classical invariant theory for finite reflection groups - CiteSeerX

Transformation Groups, Vol. 2, No. 2, 1997, pp. 145–161

c Birkh¨ ! auser Boston (1997)

CLASSICAL INVARIANT THEORY FOR FINITE REFLECTION GROUPS M. HUNZIKER Department of Mathematics, U.C. San Diego, La Jolla, CA 92093-0112, USA [email protected]

Abstract. We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley’s restriction theorem for the classical Lie algebras. In the interesting case when the group is of Coxeter type D n (n ≥ 4) we use higher polarization operators introduced by Wallach. The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of type F 4 .

Introduction Let G be a group of linear automorphisms of a finite dimensional vector space V over some field k of characteristic zero. For each integer m ≥ 1, the group G acts on the algebra P(V m ) of polynomial functions on the direct product V m := V × · · · × V of m copies of V via the diagonal action (gf )(v1 , . . . , vm ) := f (g −1 v1 , . . . , g −1 vm )

(f ∈ P(V m ), g ∈ G).

Let P(V m )G denote the subalgebra of invariant functions. If G is linearly reductive (e.g. if G is finite), then P(V m )G is finitely generated. The first fundamental problem in classical invariant theory in this situation is to find for all m an explicit finite system of (homogeneous) generators for P(V m )G . Following Weyl, we refer to a solution of this problem as a “first fundamental theorem”. In this paper we consider the case when V is an n-dimensional Euclidean space over R and G = W is a finite group generated by reflections. (Here by a reflection we mean a linear automorphism of V of order two which fixes pointwise a hyperplane.) If m = 1 then the algebra P(V )W of invariants in Received September 11, 1997. Accepted April 11, 1997. Typeset by AMS-TEX

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one vector variable is very nice. A celebrated theorem of Chevalley implies that P(V )W is generated by n algebraically independent homogeneous invariants u1 , . . . , un . We will refer to such a system of generators of P(V )W as a system of basic invariants. Explicit systems of basic invariants are known for each type of irreducible W (c.f. [Meh88]). We are now interested in the following problem: Problem. Given basic invariants u1 , . . . , un for P(V )W , is there an algorithm for constructing a system of generators for P(V m )W ?

The classical approach to this problem is the method of polarization. Let f ∈ P(V ) be homogeneous of degree d and let t1 , . . . , tm be indeterminates. Expanding f (t1 v1 + . . . + tm vm ) formally in terms of powers of the ti ’s we obtain ! f (t1 v1 + . . . + tm vm ) = tr11 · · · trmm fr1 ,... ,rm (v1 , . . . , vm ), r1 +...+rm =d

where the coefficients fr1 ,... ,rm (v1 , . . . , vm ) are multi-homogeneous functions on V m . The functions fr1 ,... ,rm are called the polarizations of f . Clearly, if f ∈ P(V )W then fr1 ,... ,rm ∈ P(V m )W . The obvious question is whether the polarizations of a set of basic invariants for P(V )W generate P(V m )W . Weyl proved in [Wey46] that this is indeed the case when W = Sn is the symmetric group on n letters acting on V = Rn by permuting the coordinates. We will see that it is also true for the Weyl group of type Bn (resp. Cn ) and the dihedral groups. However, as Wallach observed in [Wal93], it is false for the Weyl group of type Dn (n ≥ 4). Wallach generalized the method of polarization and was able to give a method for constructing a system of generators for the Dn -invariants in two vector variables. Before we describe this method we recall the definition of polarization operators. Define linear differential operators D ij (1 ≤ i, j ≤ m) on P(V m ) as follows: " d "" Dij f (v1 , . . . , vm ) := " f (v1 , . . . , vj + t vi , . . . , vm ). dt t=0 If we choose a basis for V and write vi = (xi1 , . . . , xin ) then Dij =

n !

ν=1

xiν

∂ . ∂xjν

The operators Dij are called polarization operators. They commute with the action of GL(V ) on P(V m ). This fact can be used to construct invariants. The projections πi : V i+1 −→ V i given by πi (v1 , . . . , vi+1 ) := (v1 , . . . , vi ) induce a nested sequence of algebras P(V ) ⊂ P(V 2 ) ⊂ · · · ⊂ P(V m ).

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Applying successively operators Dij (i > j) to f ∈ P(V )W we obtain precisely (up to a constant) the polarizations of f in any number of variables. Geometrically, if f ∈ P(V m ), then Dij f (v1 , . . . , vm ) is the directional derivative of f with respect to vj in the direction of vi . We now define generalized polarization operators as directional derivatives in the direction of gradients of the basic invariants. Assume that we have chosen an orthonormal basis for V . For k = 1, . . . , n define (k)

Dij :=

n ! ∂uk ∂ (xi1 , . . . , xin ) . ∂xiν ∂xjν ν=1 (1)

(k)

Note that if u1 = 12 (x21 + · · · + x2n ) then Dij = Dij . The operators Dij commute with the action of W on P(V m ) and thus can be used to construct invariants in the same manner as the classical polarization operators. One of the motivations for understanding W -invariants in several vector variables is the geometry of commuting varieties of a semisimple complex Lie algebra g. For m ≥ 1 let C m (g) := {(X1 , . . . , Xm ) ∈ gm | [Xi , Xj ] = 0 for all i, j}. Very little is known about the varieties C m (g). In [Ric79], Richardson proved that C 2 (g) is irreducible for every semisimple Lie algebra g. This is no longer true in general. Kirillov and Neretin proved in [KN84] that C m (g) is reducible for g = sln (n ≥ 4) and m ≥ 4. It is not known (except in low dimensions) whether the commuting varieties are normal, Cohen–Macaulay, or have rational singularities. In this paper, we consider the (categorical) quotient of C m (g) by the diagonal action of the adjoint group G. Fix a Cartan subalgebra h in g and let W denote the corresponding Weyl group. Then hm ⊂ C m (g) and we have an induced map between quotient varieties hm /W −→ C m (g)//G. This map is an isomorphism if and only every invariant f ∈ P(hm)W is the restriction of an invariant F ∈ P(gm )G . For m = 1 this is true by Chevalley’s restriction theorem. The idea is then to extend this theorem to the case m > 1 by applying generalized polarization operators as described above. We will show that this method works for all classical Lie algebras and G2 . After finishing this paper we received a preprint by Joseph [Jos96] in which he proves that h2 /W −→ C 2 (g)//G is an isomorphism for every semisimple Lie algebra g. His method however does not provide any algorithm for constructing diagonal Weyl group invariants in two copies of a Cartan subalgebra. The paper is organized as follows: In the first four sections we prove first fundamental theorems for the classical reflection groups. The main result of this paper is contained in Section 3 where we prove a first fundamental theorem for the reflection group of type Dn (n > 4) using generalized polarization operators. We show that the least upper bound for the degrees of a system of generators does not depend on the number of variables. In

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Section 5 we conjecture that this is true for all finite reflection groups and sketch a proof for the group of type F4 . In Section 6 we prove a variant of Chevalley’s restriction theorem in several variables for the classical Lie algebras. The author would like to thank Hanspeter Kraft and Nolan Wallach for suggesting this problem. He also thanks the referees for their remarks which helped to improve the exposition. 1. First fundamental theorem for the symmetric group Let V = Rn and let W be the group generated by the reflections about the hyperplanes xi − xj = 0 (1 ≤ i < j ≤ n).

Then W is the symmetric group Sn acting on V by permuting the coordinates. An explicit set of basic invariants is given by the power sums of degrees 1, 2, . . . , n, n ! uk := xki (1 ≤ k ≤ n). i=1

Weyl proved in [Wey46, II.3.] the following first fundamental theorem.

Theorem 1.1. The algebra P(V m )W is generated by the polarizations of the basic invariants u1 , . . . , un . We will give a proof of this theorem which is different from Weyl’s original proof. Our proof has the advantage that it generalizes immediately to the case when W is the Weyl group of type Bn . First we introduce some notation. For 1 ≤ i ≤ m let xi1 , . . . , xin denote the canonical coordinates on the ith factor of the direct product V m . Then P(V m ) = R[xij | 1 ≤ i ≤ m, 1 ≤ j ≤ n]. Let Λm := Nm×n . We write an element λ ∈ Λm as a matrix with m rows and n columns   λ11 · · · λ1n . ..  .. λ =  .. . . . λm1 · · · λmn Often we abbreviate and write λ = (λ1 , . . . , λn ) where λj denotes the jth ' λ column of λ. To each λ ∈ Λm there corresponds a monomial xλ := i,j xijij . The symmetric group W acts on Λm by permuting the columns. For λ put 1 ! wλ mλ := x . n! w∈W

The mλ form a spanning set for the space of invariants P(V m )W . Note that mλ = mλ! if and only if λ$ = wλ for some w ∈ W . Let Λ+ m ⊂ Λm be the subset of all λ such that λ1 ≥ · · · ≥ λn with respect to the lexicographic order on the columns. Then every orbit W λ ⊂ Λm contains a unique element + m W in Λ+ m . Thus the mλ with λ ∈ Λm form a vector space basis for P(V ) . Define the length of λ ∈ Λm as l(λ) := #{j | λj )= 0}.

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Lemma 1.2. Let λ = (λ1 , . . . , λn ) ∈ Λ+ m . Fix 1 ≤ k ≤ l(λ) − 1 and set $ $$ λ = (λ1 , . . . , λk , 0, . . . , 0) and λ = (λk+1 , . . . , λn , 0, . . . , 0). Then ! mλ! mλ!! = cλ mλ + cµ mµ , l(µ) 1. Put λ$ := (λ1 , 0, . . . , 0) and λ$$ := (λ2 , . . . , λn , 0). By the lemma . ! 1mλ = mλ! mλ!! − cµ mµ cλ l(µ) 1. Let λ := $$ $ $$ (λ1 , 0, . . . , 0) and λ := (λ2 , . . . , λn , 0). Then λ and λ are both even. Therefore we can use the same induction argument as in the proof of Theorem 1.1 to show that all mλ with λ ∈ Λ+ m even are contained in the algebra generated by the polarizations of the basic invariants u1 , . . . , un . By Lemma 2.2 this implies the theorem. !

3. First fundamental theorem for the Weyl group of type Dn Let V = Rn (n ≥ 4) and let W be the group generated by the reflections about the hyperplanes xi ± xj = 0 (1 ≤ i < j ≤ n).

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The group W acts on x1 , . . . , xn by permutations and changes of an even number of signs. We will often think of W as a subgroup (of index two) of the Weyl group of type Bn denoted by W $ . The functions uk :=

n ! i=1

x2k i

(1 ≤ k ≤ n − 1),

un := x1 x2 · · · xn

form a set of basic invariants (cf. [Hum90, 3.12]). It was observed by Wallach in [Wal93] that the polarizations of u1 , . . . , un do not generate the algebra P(V m )W for m ≥ 2. Explicitly, Wallach used bidegree considerations to show that the invariant n !

u :=

x1 . . . x ˆi . . . xn yi 3

i=1

cannot be expressed in terms of the polarizations of u1 , . . . , un . (Here xi = x1i and yi = x2i .) However u can be obtained from un by applying a generalized polarization operator. For odd r ≥ 1 define Pr :=

n ! i=1

yir

∂ . ∂xi

Then u = P3 (un ). The operators Pr commute with the action of W (and W $ ) on P(V 2 ). Wallach proved in [Wal93] that the algebra P(V 2 )W is the smallest subalgebra of P(V 2 ) which contains u1 , · · · , un and which is stable under the operators Pr . We now sharpen Wallach’s result by giving an explicit upper bound for the degrees of a system of generators for P(V 2 )W .

Theorem 3.1. The algebra P(V 2 )W is generated by the polarizations of the basic invariants u1 , . . . , un and the generalized polarizations Pr1 . . . Prk (un )

(ri ≥ 1 odd ,

k ! i=1

ri ≤ n − k)

of the basic invariant un . In particular, P(V 2 )W is generated by homogeneous elements of degree ≤ 2n − 2.

Proof. Let A ⊂ P(V 2 ) be the algebra generated by the polarizations of ! u1 , . . . , un . Then A contains the algebra of invariants P(V 2 )W of the Weyl group W $ of type Bn . This can be seen as follows. Let u$1 , . . . , u$n be the set of basic invariants for W $ considered in the previous section. ! By Theorem 2.1, the algebra P(V 2 )W is generated by the polarizations of u$1 , . . . , u$n . The polarizations of u$1 , . . . , u$n−1 are trivially contained in A since u$1 = u1 , . . . , u$n−1 = un−1 . Using the fact that u$n can be expressed as a polynomial in the basic invariants u1 , . . . , un for W , and the fact that

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the polarization operators are derivations, we see that the polarizations of u$n are also contained in A. We now consider the ideal I ⊂ P(V 2 ) which is generated by the homogeneous elements of positive degree in A. Let f ∈ P(V 2 )W . By the results of Wallach, f is a linear combination of monomials in the polarizations of u1 , . . . , un and the generalized polarizations of un . (The generalized polarizations of the invariants u1 , . . . , un−1 are invariant under W $ and hence can be expressed in terms of the ordinary polarizations of u1 , . . . , un by the remarks above.) Modulo I, the function f is therefore congruent to a linear combination of monomials in the generalized polarizations of un . A product of two (or any even number of) generalized polarizations of un is invariant under the Weyl group W $ of type Bn and therefore contained in A. Furthermore, any such product is homogeneous of positive degree. Hence every product of two or more generalized polarizations of un is contained in the ideal I. It follows that every function f ∈ P(V 2 )W is congruent modulo I to a linear combination (with scalar coefficients) of generalized polarizations of un . We will now show that the generalized polarizations of un of degree > 2n − 2 are in the ideal I, i.e. that f is actually congruent to a linear combination of generalized polarizations of un of degree ≤ 2n − 2. This clearly implies the theorem. Let r1 , . . . rk be odd integers. Let h be the number of ri ’s which are ≥ 3. We may assume that r1 , . . . , rh ≥ 3. In order to simplify notation we observe that Pr1 . . . Prk (un ) = c mλ , where λ=

/

r1

···

rh

1 ···

1

1 1 ···

1

0

.

(The number of 1’s in the top row of λ is n − k. Notice also that we leave a space to indicate a 0.) ,h Claim. If i=1 ri > n − k (in particular if deg(mλ ) > 2n − 2) then mλ ∈ I. Note that the claim also makes sense when h = 0. In this case mλ is an ordinary polarization of un and hence mλ ∈ I. We prove the claim on a downward induction on k = k(λ), i.e., the number of nonzero elements in the second row of λ. If k = n then mλ is a function in the y-variables. Thus mλ ∈ I. (Since u1 (y), . . . , un (y) can be obtained from u1 (x), . . . , un (x) by polarization, every invariant in the y-variables is in I.) Suppose k < n. We begin a second induction on h = h(λ). If h = 0 then mλ ∈ I, as was already observed above. Suppose h ≥ 1. For i = 1, . . . , h, let (i)

λ

:=

/

r1

···

rˆi

···

rh

1 1 ···

1 1 ··· 1 ri − 1

1

0

CLASSICAL INVARIANT THEORY

and (i)

µ

:=

/

r1

···

rˆi

···

rh

1 1 ···

1

1 1 ···

153

1

0

.

(Here the number of 1’s in the top rows of λ(i) and µ(i) , respectively, is n − k + 1.) A straightforward computation (similar to the one in the proof of Lemma 1.2) shows that n ! ! mµ(i) yjri −1 = (k − h + 1)mλ + (n − k)mλ(i) + cν mν , j=1

h(ν) k(λ) and hence by induction mµ ∈ I. Applying the polarization operator D12 to mµ we obtain h ! D12 mµ = (k − h + 1)mλ + ri mλ(i) ∈ I. i=1

We now have the following homogeneous system of h + 1 linear congruences (modulo I) in the h + 1 indeterminates mλ , mλ(1) , . . . , mλ(h) : (k − h + 1)mλ + (n − k)mλ(i) ≡ 0 (1 ≤ i ≤ h), (k − h + 1)mλ +

h ! i=1

ri mλ(i) ≡ 0.

The determinant of this system is equal to det = (−1)h−1 (k − h + 1)(n − k)h

h 1! i=1

2 ri − (n − k) .

By the hypothesis of the claim it follows that det )= 0 and so mλ ∈ I. This proves the claim and hence the theorem. ! We now study invariants in arbitrary many vector variables. Again we retain the notation of the previous sections. Recall that λ , ∈ Λm was called m even if for all columns λj of λ the sum of matrix entries i=1 λij was an even integer. We ,mcall λ ∈ Λm odd if for all columns λj of λ the sum of matrix entries i=1 λij is an odd integer. The following lemma is proved in the same way as Lemma 2.2.

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Lemma 3.2. The invariants mλ with λ ∈ Λ+ m even or odd form a vector m W space basis for P(V ) . For r1 , . . . , rm−1 ≥ 1 define

Pr1 ,... ,rm−1 :=

n !

m−1 xr2ν1 . . . xrmν

ν=1

∂ . ∂x1ν

,m−1 If r := i=1 ri is odd, then Pr1 ,... ,rm−1 commutes with the W $ -action on P(V m ).

Lemma 3.3. Suppose m > 2. Let A ⊂ P(V m )W be a subalgebra which contains P(V 2 )W and which is stable under the polarization operators. Then A is stable under the operators Pr1 ,... ,rm−1 , where r1 + . . . + rm−1 is odd. Proof. Since A contains P(V 2 )W it follows that A is stable under the operators Pr for r odd. Clearly, A is then also stable under the operators in the Lie algebra generated by the Pr and the polarization operators. For differential operators P, Q put ad(P )(Q) := [P, Q]. Then Pr1 ,... ,rm−1 =

r1 ! ad(D32 )r2 . . . ad(Dm2)rm−1 (Pr1 +...+rm−1 ). (r1 + . . . + rm−1 )!

The lemma follows.

!

Theorem 3.4. For m > 2 the algebra P(V m )W is generated by the polarizations of P(V 2 )W . In particular, P(V m )W is generated by elements of degree ≤ 2n − 2.

Proof. Let A ⊂ P(V m ) be the smallest subalgebra that contains P(V 2 )W and is stable under the polarization operators Dij (i > j). We have to show that mλ ∈ A for all even and odd λ. If λ is even then mλ is invariant under the Weyl group W $ of type Bn . Hence mλ ∈ A by Theorem 2.1. Now let λ be odd. There is one case where it is easy to prove that mλ ∈ A. Assume that λ1j ≥ 1 for j = 1, . . . , n. Then it is possible to factor mλ in the form mλ = un mλ! with λ$ even. We are now going to prove that mλ ∈ A for general odd λ by downward induction on the number q(λ) of columns of the form (1, 0, . . . , 0)T . If q(λ) = n then mλ = un and there is nothing to prove. So we may assume that q(λ) < n and that mµ ∈ A if q(µ) > q(λ). By the remarks above we may further assume that λ is of the form 

0 r  1 λ=  ...

rm−1

1 ··· 0 ··· .. . . . . 0 ···

1 ∗ ··· 0 ∗ ··· .. .. . . . . . 0 ∗ ···

 ∗ ∗ ..  . ∗

.

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155

˜ by replacing the column (0, r1 , . . . , rm−1 )T of λ by a column We define λ ˜ = q(λ) + 1 and by the induction of the form (1, 0, . . . , 0)T . Then q(λ) hypothesis mλ˜ ∈ A. We now apply the differential operator Pr1 ,... ,rm−1 to mλ˜ and obtain !

Pr1 ,... ,rm−1 (mλ˜ ) = cλ mλ +

cµ mµ

q(µ)=q(λ)+1

with cλ )= 0. Again by the induction hypothesis and the fact that A is stable under the action of Pr1 ,... ,rm−1 we conclude that mλ ∈ A. ! Corollary 3.5. The algebra P(V m )W is generated by the generalized polarizations of the basic invariants u1 , . . . , un . 4. First fundamental theorem for the dihedral groups Let V = R2 . The dihedral group W of order 2k (k ≥ 2) is generated by the reflections about the lines x1 = 0,

x1 cos(π/k) + x2 sin(π/k) = 0.

A set of basic invariants is given by u1 :=

x21

+

x22 ,

u2 :=

%k/2& /

! j=0

0 k k−2j (−1)j x2j . 1 x2 2j

(Different basic invariants are given in [Meh88].) In the following proof of the first fundamental theorem for W we use an adaption of Weyl’s original method for the symmetric group. Theorem 4.1. The polarizations of the basic invariants u1 , u2 form a complete set of basic invariants for P(V m )W .

Proof. Let A ⊂ P(V m ) be the smallest subalgebra which contains u1 and u2 and which is stable under the polarization operators Dij (i > j). Consider the subgroup W $ ⊂ W which is generated by the reflection about the line x1 = 0. Then !

P(V m )W = R[xi2 , xi1 xj1 | 1 ≤ i ≤ j ≤ m]

= R[xi2 , xi1 xj1 + xi2 xj2 | 1 ≤ i ≤ j ≤ m]. !

Let u ∈ P(V m )W . Since P(V m )W ⊂ P(V m )W we have an expansion of the form ! m u= ar1 ,... ,rm xr121 . . . xrm2 r1 ,... ,rm

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with ar1 ,... ,rm ∈ R[xi1 xj1 + xi2 xj2 | 1 ≤ i ≤ j ≤ m] ⊂ A. Consider the Reynolds operator R : P(V m ) −→ P(V m )W ,

f +−→

1 ! wf. |W | w∈W

Applying R to the equation above we obtain ! m u = R(u) = ar1 ,... ,rm R(xr121 . . . xrm2 ). r1 ,... ,rm

m Hence it suffices to show that R(xr121 . . . xrm2 ) is contained in A for every r1 rm monomial x12 . . . xm2 . Using the fact that the Reynolds operator commutes with the polarization operators we obtain

m R(xr121 . . . xrm2 )=

r1 ! r2 D rm . . . D21 R(xr121 +...+rm ). (r1 + . . . + rm )! m1

m This implies that R(xr121 . . . xrm2 ) ∈ A since R(xr121 +...+rm ) ∈ P(V )W ⊂ A.

!

5. A conjecture for general reflection groups Our proof of the first fundamental theorems for the classical reflection groups is a case by case analysis. It is therefore natural to ask whether there exists a conceptual method that works in general. The first guess is that generalized polarization would be such a method. Recall the definition of generalized polarization operators from the introduction. For k = 1, . . . , n define n ! ∂uk ∂ (k) Dij := (xi1 , . . . , xin ) . ∂xiν ∂xjν ν=1

If W is a classical reflection group or a dihedral group, then the algebra of invariants P(V m )W is the smallest subalgebra of P(V m ) which contains (k) P(V )W and is stable under the operators Dij . Unfortunately, this is not true in general. Consider for example the Weyl group W of type F 4 acting on V = R4 . The reflecting hyperplanes for this group are xi = 0,

xi ± xj = 0

(1 ≤ i < j ≤ n),

1 (x1 ± x2 ± x3 ± x4 ) = 0. 2

Clearly, W normalizes the set of linear forms xi ± xj (i )= j) and so the functions . ! f2k := (xi + xj )2k + (xi − xj )2k 1≤i d := max{deg(ui )}. In particular, the algebra P(V m )W is generated by the elements of degree ≤ d for all m. Sketch of proof for the Weyl group of type F4 . We restrict ourselves to the case when m = 2 and retain the notation from above. Let W $ denote the reflection group of type B4 considered as a subgroup of W . The functions ! f2 , p4 , f6 , f8 are W $ -invariant and form a set of basic invariants for P(V )W . (Here p4 = x41 +. . . +x44 .) The first fundamental theorem for the Weyl group ! of type B4 implies that if u ∈ P(V 2 )W then ! rk r1 u= ar1 ,... ,rk D21 p4 . . . D21 p4 , 1≤r1 ,... ,rk ≤4

where the ar1 ,... ,rk are elements of the algebra generated by the polarizations of f2 , f6 , f8 . Let I ⊂ P(V 2 ) denote the ideal generated by the polarizations of f2 , f6 , f8 , f12 . Then modulo this ideal we have the congruence ! rk r1 u≡ cr1 ,... ,rk D21 p4 . . . D21 p4 . 1≤r1 ,... ,rk ≤4

where the cr1 ,... ,rk are now constants.

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Claim. For k ≥ 4

rk r1 D21 p4 . . . D21 p4 ≡ 0.

The proof of the claim is rather tedious and we do not include it here. Since it is a finite problem (it suffices to consider the case k = 4) we were also able to check the claim (at least in finite characteristic ≤ 31991) by using the computer program MACAULAY. ! By the previous remarks the claim implies that if u ∈ P(V 2 )W ⊂ P(V 2 )W is homogeneous of degree > 12 then u ≡ 0. ! 6. Chevalley restriction in several variables ¿From now on the base field is C. Let G be a connected semisimple linear algebraic group over C. The group G acts on its Lie algebra g by the adjoint action. We fix a Cartan subalgebra h of g. Then the Weyl group W := NG (h)/ZG (h) acts on h as a finite (complexified) reflection group. The celebrated restriction theorem of Chevalley implies that restriction of functions on g to h induces an isomorphism of algebras ρ : P(g)G −→ P(h)W . We will discuss an analogue of this theorem in several variables. For m ≥ 2 define the mth commuting variety of g as C m (g) := {(x1 , . . . xm ) ∈ gm | [xi , xj ] = 0 for all i, j}. Then C m (g) is an affine subvariety of gm which is stable under the action of G. Clearly, C m (g) contains the hm . We will need the following lemma due to Richardson [Ric88]. Lemma 6.1. The closure of each orbit in C m (g) intersects hm nontrivially.

This lemma implies that if two invariant function f, h ∈ P(gm )G are identical on hm then they are identical on C m (g). In other words the restriction map ρ : P(C m (g))G −→ P(hm) is injective. (Here P(C m (g)) denotes the ring of regular functions on C m (g), i.e., the functions which are restrictions of polynomials on gm .) We now prove a generalization of the Chevalley restriction theorem for the case when g is a classical group. Theorem 6.2. Let g be a complex simple Lie algebra of type An , Bn , Cn , Dn or G2 . Then the Chevalley restriction map ρ : P(C m (g))G −→ P(hm)W is an isomorphism of algebras for all m ≥ 2. In particular, P(C m (g))G is a normal Cohen–Macaulay domain with rational singularities. Proof. Choose an orthonormal basis e1 , . . . , eN with respect to the Killing form for g in such a way that e1 , . . . , en form a basis for h. Let X1 , . . . , XN denote the corresponding coordinates on g and x1 , . . . , xn the corresponding

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coordinates on h. (Note that xi is the restriction of Xi .) Let u1 , . . . , un be basic invariants for P(h)W . By Chevalley’s theorem there exist u ˜1 , . . . , u ˜n in P(g)G such that ρ(˜ ui ) = ui for i = 1, . . . , n. We now consider generalized polarization operators on P(hm) resp. on P(gm ). For k = 1, . . . , n let (k) Dij

and similarly

n ! ∂uk ∂ := (xi1 , . . . , xin ) ∂xiν ∂xjν ν=1

˜ (k) := D ij

N ! ∂u ˜k ∂ (Xi1 , . . . , XiN ) . ∂X ∂X iν jν ν=1

(k) ˜ (k) are W -equivariant and G-equivariant, respecThe operators Dij and D ij tively. They satisfy the following compatibility condition

˜ (k) = D (k) ◦ ρ. ρ◦D ij ij Hence the generalized polarizations of the basic invariants u1 , . . . , un are restrictions of elements in P(gm )G . Explicitly, 1 (kr ) 2 ˜ ˜ (k1 ) ul ) = D (kr ) . . . D (k1 ) (ul ). ρ D ir jr . . . Di1 j1 (˜ ir jr i1 j1 Now if g is of type An , Bn , Cn , Dn or G2 then the results of the previous sections imply that the generalized polarizations of the basic invariants u1 , . . . , un generate the algebra P(hm )W and hence ρ : P(gm )G −→ P(hm)W is surjective. !

We return to the general case. Let C m (g)//G and hm //W = hm /W denote the affine varieties spec P(C m (g))G and spec P(hm)W , respectively. The inclusion map ι : hm −→ C m (g) induces a morphism ¯ι : hm /W −→ C m (g)//G of the quotient varieties. Note that ¯ι∗ = ρ and that ρ is an isomorphism of algebras if and only if ¯ι is an isomorphism of affine varieties. The following theorem implies that ¯ι is at least a homeomorphism. Theorem 6.3. The map ¯ι : hm /W → C m (g)//G is a finite bijective morphism. In particular, hm /W is the normalization of C m (g)//G. Proof. The lemma implies that ¯ι is surjective. In order to prove that ¯ι is injective we show that if (h1 , . . . , hm ) ∈ hm then Ad(G)(h1, . . . , hm ) ∩ hm = W (h1 , . . . , hm ).

Choose h0 ∈

,m

i=1

Chi such that ZG (h0 ) =

m 5

i=1

ZG (hi ).

160

M. HUNZIKER

Let g ∈ G such that Ad(g)(h1, . . . , hm ) ∈ hm and so Ad(g)h0 ∈ h. By the Chevalley restriction theorem (in one variable) there exists an element u ∈ NG (h) such that Ad(u)h0 = Ad(g)h0 . Now u−1 g ∈ ZG (h0 ) ⊂ ZG (hi ) and therefore Ad(u)hi = Ad(g)hi for all i. We now prove the finiteness. This follows from a general observation which is due to Hilbert (see [Kra84, II.4.3, Satz 8]). For sake of completeness we include a proof in our special case. Recall that ¯ι being finite means that P(hm)G is finitely generated as a P(C m (g))W -module via ρ = ¯ι∗ . Put A := P(hm )W and B := P(C m (g))G . We may think of B as , a graded subalgebra of A via ρ. Consider the augmentation ideals m := n>0 An , , n := n>0 Bn . Choose homogeneous elements f , . . . , f ∈ A of respective 1 k , degrees d1 , . . . , dk such that n = i Bfi . , By Hilbert’s Nullstellensatz there exists an integer n0 ≥ 1 such that mn0 ⊂ i Afi . Moreover there exists an integer n1 ≥ 1 such that An ⊂ mn0 for n ≥ n1 and so An ⊂

k !

fi An−di

i=1

(n ≥ n1 ).

This implies that A as a C[f1 , . . . , fk ]-module (and hence as ,a B-module) is generated by the finite dimensional vector space A≤n1 := n≤n1 An . !

Remark. The fact that hm /W −→ C m (g)//G is the normalization also follows from a more general result of Luna in [Lun75]. Let T ⊂ G denote the maximal torus with Lie algebra h. Then T is the isotropy group of a generic closed orbit in gm . Since (gm )T = hm , we have by Proposition 2.2 in loc. cit. hm /W −→ Ghm //G is the normalization. By Lemma 6.1, Ghm //G = C m (g)//G. References

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CLASSICAL INVARIANT THEORY

161

[Kra84] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik D1, Vieweg Verlag, Braunschweig, 1984. [Lun75] D. Luna, Adh´erences d’orbite et invariants, Inventiones Math. 29 (1995), 231–238. [Meh88] M. L. Mehta, Basic sets of invariant polynomials for finite reflection groups, Comm. Algebra 16 (1988), 1083–1098. [Ric88] R. Richardson, Conjugacy classes of n-tupels in Lie algebras and algebraic groups, Duke. Math. J. 57 (1988), 1–35. [Ric79] , Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), 311–327. [Wal93] N. R. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), 779–816. [Wey46] H. Weyl, The Classical Groups, Princeton Math. Series 1, Princeton University Press, Princeton NJ, 1946. Russian translation: G. Ve!l", Klassiqeskie gruppy, ih invarianty i predstavleni!, Moskva, IL, 1947.