Representations of simple lie groups with regular rings ... - Springer Link

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Inventiones math. 49, 167- 191 (1978)

mathematicae ~) by Springer-Verlag 1978

Representations of Simple Lie Groups with Regular Rings of Invariants Gerald W. Schwarz* Brandeis University, Department of Mathematics, Waltham, Massachusetts 02154, USA

w O. Introduction In the theory of compact transformation groups, an important class of actions have been those which locally look like the "'simplest" representations of the classical groups. These simplest representations all share the property that their rings of invariants are regular, i.e. are isomorphic to a polynomial ring in some number of indeterminants. (We will call such representations coregular.) One is naturally led to the question : Which representations of the compact Lie groups are coregular? More generally, which representations of the reductive complex algebraic groups are coregular? Wu-Yi Hsiang has answered the first question. In this paper we determine the coregular representations of the connected simple complex algebraic groups. This classification has also been carried out by O.M. Adamovi6 and E.O. Golovina, and their work is to appear in a collection of the University of Yaroslav. In w1 we discuss necessary conditions for coregularity. One such condition is that every irreducible factor of a coregular representation must be coregular. Thus it is important to know the list of irreducible coregular representations of the connected simple complex algebraic groups, and this list was determined by Kac, Popov, and Vinberg [12]. Not surprisingly then, the difficult part in our classification is the determination of the rings of invariants of those representations satisfying our necessary conditions. For most representations of the groups A,=SL,+I=SL(n+I, C), classical invariant theory is sufficient for our calculations. These cases are discussed in w2. For spin representations and representations of the exceptional groups, there is no analogue of classical invariant theory, but the techniques developed by the author in [29] suffice. These matters are discussed in w3. In w we consider several miscellaneous topics related to coregularity. In particular, we discuss the question of when the ring of all polynomial functions on a representation space is a free module over the ring of invariants. (We call such a representation c@ee. In a forthcoming paper we classify the cofree * Research partially supported by NSF grant MCS 77-04522. AMS (MOS) subject classifications 1970. Primary 15A72, 20G05

0020-9910/78/0049/0167/$ 05.00

168

G.W. Schwarz

representations of the connected simple complex algebraic groups.) In an appendix we comment upon the techniques and results of [12].

w1. Necessary Conditions and Main Results G will always denote a reductive complex linear algebraic group. Recall that a

representation of G is a complex vector space V (the representation space) together with a morphism of algebraic groups ~p: G ~ GL(V). We shall denote a representation as above by ~p or by the pair (V, G). V will always denote a representation space of G. The algebra of invariants C [ V ] ~ is always noetherian [21]. We say that a coregular representation (V, G) is maximally coregular provided that V~= {0} and that (V| V1, G) is not coregular for any non-trivial representation space V~ of G. It is "obvious" that every coregular representation (V, G) with VG= {0} is a subrepresentation of a maximally coregular representation. For G simple and connected this fact follows from:

Classification I. Let G be connected, simple, and simply connected. Suppose that V G= {0} and that (V, G) is coregular. Then, up to an outer automorphism of G, (V, G) is a subrepresentation of one of the representations listed in Tables la, 2a, 3a, 4a, and 5a, and the representations in these tables are maximally coregular. We now recall some standard facts about representations of G; proofs can be found in [18] or [29]. Let x e V. The conjugacy class (Gx) of the isotropy group Gx is called an isotropy class of tp=(V, G). We say that the isotropy class (or isotropy group) is closed if the orbit Gx is closed. In this case Gx is reductive and we call the representation q~x of Gx on T~ V/Tx(Gx) the slice representation at x (or the slice representation of Gx). The slice representation ~o~ and isotropy class (Gx) are said to be proper if Gx+ G. G~ determines q~ since q~x@(Ad G)lGx=tplax@Ad Gx. The (finitely many) closed isotropy classes of tp are partially ordered, where (L)=2,

((ol(A,) , A . _ 2 ) = q ) I + 0 2.

(1.s) (~o~+q,~+,q,, +b~,T, A,,)~(~o2+~o~ +~,~o, +b~o~'+0, a+b>l,

n>4,

A._~);

~__>4,

(~oi(A,),A,_z)=q01+0 2.

(1.9) (~p3+qo~, A{,) --+(qOl|

+01, A 2 x A2);

(qol(Ao), A 2 x A2)=q01+ ~ ; + 0 , .

w 2. Calculating lnvariants Using Classical lnvariant Theory In this section we indicate how one can compute invariants using classical invariant theory (abbreviated CIT henceforth). This method allowed us to establish the coregularity of " m o s t " of the representations of SL,, satisfying our necessary conditions. We also derive information about the decomposition of symmetric powers of certain representations of SL,. CIT for S L , (see [321) says that the invariants of copies of C" (typical points denoted x 1, x 2, ..) and copies of (C")* (typical points denoted gl, g2,,..) are generated by determinants [xi~, .... xi,,] and [gil,--.,~i.] and by contractions (xl, ~}.The relations of these invariants have generators of the following form: (2.1)

Ix 1..... x,] [Xo,X,+ 1..... x2,_1] = [ X o , X 2 , ..., Xn] [X1, Xn+ 1. . . . ,X2n_l]-]-"" -}-IX1, X2, . . . , X n _ I , X o ]

[Xn,Xn+ 1, . . . , X i n - 1 ] .

(2.2} [xl, ..., x,] (Xo, go} = [Xo, x2, ..., x,] @1, ~o5 + ' " + [xl . . . . . x, ,, Xo] ( x , , ~o}.

(2.3)

[x 1..... x,] [~1

....

,

{,] =det((x,, {j)).

One also has relations (2.1') and (2.2') where the roles of the x i and ~j are interchanged. As an example of a typical computation we prove:

Proposition 2.4. For the groups A,, n> 2, one has: (1)

~o21+qo*+q) * is coregular.

(2) sJ(v,~,)=@{v,~",

9 ~p,2~ . a l + 2 a 2 + . . . + n a , , + m ( n + l ) = j

for some meZ+}. (3)

sJ (,o2 ~-..~-@{ q)~'.., (P2k" ak" a l + 2 a 2 + " ' + k a k = j } ,

(4)

SJ(D2 =@{q0~' ... q)zk" ak. al +2a2 +"" +k a k + m ( k + 1)=j forsome meZ+};

n=2k.

n=2k+l.

Proof Let f be a generator of the invariants of (V, G)=(~oZ+~p~ +q)T, A,). We may assume that f is homogeneous in each of the three irreducible factors of

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G.W. Schwarz

(V, G), and polarizing f we obtain a multilinear invariant f(s~, S2 . . . . . 0)1, 0)2 . . . . . rh, r/2 .... ) where the sl, 0)j, and Ok are in copies of ~p~, ~p*, and ~o*, respectively. Now f is determined by its action on decomposable tensors, so we replace each s i by X i l | 1 7 4 and each 0)j by ~ l | 1 7 4 where the x n and (~m lie in new copies of ~oa and ~o*. We thus obtain a multilinear invariant f ( x , , ~ , . , rlk) which is symmetric in i, j, and k; symmetric in 1 for fixed i; and skew in m for fixed j. By CIT, we may express f a s a sum of products of determinants and contractions of the x , , Cjm, and rlk. Let P denote one of these products. Case I. P is of the form [xol .... ,x,1] Po: Suppose that Po has the form [x02 .... ,x,2, ~ ' " , ~ , ] [x~r+,t2, ""] Pl or [x02 ..... xr2, 0%1 .... ,e,] (xl,+,12, "5 P2 Where no ~s is an x~2 for i < n , r + l < s < n . In the former case the symmetries of the x n force f to contain the term

P-/]1 [x01 . . . . .

Xnl] (IX(r+ 112' X12 . . . . . Xr2, 0{r+l,

" " , 0{hi IX02, "" '] -it-...

+ Ix02, ..., x~-1t2, xlr+,2, ~+1 . . . . . ~,] [xr2, ..-]), and using (2.1) we see that this term is a sum of terms of the form [Xo~ ..... x , , ] [Xo~, ..., x~+ll~,/3,+ ~, ...,/3,] g . The case Po = [...] (x{,+ltz, .)P2 leads to terms of the same form; one uses (2.2) rather than (2.1). Continuing inductively we may reduce to the case where P = [ x m . . . . . Xna][Xoz,...,Xnzl]P4. But [Xol,...,Xnl'][Xo2,...,Xn2 ] plus terms guaranteed by symmetry is the polarization of the degree n + l generator of C [q02]A", so P plus all terms demanded by symmetry corresponds to an invariant divisible by this generator. There remains the case P = [xm, ..., x,1 ] (X02 , 0{0) ... (Xn2 , O~n) P1 where the cq are {'s or r/'s, O < s < n . But the X~l are skew in i, hence so are the xi2, and from (2.3) we see that P plus other terms of f equals [ x o l . . . . . x , , ] [x0~ . . . . . x , ~ ] [So, .. , ~.] 8 .

Thus we find no new generators. Case 2. P contains a determinant of the {i,, and qk: Using (2.1') and (2.2') as in ([29] w15) we may reduce to the case that P is of the form [{11, {12, -.-, {~1, {k2] Po ( n = 2 k - 1 ) , or of the form [{n, {12 . . . . . {tl, {k2, r/i] Po (n=2k). Thus we find one new generator. Case 3. P is a product of contractions: If no ~s.~ appears in P, then the only type of invariant one sees is of the form ( x n , ql) (X12, q2) + (X12' ql) (Xll' /~2)" If some ~j,. appears in P, then the symmetries of the x . , ~jm, and ~k show that P must contain a factor like (2.5)

( X l 1 ' ~11) (X21' ~12) (X31' ~21) (X41' ~22) "" (X(2r)l, ~r2)

Regular Rings of Invariants

173

or (2.6)

(Xll, ~11) ((x21, ~12) (x31, ~21) "'' (X(2r}l, ~r2) (X{2r+ 1)1, g]l) '(X22' r

1) (X32, ~(r+1)2) (X42, r

""

(X(2r+1)2, ~(2r)2) (X12, ~2)" If 2 r = n + 1 in (2.5), then (2.3) shows that we m a y re-express (2.5) plus other terms of f as a product of determinants, and if 2 r > n + 1 then the result is zero. T h u s (2.5) can only correspond to generators of C [ V ] a for 2 r < n + 1. Similarly we m a y restrict to 2 r + 1 < n + 1 in (2.6). We have shown that C I-V] a has at most 3 + In/2] + [ ( n - 1 ) / 2 1 = n + 2 generators, where dim C I V ] G= n + 2 . Thus (V, G) is coregular, as claimed in (1). We now prove (2). Let (V, G)=(~o 2 + s qh, A,), s arbitrarily large. Then C [ V ] a is generated over C[s~ol] ~ by contractions 01c~0*| where q~*~C[cp~] and ~o c C Is % ] are irreducible representations of G. We use CIT to determine C [ V] G, and then we are able to determine the G-module structure of C[q~ 2] and its dual S* (q~). As before, to c o m p u t e generators of C [ V ] ~ we reduce to considering multilinear invariants f(xgk, yt) where f is symmetric in i and symmetric in k for fixed i, k = 1, 2. Here the xik and y~ lie in copies of ~0l. N o w f i s a sum P~+ ... +Pt where each P~ is a product of determinants. Using (2.1) (see [8] p. 328) one can reduce to the case that each P~ is a product of factors like (2.7)r

[ x n . . . . . xrl, Yl .... , Y,-r+l] [-X12

....

'

Xr2, Yn-r+2,

"",

Y2n-Zr+2]

where 0 < r < n + l . Let P denote one of the P~. We m a y assume that f equals P plus all terms guaranteed by symmetry, and we assume that P contains no factor which is a determinant of the Yr. Let a~ denote the number of factors of type (2.7)~ occurring in P, l < r < n + 1. Then clearly the representation q~ corresponding to f is determined by the sequence (ax .... , a , + 0 , and ~ocsJ(~o~) where j = a~ + 2 a 2 + . . . + ( n + 1) a,+ ~. Letting the x~k be weight vectors for the action of a maximal torus of G one can see that q~= ~o~~ ... r "" We have proved (2), and similar arguments prove (3). []

w 3. Other Methods We present three other techniques for calculating rings of invariants. In case the representation (V, G) has non-trivial principal isotropy class, one can apply the following result of Luna and Richardson:

Proposition 3.1. Let H be a principal isotropy group of (V, G), and let N denote No(H)/H. (1) resH: C[V]O--.C[Vn] N is an isomorphism [19]. (2) Let (L) be a closed isotropy class of(V, G) where we may assume that H c L .

174

G.W. Schwarz

Then (M = NL (H)/H) is a closed isotropy class of (V •, N), and the correspondence (L)~-*(M) is a bijection ([29] w 11). [] Remark 3.2. Let H be as above, and let W be a representation space of G. It is not in general true that the inclusions VH --, Vand W H ---, Winduce an isomorphism of (C [V] | W)G with (C [ V~] | WU)N. Sufficient conditions are given in ([29] w11). Example 3.3. Let (V,G)=(3(0 z, As). The principal isotropy class is ( H = C * ) where ((01(As), C*)= 3 vI + 3 v 1. The universal cover N of No(H)/H is the semidirect product A 2 )< A2 x r t Z 2 , where Z 2 interchanges the two factors of type A. By 3.1, C[V]~_~C[Vn] N where (V n, N)=(3(01| A 2 x A 2 )-4 2q~2+3~o* ; n__>4, neven 2~o2+3~o* ; n_>_7, n o d d ~%+qg*+3q~; n > 4 , n e v e n ~o2 + q~* + 2 q~l + ~p'~; n=>4 ~o2 + q~, + ( n - 2 ) qo~" r +(n-- 1) ~p* ~p~Z+ {p2 + q91; n > 4 ~o~+~o2+~o* ; n > 4 , neven rp~Z+~p2+~o*; n > 5 , n o d d ~o~+q~*+~oa; n > 4 , neven q)2 + qo~+ q~*; n ~ 4 2~

nZ--n+l

(n+ 1) (D, n(n- 1)

1~02 -1- (p2 1

'-01 ~o._ ~ + qo1

89 1)+1 89 l)+l 89 2n+l 2n+l 2n+l 2n+l

2n+l 2n+l 2n+l 89 89 n+l n+l n+l n+l n+l n+l n+l n

1)+ 1 1)+1

(3n - 6 ) (2), 4 ( 3 ) , 89 - 2 ) ( n - 3) ( 4 ) ( 2 n - 2 ) (2), 2 ( 3 ) , 8 9 1)(n-2) (4) , ~ n ( n - 1 ) ( 4 ) 8 , ( 2 n - 5 ) (5> 2 , 2 , 2 (4>, ( 2 n - 5 ) 2 (3>,6 ( 4 > , ( 2 n - 7 ) 6 , ( 2 n - 5) (5} 5 , 3 , 3 , , (7>, (8>, 89 1)(n - 2 ) (9> (7), 89 1) (3), (7), (8>, ( n - 2 ) (7>, (9>,(n--2)(10>, (11> (3>, 2 < 7 > , ( n - 1) 2 < 7 > . ( n - l) (14) (2, 0), (3, o) ..... (n, 0), < 16>

Table l a ' (G=SL.=A._fl

qo

d

Generators

1

( n + 2 ) (&

2 3 3' 4 5

(n + 1) ~o~+ ~o~' q~2 +4q~; n=4 ~o2 +4~p~; n>5 qg~ +2q)~; n>3 2q~z+3~o*; n = 5

2n+ I 2n+l 7 7 4 11

(n+ 1) ( l ) , ( n + 1) (1>,7 ( 3 ) 8(3> (7), 3 (8), 6 (4>, 6 (5>

89

l)(n+2) (1)

Table I b (G =SL. = A._ i) Invariants

01 = A" % c S" (n %) O, c % | 01~q~21| 01 ~ q)2|

2iczS~q)e@s"-ei((n-2i)q)O;

2I

c-S'q~2|174174

qo*);

2i+j=2k+ll>=l;

11

2 * 2 0~ c_ s174163 |163 * =~Pl|

12

_ 2 O1 (--s163

2

s

*

i+j=2k+l2l

s174163* .

n=2k+ 1

c2S'(s163176

ls

i,k>=l

13

o, ~=s174163 ,~_s'(s174

14

0,=s163

15

2 2 01=s174163

16

01cs163

1 5l+tp,, ~f~01>6) $2 tp.= tp2 + q~._ 4 (n>4);

qO2 I +q~2 =Anq) 1

3NnN7

A2~O.-I=AZ(P,,=q).-2+tP. 6 (n>6) r n 3+(,0._5 (n>5)+fign_v 0!>7)

B,, or D.

01cS2(P1 or 9q|

B.,

In 2, 4, and 6 below, cr=q%_l(D.), ~o.(D.), or q~.(B.)

32. If G is semisimple and C [ V ] ~ = C , then (V, G) is strongly prehomogeneous ([20] Cor. 2 and Lemma 4). Corollary 4.7. Let 1/1, H, p, etc. be as in 4.6. Suppose that (1/1, H) is strongly prehomogeneous with open orbit H x. Then mR= dim (W*) u~.

Proof The hypotheses imply that C [G x n VI] -~ C [G x n H x], and using Frobenius reciprocity we see that Hom(W, C [ G x u V1])G~ Hom(W, C[G/H~])~

u~.

[]

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G.W. Schwarz

Remark 4.8. Let (F, G) be coffee, and let S be as above. One can sometimes use properties of S to prove the coregularity of a representation (V+ V~, G). For example, consider (V+ V1, G)=((01 ~P,,-x +qh, A , - 0 = la-20- From the slice representation of a maximal torus (C*)"-1 one sees that all bihomogeneous elements of C [-V + V1]G occur in degrees (k, In); k, Ie Z +. Since Sl" (qh)=(~o~) * , t, , c [ , v + v~]~ is generated by C[V] G and the contractions of occurrences of q~= in S with (%*)~"=C[V1]. But (p~" occurs with multiplicity one in S, and the generator is in degree / ( ~ ) ( [ 1 3 ]

Thm. 0.11). Hence C[V+V1] c' is generated by CEV] a

and an invariant of degree

(("))

2 ' n , and C[,V+ V1]~ is a regular ring.

Remark 4.9. One can often generate new coregular representations from old ones by "increasing the group" and using Proposition 3.1. For example, p = (2~%+2rpl, Azk_0 is coregular for all k>2, and adding the AI factor of p'= (2r + r174 A2k_ 1 x A'0 does not change the ring of invariants. Applying 3.l to p' one obtains the coregular representation (2S2Ck+Ck,SL~) where SL~ denotes the elements of GL k of determinant _+ 1. Finally, to help "explain" why the principal isotropy groups of all the representations in 1.2.1 are connected, we offer the following:

Proposition

4.10. Let G be connected and semisimple, and let H be a principal isotropy group of(V, G). Assume that (V, G) has generically closed orbits, dim V/G = 1, and that ZG(V ) is non-singular in codimension one. Then H is connected.

Proof If dim V 3. From the homotopy sequence of the fibration H ~ G ~ G/H we see that nl(G/H ) maps onto no(H), so it suffices to prove that nl(G/H) is trivial. Let J' be a form generating C [ V ] G, e = d e g f . Let C* act on V with weight 1 and on C with weight e. Luna's slice theorem shows that we have a fibration G/H

, v: ~

c - (0},

and quotienting by C* we see that (G/H)/Z e ~-P(V): where P(V) is the projective space (V-{0})/C* and P ( V ) : = VJC*. It thus suffices to show that n~(P(V):)~_Z,,. If L is a generic hyperplane in V, then Lc~Z~(V) will be non-singular in codimension one. We may choose L so that (providing dim V > 4 ) we have na(P(V):)~-n~(P(L):O ([9]). By induction n~(P(V):)~-n~(P(M):) where M is a 3-dimensional subspace of V and the zero set of J'IM has at most an isolated singularity at 0. Hence flM = hj where j~ Z + and h a C [M] has at most an isolated singularity at 0. Then by a theorem of Zariski (see [3]), nl(P(M):)~Ze/j. But n I(P(M):) has quotient Z~, hence j = 1. []

Remark 4.11. Suppose that dim V/G=l. Then ZG(V ) is non-singular in codimension one if and only if it is normal ([-22] p. 391). If Z~(V) contains finitely many G-orbits of the same dimension whose complement has codimension > 2 in Z~(V) (e.g. (V, G) is exceptional--see [26]), then ZG(V ) is normal.

Regular Rings of Invariants

187

w5. Appendix The completeness of our classification of coregular representations depends upon the completeness of the classification announced by Kac et al. in [12]. Because of this we have verified their results. Not all of the calculations involved are straightforward, and below we indicate how we handled most of the non-routine cases, e.g. those involving the exceptional groups. We first recall and illustrate the use of the methods of [12]. Let T denote a copy of C* which is contained in a simple 3-dimensional subgroup of G. Then for any representation p=(W, G) the non-zero weights of (I4/,, T) occur in pairs _+/~, and we let q(p) or q(W) denote the number of such pairs. Assume that (V r, NG(T)/T ) has finite principal isotropy groups (abbreviated FPIG). Then the argument of Example 1.4 shows that there is a closed orbit G v, ve V r, such that rank G,,= 1. Hence the identity component (G,.)~ of G,. is either T or a simple 3-dimensional subgroup of G. Let ~ denote the slice representation of G,,. Then q(~b)=q(V)-q(Ad G)if (G,,)~ = T, else q ( O ) = q ( V ) - q ( A d G)+ 1. A careful analysis shows that ~ (hence (V, G)) is not coregular if either q(O)>4, q(O)= 3 and (G~,)~ = T, or q(O)= 2 and G~, centralizes T ([12]). To sum up, we have the following result: Theorem 5.1 ([-12]). Let T be as above. Suppose that (1) (V "r, NG(T)/T ) has FPIG. (2) q ( V ) - q ( A d G ) > 3, or q ( V ) - q ( A d G ) = 2 and there is a closed orbit Gv such that G,: centralizes T, (G,,)~ T. Then (V, G) is not coregular. []

Remark 5.2. Suppose that (V, G) is irreducible and not coregular, and suppose that G is connected and simple, rank G>2. Then there is almost always a T satisfying 5.1.1 such that q ( V ) - q ( A d G)>3. The only exceptions are (qo1 ~0z, A3) (here q ( V ) - q ( A d G ) = 2 and G,,= T), and (q~, A3) (in this case one finds a finite closed isotropy group with non-coregular slice representation; see [12]). Note that one may replace the group NG(T)/T of 5.1.1 by N, where N denotes (N~;(T)/T)~ or a finite cover thereof. Then to apply Theorem 5.1 one must be able to show that the representations (V, T) and (V r, N) are "large". The following result (a special case of which is stated in [12]) allows one to estimate (V, NG(T)~ hence also (V, T) and (V r, N). Theorem 5.3 ([16]). Let G be connected, and let H be a connected reductive algebraic subgroup of G. Suppose that Pl, D2 p,, are irreducible representations of G such that (Pl, H) contains the irreducible representation zi of H with multiplicity n~, i= 1, ...,m. Then (PIP: ... Pro, H) contains rl r 2 ... r,, with multiplicity at least max{n 1..... n,,}. [] ....

,

Remark 5.4. In our applications H=NG(T) ~ for some T = C * c A ' l c G , and it usually turns out that H e l l ' = Nc;(A'0 ~ It sometimes gives better results or is notationally more convenient to apply 5.3 to estimate the restriction of a representation of G to H', and then to restrict to H.

188

G.W. Schwarz

Example 5.5. Let T c A' 1 c A, = G where (~ol(G), A'I) = q~'l + 0,_ 1. Let j e Z +, j > 1, let H=Na(T) ~ and let H ' = N G ( A ' 0 ~ Then H=(T• and H'=(A'l x A , _ z x C * ) / Z z , _ 2 where (tpl(G),H')=q~'x| +tpl| 2. By 5.3, (~p{(G), H') contains each term (q~'l)k| j- k| Vk,+ k-2j with multiplicity at least 1, 0 < k 9 . We show that tp~o5 is not coregular: Let T, ,6,'1, and H ' = (A' 1 x An_ 2 xC*)/Z2n_2 be as in Example 5.5. Then N = A n _ 2 x C * is a finite cover of(NG(T)/T) ~ If (p = ~p/, (P3, ..., (P,- 1, (P], or ~0k,,k > 2 , then (tp r, N) contains a factor ~ | 1 7 4 r where z~ and p are irreducible representations of An_ 2 and p, r > 0 . In particular, (q~r, N)=tP3|174 If q~=~p~' ... ~p,~"then it follows from 5.3 that ((~oq~5)r, N) contains a factor aq~3| 8 (one has estimates s, t > 10 if a I and a 2 are not 1). Using the tables of [1] and [7] one sees that both a q~3 and zip5 have non-trivial rings of invariants and that a~o 3 + r q~s has F P I G . It follows that a t P 3 | tP5| t and ((tp ~o5)r, N) have F P I G . Now ( ~ o s ( A , ) , A ' l ) = - ' ( n : l ) t p ' l + 0

and (AdAn, A 0' =(q~ ,0z + ( 2 n - 2) q~'l+ 0.

Using 5.3 one sees that q(~p\'~ : % )> q(% ), so q(q~q~5)- q(Ad An) > -(n 4 1)- _ (2 n - 1) > 53 (since n>9). Thus q3q~5 is not coregular. A similar argument establishes that (~o q~, A,) and (~o ~o,_ ~, An) are not coregular, 5 < i < (n + 1)/2. In a few cases one needs to use tensor product identities to improve upon the estimates of 5.3. (All the required identities can be found in [15].) For example, suppose that (V, G)=(~p z, B3) and that T e A ' I where (tP3(B3), A'0=2q~'l+04. T h e n 5.1.1 holds and q(Ad G)=7. The estimate q(V)> 10 is sufficient to establish that (V, G) is not coregular, while 5.3 only shows that q ( V ) > 9 . However, using the identity SZq~3 =~p~ +01 one can see that q ( V ) = 11. Sometimes one has to explicitly exhibit invariants in order to show that 5.1.1 holds:

Example 5.7. Let (V,,G)=((pltP2 , A5), and let T and N be as in 5.6. Using 5.3 (resp. the identity ~o1| 2 = ~01(~02At-(~3) one can show that ((q~ q~2)r, N) contains (resp. equals) (2 q~l | v6 + ~0~ @2@ V 6, A 3 X C*). Since (~o1 q~2, A 3) has F PIG, to show that ((tp~ q~:)r, N) has F P I G it suffices to find an invariant of(2 ~o1+ ~Pl ~o~, A3) of degree (a, b, c), a + b > c. Now T h e o r e m 0.11 of [13] implies that S 3 (q~l q~2(A2)) ~o2 a , and then Theorem 1 of [ 17] implies that S a (q~t q~z(A 3)) = ~o~ ~o3 . Thus there are invariants of (2~p~ +(Pl q~, A3) of degrees (4, 3, 3) and (3,4, 3), and (V r, N) has F P I G . We now comment upon the T's needed for the classification of [12]. We also mention any unusual aspects of the computations involved. G = A,, n > 2 : T h e T of Examples 5.5 through 5.7 suffices for almost all irreducible non-coregular representations. Let T' = C * ~ A, where (~p~(An), C * ) = k v~ + k v 1+ 0,,, m = k + 1, k +2, or k + 3. As pointed out in [12], it is necessary or easier to use T' for the representations tP3(An), n > 9 ; ~o4(A,), n > 8 , n even;

Regular Rings of Invariants

189

and q0~(A,), n > 6 . To this list we would add (PlqO2(An), n > 6 . It takes a little work to show that (V r', No(T')/T') has F P I G in the cases of ~03(A9), q%(Alo), q~3(Alz), and q~4(As). Using T and T' one can handle all the irreducible noncoregular representations of G except for q~3(A,), 3 < n < 5. Using T" = C* c A, where (~0I(A.), C*) = vI + v 1 + v2 ~- l'- 2 -~ 0 one can handle ~o3 (A4) and q~ (A 5). G = B , , n > 3 or D,, n > 4 : Here one always can use T c A ' l c G where (~ol(G), A'0 =2~o'1 +0. Note that in this case (and almost all other cases to follow) N is semisimple, so the fact that (V T, N) has F P I G usually follows quite easily from the results of [1] and [7]. G = C , , n > 2 : Mainly one uses T c C ' 1 c O , where (~o1(C,), C'l)=~o'1 +0. For ~%(C4) one instead arranges that (qh(C4), C'0=3~0'1 + 0 2 , and for (~o~, C2) one arranges that (q~](Cz), C'0 = 2 q~'lIf G is an exceptional group we proceed as follows: Let [3 denote the highest root of G. We choose T c A'1 where A'1 is the 3-dimensional simple subgroup of G corresponding to /3. Let N denote the connected subgroup of G whose Dynkin diagram consists of the simple roots of G orthogonal to [3 (see Ch. II w5 of [5]). Then N~(T) ~ (resp. N~(A'I)~ has finite cover N x T (resp. N x A]). We list the groups N and describe the decomposition of the adjoint representation and lowest dimensional representation of G with respect to N x A'1. (One can determine these decompositions from the tables of [5]. We also show how one can read off the decompositions from the Dynkin diagram of G.) We list identities which allow one to express any basic representation in terms of the representations whose decompositions are known. One can then easily obtain enough information to apply Theorem 5.1. We were unable to find enough identities for Es, but we get around this difficulty as indicated below. G--= G2: N = A 1 where (~01(G2) , A 1 x A'I)~-qOl @q)'l ~-q~2. AZqol=(Ad G2=q02) +(Pi" G = F4: N = C a where (q~l(F4), C 3 x A'I)= q~,| 1 +q~z; (Ad F4=~o~, C 3 x A'0 = A d C 3 + A d A] +qo3| 1. AZqol = qo2-t'- qo4; A2 q04=(P3-t-q) 4. G = E6: N = A 5 where (qo~(g6), A s x A ] ) = qo~| +~P4; (Ad g6=q~6, A 5 x A'0 = A d A s + A d A]+~03| ~. AZ~ol=(p2; A3~01=~03; qo4=(p*; q~s=(p*. G = Ev: N = D 6 where (~o](gv), D 6 x A'~)= qo~| +qo6; (Ad E7 = q~,, D 6 x A'0 = A d D 6 + A d A'1 + q~s| AZq01 = ~02 + 01; A3~o] =q~3 +~0~; A~'q01 = q04 + (p2 + 01; A z q~ = q)~ + ~o6 ; q)~ | = ~oI q% + q)v + q~l. (To compute the decomposition of ~o7, we first computed the decomposition of ~o1| %. Using 5.3 we estimated the decomposition of q~ ~o6, and then using dimension arguments and the last identity above we found that

(e~(E~), D,, x A i ) = e l e., + e3|

+ ~ol|

, + ~06|

G= Es: N = E~ where (Ad Es=q~, E7 x A ' 0 = A d E v + A d A' t +~Pa|

We

determine two of the components of the restrictions of qo2, . . . , (])8 from the Dynkin diagram of Es, and these components are (with one exception) enough to establish the needed non-coregularity results.

190

G.W. Schwarz

Let co1. . . . . 0) 8 denote the highest weights of ~Pl (s -.., ~P8(s and let el, .-., % denote the corresponding simple roots. Then fl = 2 el + 3 c~2 + 4 % + 5 c%+ 6 c%+ 4 c ~ 6 + 2 ~ 7 + 3 0 %. Changing the canonical inner product on the weights by a scalar, we can arrange that (a,i,09j)=6ij. Each 09i then restricts to the highest weight of (~pi_l| x A'I) where ki=(09i, fl). For 1 < i - < 7 , let d i denote 0 9 1 - e l - e 2 . . . . . ~i. Then each 09'i restricts to a highest weight of (q)i(Fs), E T x A ' 0, and the corresponding representations are, respectively, qh|

q)2|

2, q93|

3, q~4|

4, q)5 (p7|

5, q96 q)7|

3, and ~o7|

i. From

09~ = 098 - cq - cq - ~3 - e4 - as - ~8 one similarly obtains that ~Ps| 2 ~ (~P8(Es), E 7 x A't). Our estimates so far are sufficient to handle all irreducible non-coregular representations of I:8 except for ~07. In this case we only have the estimate (V r, N ) D A d N. But using Theorem 5.3 and the identity SZq?l=p~+q)7+O1 one can compute that (P7(1:8), Ev x Ai)=q02+01 + qot @~0 i + (pT| q')6| 2, and Theorem 5.1 applies. Our method above estimated that (Ad 1:8, 1:v x A]) contains Ad A] +qh | Clearly the factor Ad 1:7 must also be present, and then a dimension argument yields our claimed decomposition for Ad 1:8. One can similarly prove the decompositions claimed for 132, F4, etc.

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18. guna, D.: Slices 6tales. Bull. Soc. Math. France, M6moire 33, 81-105 (1973) 19. Luna, D.: Adh6rences d'orbite et invariants. Invent. Math. 29, 231-238 (1975) 20. Luna, D., Vust, T.: Un th6or6me sur les orbites affines des groupes alg6briques semi-simples. Ann. Scuola Norm. Sup. Pisa 27, 527 535 (1973) 21. Mumford, D.: Geometric Invariant Theory, Erg. der Math. Bd. 34, New York: Springer-Verlag 1965 22. Mumford, D.: Introduction to Algebraic Geometry, preliminary version. Cambridge: Harvard University 1966 23. Popov, A.M.: Irreducible simple linear Lie groups with finite standard subgroups of general position. Functional Anal. Appl. 9, 346 347 (1976) 24. Popov, A.M.: Stationary subgroups of general position for certain actions of simple Lie groups. Functional Anal. Appl. 10, 239-241 (1977) 25. Popov, V.L.: Stability criteria for the action of a semi-simple group on a factorial manifold. Math. USSR-Izvestija 4, 527-535 (1970) 26. Popov, V.L.: The classification of representations which are exceptional in the sense of Igusa. Functional Anal. Appl. 9, 348-350 (1976) 27. Popov, V.L.: Representations with a free module of covariants. Functional Anal. Appl. 10, 242-244 (1977) 28. Rallis, S.: New and old results in invariant theory with applications to arithmetic groups. In: Symmetric Spaces, pp. 443 458. New York: Marcel Dekker 1972 29. Schwarz, G.W.: Lifting smooth homotopies of orbit spaces. To appear in Inst. Hautes 12tudes Sci. Publ. Math. 30. Vinberg, I~.B.: The Weyt group of a graded Lie algebra, Math. USSR-Izvestija 10, 463 495 (1976) 31. Wallach, N.: Minimal immersions of symmetric spaces into spheres. In: Symmetric Spaces, pp. 1-40. New York: Marcel Dekker t972 32. Weyl, H.: The Classical Groups, 2nd edn., Princeton: Princeton University Press 1946

Received November 25, 1977; May 26, 1978