2 8 AKCH. MATH.

Multiplicity Free Subgroups of Compact Connected Lie Groups

By

t~ANFR ED KRAI_WIER

Let G be a compact Lie group. By a representation of a group (or a Lie algebra) we always mean a finite dimensional complex representation unless we explicitly say otherwise. A closed subgroup H of G is called a multiplicity bounded subgroup if it has the following property: There is a constant K > 0 such that for any irreducible representation @ of G, all the irreducible H-components of the restriction @I~ of @ to H have multiplicity at most K. I f we can choose K = 1, we call H a multiplicity ]ree subgroup. In [5] the authors give a necessary and sufficient condition for a sub- group to be multiplicity free (Theorem 1 in [5]). They also discuss the significance of multiplicity free subgroups in physics. The condition in [5] is closely related to an earlier criterion of Godement ([4], 13, Coroll. of Theor. 8) and is surely the ,,right" condition from a geometrical point of view. For the purpose of classification however it is not so practicable. In the present paper we give a classification of the multiplicity free subg-{oups of compact connected Lie groups (see our Theorem). I t turns out tha t one gets only such subgroups which are classically known to be multiplicity free. Our approach is quite different from that in [5]. To do the classification, we derive a necessary condition for the closed subgroup H to be multiplicity bounded (Proposi- tion 1) which a posteriori turns out to be also sufficient. The way of proving Proposi- tion 1 may have an interest of its own: By means of general estimations on the ,,asymptotical" behaviour of dimensions of representations, of multiplicities of weights and so on, we obtain a concrete condition which is powerful enough to deliver the desired classification rather quickly.

Our arguments are partially given in the language of Lie algebras. The standard facts about semisimple Lie algebras, compact Lie groups and their representations can be found in [6], [7], [11] and [12].

Let L be a complex semisimple Lie algebra of rank r, C a fixed Cartan subalgebra. Roots and weights are always taken with respect to C. Let B = {al, a2, . . . , at) c C* (the dual of C) be a fixed base of the root system of L, {~rl, ~r2 . . . . . ~rr} the system of fundamental weights with respect to B, i.e. the base of C* which is dual to the base {a~, a~, . . . , aT) c C inverse to B (in the sense of [11], u 2.).

The set of the weights is the lattice A = ~4 ~ l ~n~ ~ Z for all i c C'* and the

Yol. XXVII, 1976 Multiplicity Free Subgroups 29

set of dominant weights is the monoid A + = mi=~ I m~ e 25 and m~ => 0 for all i c A. i

For M e A + we denote by ~M an irreducible representation of L with highest weight M, by Z M the set of weights of ~M and by d imM the dimension of 0M.

Let L' be a reductive subalgebra of L of rank s, C' a Cartan subalgebra of L'. We assume C' r Let L~ be the semisimple part of L' and let C~ denote the Cartan subalgebra C' • L~ of L~.

By B' = {bl, b2 . . . . . b~}, t =< s, we denote a base of the root system of L s with respect to C~, by {31, ~z, . . . , ~t} the system of fundamental weights of L~ with respect to B', by A '+ the set of dominant weights of L~ with respect to C~ and B'.

For reduetive subalgebras L' of L we also have the notion of a multiplicity bounded respectively a multiplicity free subalgebra: The subalgebra L' is called multiplicity bounded if there is a constant K > 0 such that the multiplicity of all the irreducible L'-eomponents of all the finite dimensional irreducible representations of L are bounded by K. I f K can be chosen to be 1, we call Jh' a multiplicity free subalgebra.

Let g be the number of positive roots of L and h the number of positive roots of L'. By definition, r and s are the rank of L and L' respectively.

Proposition 1. I f L' is multiplicity bounded in L, then h -~ s >= g.

P r o o f . The idea of the proof is rather simple and is the following: The function

dim: A + ~ Z, M = ~ M~ z~ ~-> dim M, is known to be a polynomial of degree g in i = l

the M~ (ef. [8], VI I I 4, p. 257). Now, i fL ' is multiplicity bounded, d imM -- calculated as the dimension of the restriction of ~M to L' - - turns out to be majorized by a polynomial of degree h ~ s in the Mi. Thus, h + s must be at least g. We give the arguments for semisimple L', i.e. we assume L ' = L~. The general case can be done by minor technical modifications.

The proof is based on certain estimations. 0n ly the qualitative aspect of the estimations plays a role. Therefore we do not worry about quantitative aspects and

make the estimations rather coarse. We give some details. For M---- ~ M~ g~ e A + r i = l

let I M I be defined as l M I ----~IM"

A s s e r t i o n . There is a constant A > 0 such that for all M e A + and all M ' e A '+ the following is true: I f QM is an irreducible representation of Z with highest weight

M and 31" = .M i ~ is the highest weight of an irreducible L'-component of QMIL', i = l

then M~ < = A ' I M I for all /----1,2 . . . . . s.

P r o o f o f t h e a s s e r t i o n . Let B,, i e (1, 2 . . . . , s} be the three dimensional simple subalgebra of L" generated by the eigenspaces of the roots b, and -- bi of L' and let b~" e Bi be the root inverse to bi. Consider the adjoint representation ad of L and its restriction adlL, to L'. For every weight of ad, i.e. for every root a, the restric- tion a' = ale, is a weight of halL,. There, fore a (b~) = a' (b~) is an integer and l a (b~')I

30 M. KR~MBR ARCH. MATH.

is at most d - - 1, where d is the maximum of the dimensions of irreducible/~i-compo- nents of adiB ~ . I t follows that [ a (b~) ] ~ Ai ---- dim/}. We also introduce the functions x~: A +--> O, k ---- 1, 2 . . . . . r, defined as follows: The x~ (M) are the coefficients in the

expression M---- ~ xk (M)a~, where M is written as a linear combination of the / r

simple roots a~. I t is easily checked that there is an A2 > 0~ such that x~ (M) _~ A2 ]Mt for all M t A + and all k = l , 2, . . . , r (eft also Lemma 2 (i) in [9]). For a / z tZ]M,

written as ~---- ~ / ~ a ~ , it is known tha t Iju~l ~ xj(M) for some i t {1, 2 . . . . , r}. k=l r

On the other hand, we know from the above considerations that [/~ (b~') ] S A1 E] ju~ ]. Putting all this together, we obtain the inequality [/~ (b~)J =< r . Xi . Ap.[M I = : A " I M[.

$

Now, a highest weight M' ---- ~ M~Ti of an irreducible L'-component Of~M[L" is of the i = l

form #' ---- /~lc' for some # tAM. I t follows that M~=M'(b~)-=/z(b~)~A. ]M] and the assertion is proved.

8 /

The dimension of M' = ~ M~ T~ t A '+ is a polynomial of degree h in the M~. i = l

Therefore the assertion implies that there is a constant Ki such tha t dim M' ~ Ki ] M ] for all M t A + and all those M ' t A '+ which occur as the highest weight of an irredu- cible L'-component in QMIL'. Let us denote by y(M, L') the number of the various M ' t A '+ which are the highest weight of an irreducible L'-component of ~M]L'. Using similar arguments as in the proof of the assertion, it is not difficult to prove that there is a constant K2 such that for all M t A + the number y(M, L'). is not greater than K2 ]M I s. Now, if we suppose that L' is multiplicity bounded in L, i.e. tha t there is a constant Ks such that the multiplicity of any irreducible L'-eomponent in ~MIL" is at most Ks, we obviously have the following: I f we denote by z(M, L') the maximum of the dimensions of the various irreducible L'-components of ~MJL', the dimension of ~M is at most K 3 �9 y(M, L') �9 z(M, L'). Thus, d i m M ~ K s . Kp" �9 ]M[s" Ki" I M]a = g " [M] a+s for K---- K i K p K 3 . However, as already stated,

dim Z M, g, is a polynomial of degree g in the M,. Such a polynomial can only be i = 1

majorized by the polynomial K . IM [a+s if h Jr s ~ g. This proves the proposition.

Let us call a reductive subalgebra L' of L a huge subalgebra for short, if h Jr s ~ g. We now return to compact connected Lie groups. Let G be a compact connected semisimple Lie group and H a closed subgroup. The notation "huge" carries over: The subgroup H is called huge in G, if the complexification of the Lie algebra of H is huge in the complexification of the Lie algebra of G. We call a property of a con- nected subgroup H of G a property o/the Lie algebras if for all connected G' locally isomorphic to G, the connected subgroup H ' of G' which has (after having identified the Lie algebras of G and G') the same Lie algebra as H also has the property in question. Obviously, to be huge is a property of the Lie algebras.

Proposition 2. For closed connected subgroups o/compact connected Lie groups, to be multiplicity/tee and to be multiplicity bounded are both properties o/the Lie algebras.

Vol. XXVII, 1976 Multiplicity Free Subgroups 31

P r o of. We give the proof for the case of multiplicity free subgroups. For multipli- city bounded subgroups one can use analogous arguments. Let G be a direct product G---- Gs • T of a compact semisimple simply connected Lie group and a torus T. Let G' be a Lie group locally isomorphic to G and g : G--->G' a covering homomorphism. Take a closed connected subgroup H of G and the connected subgroup H'--~ ~ (H) in the covered group G'. We prove that H is multiplicity free in G if and only if H ' is multiplicity free in G'.

The only if part is clear. For, every irreducible representation ~' of G' naturally induces an irreducible representation of G, namely ~ ~ ~ 'o z, and equivalent H'- components of ~' induce equivalent g-1 (H')-components of ~ and thus give rise to equivalent H-components of ~. The if par t can be proved in the following way: Suppose ~ is an irreducible representation of G and ~ is an irreducible H-component of ~ which has multiplicity k > 1. Let M be the highest weight of ~ (with respect to a maximal torus in G and an usual ordering of the roots of G). There exists a power M m of M (here, in the case of groups, we write the weights multiplicatively), such that M m is the highest weight of an irreducible representation ~m of G which factorizes through G', i.e. which can be regarded as a representation of G'. I f the highest weight of ~ {with respect to a maximal torus in H and an usual ordering of the roots of H) is M', it follows from Corollary 6 in [10] tha t in ~m, considered as a representation of G', the multiplicity of an H'-component with highest weight (M') m is at least k > 1. This means that H ' is not multiplicity free in G' if H is not multiplicity free in G. Since any two locally isomorphic compact connected Lie groups have a covering group in common, the assertion of the proposition follows.

I f one wants to classify subgroups with a certain property which is a property of the Lie algebras, it suffices to give the classification up to local isomorphy, i.e. for only one locally isomorphy type of the large group. Our notation for "G is locally isomorphic to G' " is "G ~ G' ". We also speak of G as of type Ar, -Br,.Dr, Cr, G2, F4, E6, ET, Es as used in the classification theory of simple Lie groups.

Proposition 3. Let G be a compact connected simple Lie group and H a proper closed connected subgroup. Then H is huge in G i/ and only i/, up to local isomorphy,

G = S U ( n ) and H----S(U(1) • U(n--1))_-~'~U(n--1) , n ~ 2 ,

o rG- - :SO(n ) and H ~ S O ( n - - 1 ) , n ~ 3 , n ~ 4 ,

or G ~ S0(8) and H ~-- Spin(7).

P r o of. We use the classification of simple Lie groups and also Dynkin's paper [3]. (i) Let G ~ S U ( n ) , n ~ 2 . Firstly, let H be the subgroup

H = S(U(n l ) • U(n2)) :-- ~U(n) n (U(nl) • U(n2)),

nl + n2 = n, nl ~ n2 ~ 1. One computes, tha t we have h + s ~ g unless ne---- 1. In this latter case, i.e. in the case U ( n - - 1 ) r S U (n), we have h + s = g. Secondly, let H ~_ H' ~ SU(nl) • su(n2) , nln2 = n, nl , n2 ~ 2, and let the representation of H ' in S U (n) be given by the tensor product of the two projections prl : H ' - > S U (nl) and pru: H'--->SU (n~). Again, we have h + s ~ g. Thirdly, if H is simple, maximal and the

32 M. KRXMER ARCH. MATH.

embedding H--+SU (n) is an irreducible representation, the inequality h q - s ~ g also holds except in one case, the case Sp(4) c SU(4). This may be proved case by ease: Let H~_SU(m). Then n is at least (~) if m > 4 (we assume of course tha t H is a proper subgroup) and is a t least (m+ 1) if m __< 3. In all these cases we have h- t -s ~ g. Let H~_SO(m), m-->3, m~=4. Then either H is an usual SO(n) cSU(n) in which case we have h + s ~ g, or H is represented by an irreducible representation which is not real. In this lat ter situation, the value of g - - (h + s) is minimal for m fixed, if, in the case m = 2 k, we have n = 2 ~-1 and H is represented by a half-spin represen- tation, or if, in the ease m : 2k q-1, we 'have n : 2 ~ and H is represented by a spin representation. The half-spin representations are not real in ease k - 1, 2, 3 rood 4 and the spin representations are not real in case k - 1, 2 rood 4. We consider these eases. Firstly, let H ~_ SO (2k), k => 3. I f k = 3, the image of the half-spin representa- tions is all of S U (4). I f k > 5, then h q- s ~ g. Secondly, let H ~_ SO (2 k + 1). I f k = 1, the image of the spin representation is all of SU(2). I f k = 2, we have h + s : g, and we obtain the huge subgroup ST (4) c S U (4). In the formulation of our proposition, this ease appears as the locally isomorphic ease SO (5) c SO (6). I f k _> 5, there is h~s<~g again. Now let H~_ST(m). Then H is maximal, if m : n = 2 1 c and H is the usual Sp(n) c SU(n). I f m = 4 , we have the huge subgroup ST(4) c SU(4) which we have already met. I f m ~ 4, we have h q- s ~ g. Finally we consider the exceptional simple irreducible H c S U(n). Each H of type G2, F4 or Es is contained in SO(n) cSU(n). Each H of type E 7 is contained either in SO (n) or in Sp (n). I f H _~/~ s, then n --> 26. In all these cases, we have h q- s ~ g.

According to Chapter I in [3], we have examined all the maximal subgroups H of SU(~) and we found tha t in all the cases, we have h q - s ~ g , except in the eases U (n- - 1) c SU(n) and ST(4) c SU (4). Thus: I f H c S U ( n ) is huge, either H-= U(n-- 1) or n = 4 and H = Sp (4).

(ii) Let G=SO(n), n>=3, n=#=4. Consider the subgroups H = S O ( n l ) • SO(nz), nl + nz = n, nl _--> n2 => 1. We have h q- s < g if n2 > 1 and h q- s = g if n2 = 1. Consider next the H __ H ' = SO (nl) • SO (n2), h i" n2 = n, h i , nz => 3, where the representation H'--+SO(n) is given by the real tensor product of the real representations prx: H'-+SO(nl) and prg,: H'--+SO(n2). One computes tha t h q - s < g . Consider now the H __ H ' =ST(nz) • ST(n2 ), nl" n2=n, hi, n2 > 2 , and the representation given by a real form of the complex tensor product of the (complex) representations prl :H' -+ST(nl ) and pr2:H'->ST(n2). Again, we have h + s < g . According to Chapter I of [3], the subgroups H hitherto considered are the maximal subgroups of G = SO (n) which are reducible or for which the semisimple par t is not simple. Consider next the usual U (m) c S 0 (n) with n = 2 m. One checks tha t h q- s < g. For all other possible H ~_ SU(m), m is less than n/2 and therefore h q - s < g again. Next we look at the H ~_ SO (m) (real) irreducibly embedded as proper subgroups of SO (n). One case is H = Spin (7) c SO (8). In this case we have h + 8 = g! I t is easily seen tha t in all other eases h q- s < g. Consider next H ~_ S T (m) embedded by an irreducible real representation into SO(n). One verifies without difficulty tha t this yields no huge subgroups. The case of the classical subgroups of SO (n) is done. Let us turn to the exceptional H. The lowest dimensional real representation of G2 has dimension 7. This leads to the ease H =G2cSO (7). We have h q-s = 8 < 9 = g. The lowest dlmen-

u XXVII, 1975 Multiplicity Free Subgroups 33

sional real representations of F4, E6, E?, E s which are absolutely irreducible have dimensions n -- 26, 78, 133 and 248 respectively. One computes that in all these cases the images os the representations are not huge in the corresponding SO (n). We should observe that a H which is embedded in SO (n) by an irreducible real representation which is not absolutely irreducible is not maximal but is contained in U (n/2) c SO (n) and is therefore not huge.

Resumd: I f G = SO (n), then H c G is huge if and only f f H = SO (n - - 1) or G = SO (8) and H = Spin (7).

(iii) The case G-- Sp (n) , n ~ 6, can be carried out without difficulty with simi- lar arguments as in the case G = SO (n). I~o huge subgroups occur.

(iv) Let us now turn to G an exceptional Lie group. First, we consider the maximal subgroups of maximal rank in G. They are well known, cf. [2], and by direct checking, we see that h -{- s < g in all the cases. Next, let rank H be less than rank G. For semi- simple H with only classical simple factors and with fixed rank, the number h os positive roots is maximal if H is of type Bs (or Cs). I t is now easy to see that for a connected subgroup H with rank s = r - - 1 the number h os positive roots is bounded by 1 resp. 9 resp. 25 resp. 36 resp. 63 in case G is of type G2 resp. F 4 resp. E6 resp. E z resp. Es. In all these cases, h ~-s is less than g. That means, that in an exceptional G no huge subgroups exist. Thus our proposition is proved.

R e m ark . Our classification of the proper huge subgroups of the compact connected simple Lie groups yields only subgroups for which h-~ s = g. Generally, i fG is compact connected semisimple and H is a closed connected subgroup, we call H an extremal huge subgroup if h ~-s = g. Extremal huge subgroups are obviously minimal huge subgroups.

Corollary 1. For connected subgroups o/compact connected simple Lie groups, the notions "huge", "multiplicity bounded" and "multiplicity free" are equivalent.

P r o o f . The huge subgroups classified in proposition 3 are known to be multiplicity free by classical representation theory, see [1], V w and VII w and all the notions in question define properties which are properties of the Lie algebras. We recall tha t the cases Spin(7)c S0(8) and S 0 ( 7 ) c S0(8) are locally the same up to an outer automorphism os Spin(8).

Corollary 2. Let G be a compact connected simple Lie group, H a closed subgroup and Ho the connected component o/the identity in H. Then H is multiplicity/ree (resp. multiplicity boundedJ in G i /and only if Ho is multiplicity/tee (resp. multiplicity bounded) in G.

P r o o f . According to Corollary 1 multiplicity free is equivalent to multiplicity bounded for connected subgroups. Now, if H is multiplicity bounded, Ho must also be multiplicity bounded because H/Ho is a finite group. I t follows that H0 is even multiplicity free. The other direction of the assertion is trivial.

In essence, the work is now done. We summarize our results thus far and ~ve technical generalizations resulting in a classification of the multiplicity free subgroups of an arbitrary compact connected Lie group.

Arddv der iYlathematik XXVII 3

34 M. K ~ i i ~ AI~CH. MATH.

Theorem. Let G be a compact connected Lie group, H a closed subgroup o] G and let Ho be the connected component o/the identity in H.

(i) Let G be simple. Then t t is multiplicity/ree in G i] and only i], up to local iso- morphy, either G ---- S U (n) and H = U (n - - 1 ), n ~ 2, or G = SO (n) and Ho ~- SO (n -- 1), n ~ 3 , n~=4, or G= SO(8) and H0-----Spin(7), or, trivially, H ~- G.

(ii) Zet G be semisimple. Then H is multiplicity ]ree q and only i] the ]ollowing holds: There are simple groups G~, i = 1, 2 . . . . . . r, with multiplicity/ree connected subgroups

Hi and a certain number GI, ] = r-~ 1, r ~- 2 . . . . , t, o] copies o /SO (4) with subgroups H t = SO (3), such that G ks locally isomorphic to the product G1 X G2 X "" X Gt and Ho is the subgroup o/G which has the same Lie algebra as H1 X H2 • "'" X Ht.

(iii) .Let G be the direct product G-~ Gs x T o/ a semislmple group Gs and a torus T. Then H c G is multiplicity ]ree i] and only i] the projection o/ H onto Gs is multiplicity /ree.

(iv) Let G be an arbitrary compact connected Lie group and ~-~ Gs x T a connected covering group which is the direct product o /a semisimple group Gs and a torus T. Let ~: G--->G be the covering homomorphism. Then H is multiplicity/ree in G i /and only i /~ -1 (H) is multiplicity/ree in ~.

(v) The subgroup H is multiplicity/tee in G i] and only i/ it is multiplicity bounded in G.

P r o o L (i) Is already proved, see Proposition 3 and Corollaries 1 and 2.

(ii) According to Proposition 2, it suffices to prove the assertion for G----F1 X X 2'2 • --- • a direct product of simple simply cpnnected compact Lie groups. For ( i l , i2 . . . . . ik) c {1, 2, . . . , 8} let Gil ..... i~ be the subgroup Fi~ X -~i2 • x Fi~ of G and let gi ...... i~ be the projection. Assume H r G is multiplicity bounded. Then gi ...... i~ (H) obviously must be multiplicity bounded in Gi ...... i~. In particular ~ (H) is multiplicity bounded in 2'i for all i = 1, 2 . . . . , s, i.e. either gi (H) = E~ or ~ (H) is a proper multiplicity free subgroup of G as described in assertion (i).

Now, let 1 ~ p ~ q ~ r ~ s be natural numbers and let the enumeration of the Fi be chosen in such a way tha t the following holds: The F1, F2, . . . , Fp are those factors for which H n F~ = F~ and /~+1 . . . . . Fs are those factors for which H n F~ ~ F i . In addition, we suppose tha t ~ (H) - - 2'~ for i---- q ~- 1, q + 2, . . . , s and g~ (H) =~Fi for i-----p-~ 1, 1o+2 . . . . . q. ~inally we assume tha t F~c~A1 for i = q ~ 1 . . . . . r and tha t F ~ A 1 for i = r ~ l , . . , s .

Look at gp+~ ..... q: G->-~+~ x ~ + 2 • "'" x 2'q. The projection g~+~ ..... q(H) of H is contained in g~+~ (H) x g~+2 (H) • .-. X ~ ( H ) = :H*. On account of the remark af ter proposition 3 our assumptions imply tha t the component H~ of the identi ty in H * is extremal huge. Thus, no proper subgroup of it can be multiplicity bounded. I t follows tha t g~+~ ..... q(H), which is multiplicity bounded, must contain H0*. This means tha t H n ~vt is multiplicity bounded or, what amounts to the same thing, is multiplicity free in F~ for i = p ~ 1, T ~ 2 . . . . . q. Assume now tha t there is an i such tha t q ~ i ~ r. Since g~ (H) = F~ but H n ~ ~=_~, there must be a ] =4= i, q ~ ] ~ r, such tha t the following holds: We have F~___~I and there is an isomorphism :r F~--~F i such tha t ~ z ( H ) contains the diagonal subgroup D~ 1 = {(x, a(x))[xeF~) of F~ x F~-. Now, on the one hand, one computes tha t D~I is not huge and thus not multiplicity

Vol. XXVII, 1976 Multiplicity Free Subgroups 35

bounded in lvi • (recall tha t I ' i is assumed not to be of type A1). On the other hand, it is easily seen that Dis is a maximal connected subgroup in _~ • F t. I t follows tha t gi, j(H) has Dlj as its connected component of the identi ty and is therefore not multiplicity bounded (Coroll. 2). This yields a contradiction. Thus, we must have q = r.

We now come to the remaining factors _~, i = r ~- 1, r-~ 2 . . . . , s. They are all iso- morphic to SU(2). This time, if there is an i with r + 1 ~ i --<s, there must be a ]=~i, r-~ 1 <= ] ~ s, such that g~, j (H) contains (up to conjugaey within F~ • Ft) the diagonal subgroup D = {(x, x) ] x e S V (2)} c S U (2) • SU (2) = _Fi • F i . One computes that the diagonal 1)= {(x, x, x) lx e SU(2)} in a product of three copies of SU(2) (and then afor t ior i in a product of more than three copies) is no longer huge. Observing that the diagonals are maximal connected subgroups, we obtain the following: The set of the factors Fr+l, -~r+2 . . . . ,Fs falls into a set of pairs {Fl, Fj} such that the identity component of H n (F~ • Fj) is the diagonal in F~ • Fj = SU (2) • SU (2). Remember now that the subgroup D c S U ( 2 ) • SU(2) is locally the same as the sub~oup S0 (3) c SO (4). Then, ff we denote the products -~t • Ft of the pairs just considered by Gr+l, Gr+2 . . . . . Gt, t - - r = �89 (s-- r), ff we rename the _~, i = 1, 2 . . . . . r into G1, G2 . . . . . Gr and if, finally, we let Hi be the identity component of H ~ Gl, we see that we have proved the " i f " part of the assertion even under the weaker assumption that H is multiplicity bounded in G. The "only if" part is clear as long as we remember that SO (3) c SO (4) is classically known to be multiplicity free ([1]).

(iii) This can be easily derived if one remembers tha t the torus T operates iso- typically in the case of an irreducible representation ~ of G, i.e. all the irreducible T-components of Q are equivalent.

(iv) See Proposition 2.

(v) This is already proved in (ii) for semisimple groups G. For general G it follows from (iii).

R em ark . (i) In view of (i) and (ii) of the theorem, the subgroups U (n- - 1) r S U (n), n ~ 2 , SO(n--1)r n >'3, and Spin(7)r S0(8) together with the trivial cases GcG of non proper subgroups can be regarded as the basic connected multiplicity free subgroups. All other multiplicity free subgroups of a semisimple G are locally the products of these basic examples.

(ii) We should say a few words on the question of the connected components. Recall tha t U (n -- 1) is maximal in S U (n) for n ~ 3, tha t the normaliser of SO (n -- 1) in SO (n) is an 0 (n- - 1), n ~ 3, and that Spin(7) also is of index two in its normalizer in SO (8). Thus, we see that our classification of multiplicity free subgroups is not only complete for connected subgroups, bu t tha t the general ease of not necessary connected subgroups is also under control.

(iii) The fact tha t "multiplicity free" and "multiplicity bounded" are equivalent notions can also be derived directly by the methods of 1. in [10].

Reterenees

[1] H. BOER~ER, Darstellungen yon Gruppen. Berlin-GSttingen-Heidelberg 1955. [2] A. BOREL et J. SI~BV.~T~AT., Les sous-groupes ferm~s de rang maximum des groupes de Lie

clos. Comment. Math. Helv. 23, 200--221 (1949).

3*

36 M. KR~MER ARCH. MATH.

[3] E. B. Dv~xn% The maximal subgroups of the classical groups. Transl. Amer. Math. Soc. set. 2, vol 6 (1957).

[4] R. GODV.~-~NT, A theory of spherical functions. Trans. Amer. Math. Soc. 73, 496--556 (1952). [5] F. E. GOLD~IC~ and E. P. WIG~ER, Condition that all irreducible representations of a com-

pact Lie group, if restricted to a subgroup, contain no representation more than once. Canad. J. Math. 24, 432--438 (1972).

[6] G. HOCHSCHH,D, The structure of Lie groups. San Francisco, London, Amsterdam 1965. [7] J. E. Hu-~92B~v, YS, Introduction to Lie algebras and representation theory. New York-

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gruppen. Invent. Math. 16, 15--39 (1972). [10] M. Kai~IzR, t~oer Untergruppen als Isotropiegruppen bei linearen Aktionen. To appear. [11] J.-P. SER~E, Alg~bres de Lie semi-simples complexes. New York-Amsterdam 1966. [12] J. TITS, Tabellen zu den einfaehen Liegruppen und ihren Darstellungen. Berlin-Heidelberg-

New York 1967.

Eingegangen am 10. 3. 1975

Anschrift des Autors:

Manfred Kr~mer iYiathematisehes Insti tut der Universit~t 53 Bonn WegelerstraBe 10