1}^{i}v_{λ}) ⊗ (w̃_{2}^{i}v_{μ})

lies in the kernel of π. Then all b_{i} are equal to zero.

Suppose that some b_{i} ≠ 0. By Kostant’s refinement of PRV conjecture (see [J] or [Ku1]) for any i the 𝔤-module (U𝔤) [w̃_{1}^{i}v_{λ}) ⊗ (w̃_{2}^{i}v_{μ})] containes a unique 𝔤-submodule of highest weight w_{1}^{i}λ + w_{2}^{i}μ. By Kumar’s refinement of PRV conjecture (see [J] or [Ku2]), these 𝔤 submodules form a direct sum S = ⊕_{i=1}^{r} S_{i}. Take i such that the weight ρ = w_{1}^{i}λ + w_{2}^{i}μ is minimal and b_{i} ≠ 0.

Let π_{0} denote a G-invariant projection W → W_{ρ} on the ρ-isotipical component. Then it is clear by checking the weights that π_{0}(A) is equal either to zero or to the highest weight vector v of some irreducible submodule S_{0} of highest weight ρ. But the first case is impossible by the previous discussion of PRV, because vectors π_{0} ((w̃_{1}^{i}v_{λ}) ⊗ (w̃_{2}^{i}v_{μ})) (for various i such that b_{i} ≠ 0 and w_{1}^{i}λ+w_{2}^{i}μ = ρ) are not equal to zero by Kostant’s form of PRV and are linearly independent by Kumar’s form of PRV. Denote by π_{1} a K -invariant projection W_{ρ} → W_{ρ}^{K}. Then arguing as in proof for one summand we see that π_{1}(v) ≠ 0, so that π_{1}π_{0}(A) ≠ 0. But then π(A) ≠ 0 as well, because π_{1}π_{0} is K -invariant and hence factors through π|_{(U𝔤)A}. This completes the proof.

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Moscow State University Chair of Algebra 119899, Vorob’evy Gory, Moscow, Russia tevelev@ium.ips.ras.ru

Received April 3, 1999 and in final form June 4, 1999