On Invariant Convex Cones in Simple Lie Algebras

Prec.. Indian Acad. Sci. (IV~ath. Sci,), Vol, 91, Number 3, November 1982, pp. 167-182. 9 Printed in India.

On invariant convex cones in simple Lie algebras

S KUMARESAN and AKAtlL RANJAN School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

MS received 19 December 1981 ; revised 22 January 1982

Abstract. This paper is devoted to a study and classification of G-invafiant convex cones in g, where G is a lie group and g its Lie algebra which is simple. It is proved that any such cone is characterized by its intersection with h-a fixed compact Caftan subalgebra which exists by the very virtue of existence of proper G-invariant cones. In fact the pair (g,k) is necessarily Hermitian symmetric.

Keywords. Lie algebra ; adjoint group; invariant convex cones.

1. Introducfiou

Let g be a real simple Lie algebra. Let O be its adjoint group. A closed convex cone in g invariant under the adjoint action of O will be referred to simply as invariant cone. The study of such cones was begun by Vinberg [9]. He proved (lee. cit) that for g to have an invariant cone it is necessary and sufficient that O is the group of isometrics of a I-Iermifian symmetric space. Hence we shall always assume that this is the case. In his paper [9] Vinberg poses some prob- lems. We answer two of his questions in this paperr One question (problem 4 in [9]) of Vinberg asks for characterisation of invariant convex cones via their intersection with the compact Carta~a subalgebra. Vinberg also asks whether it is true that for an invariant convex cone Q and a compact Cartan subalgebra h of g, (Q f~ h)* = (Q* t~ h). Here Q* denotes the dual of Q. (For definition and notation, see the main part of the paper.) As an answer to this question we prove that an invariant convex cone is completely determined by its intersection with a compact Caftan subalgcbra and the answer to the second part of the question is yes. To state this more precisely, we need the standard notation such as g = k @p, Wr the Weyl group of(k,h), etc., and two Wr-invariant cores C~,(g) f'l h and C~ (g) n h. (The last two objects are defined by Vinberg and for their dettni- tion, refer to w2). Let p~ stand fol the projection of g onto h via the Killing form. With this notation, we can state our main r~sult. Main result : If Q is an invariant convex cone in g, then Q is the invariant convex cone generated by p~ (Q)*Q t~ h. Conversely if C is a W~invariant cone

167 P.(A)---1 t68 S Kurnaresan and Akhil Ranjaa

such that C~ (g) A h ___ C _ C,~x (g) n h, then the invatiant convex cone Q (C) generated by C in g is such that Q (C) f3 h = C = PA (Q (C)). From this it trivially follows that whenever Cm. (g) # C~,x (g), there exists a continuum of invaFiant cones. This answers, in particular, a question of Vinberg (problem 1 [9]). Again from the main result, we can deduce for any invariant convex cone Q, we have (Q N h)* = Q* A h. In particular, in view of the main result, Q is self-dual if and onlyif Q A/,is self-dual in h. Hence to show that self-dual cores exist in g, one has only to construct self-dual cones in (the Euclidean space) h. Using our description of Ca,~ (g) N h (Prop. 4) and C~t, (g) A h, one can easily see that self-dual WK-invariant cones can be squeezed between C,a~(g) N h and C,~z(g) O h. Hence the existence of self-dual invariant convex cones in g follows. Now the natural question arises ~ Are there only tinitely many serf-dual ilavariant convex cones in g ? Again the answer is No I In SO* (6) (~ SU 43, 1)) we show that there exists an infinity of self-~tual cones. Of course, by what we said above the construction boils down to that of 8a-invariant cones in R a squeezed in between two cones (which are to be specified). The details of this last construction are given in the appendix. (We thanl~ M S Raghunathan for pointing out a gap in the proof of Theorem 11 ie the first version.)

Proof of the main result Let ~ be a connected rent simple Lie grot~p and g its Lie algebra. Let 0 be a ~rtaa involution ofg and let g = k ~p b~ the corresponddng Caftan decomposition. Let/~ be the connected subgroup whose Lie algebla is k. We wish to study closed eoav.ex cones in g which are invariant trader the adjoint action of G. g admits invariant cones if and only if there exists a K-invarilmt vector in g under the adjoint action [91. This implies that g will have invariant epees if and only if k has a (one-dimensidnal) centre. Let ko be a (non-zero) element of k so that the centre of k is Z (k) = R k0. It is a well-l~own fact that ko induces a complex structure on p via the adjoint action [6:p. 375]. Let J denote this complex structure on p. Now let ge, ka, Pc denote the complexifieations of k andp respectively. Let h be a maximal abelian subalgebra of k. Then hc is a Caftan snlmlgebra of gr Note that k, e h. Let A denote the set of roots of (go, ha). We choose a positive system P for ~ compatible with the complex strtteture on G[I~. Bythis we mea~ a positive system in which every positive non-compact root (i.e., a root a e P such that #t c pc) is greater than every compact root (i.e., a root a with ga ~). We let P, (rcsp. P,) denote the set of positive compact (resp. non-eomlmcO roots in P. Then P = P~ O P,. Then p+ = 2~ g~ is an abelian subalgebta +ttepn invariant under the adjoint action of he. Let (p, d) denote the complex vector space whose underlying real vector space is p with the complex str~t~e 3". It is well-known that (p, J) is an irreducible/r module and (p, d) and p+ are eqm- valent as /c,-modules. For what follows it is important to maderstand this equi- valence in an explicit manner which we de~cribe now. Simple Lie algebras i69

Let O ~ Z._.~ a ~ g~ for ~ P,. Set X~ = Z~ + Z-a and Y~ = i (Z,- Z-~). Then Za, Y~ Ep and are such that J,V~ = Ya and dye, = - ,V~. These follow from trivial computations and the fact that all roots are purely imaginary on the real space h. It can also be easily checked that for H~ = [Za, Z-a], we have (i) [ill,,, X,s] = 2Y~,,

(ii) VH., r,] -- - 2x. and (iii) [Y~, Ya] = - 2ill,,. Then the map Xi~ X -- id.Vis a kc-intertwining map of the modules (p, d)ar.d p+. Under this map Xa [~ 2Z~ and Y~ I~ 2iZa. Let C=,. (g) = Con O k0, the smallest closed convex cone containing ko. Here G ko stands for {Adg ko:g ~ G}. For the coadjoint representation of G on g' we can similarly define C=~ (g'). Then C=,t (g) is defined to be the cone {.V~g : (J~, Y)/>0 for Y~C.,.(g')}. Let - B(,) denote the negative of the KilLing form of g. Identifying g' with g via this non-degenerate form, we can write C,a(g) ={J~r -B(~, Y)~>0 for YEC~(g)}. Note that k 0 ~ Cffi~.(g) and this is the reason why we used - B (,) to identify g, with g. Then C=,z (g) is the unique maximal invariant cone in g : tw.iqur to within multiplication by + 1 [9 ; theorem 3]. Our main aim is to give a description of Q A h for an invariant cone Q in g. Let We denote the Weyl group of the pair (k, h).

Observation : If Q is an Ad (O)-invariantcone in g, then Q o h is a WK-invariant cone in h. That Q n h is a cone in the real vector space h is easily seen. That it is W~- invariant follows from the fact that given any w e I~e there exists k e/~ such that Ad k, restricted to h, induces w. This suggests that we should investigate what happens when we start with a l#~-invariant cone C in h and consider the smallest Ad (O)-ia~ariant closed convex c?ne in g containing C. As a first step in this investigation we prove

Lemma 2 : Let C be a W~invariant cone in h. Then Ad K(C) is a cone in ko Note that we are not sayipg that the cone gepelated by {Adk(C) :k~X} in k! To prove this lemma we mal~ use of a convexity theorem (or rather its infinite- simal analogue) due to Kostant [6]. To state this theorem we need some notation, which we now set up. Let O be any connected semisimple Lie group with finite centre. Let g be its Lie algebra. Let g = k OP be a Caftan decomposition ofg. Fix a maximal abelian subspace a of p. Letpo denote the orthogonal projection of p onto a with respect to the Killing form. Let W, denote the little Wcyl group of (g, a). Recall that B,o is defined to be th~ ncrma!izcr of a in K (the connected subgroup corresponding to k) modulo the c~ntralizer of ain K. Theorem [6]: Let HE a. Then p, (Ad K(H)) is the convex-hull of{w. H : w ~ Wo}. 170 S Kumaresan and Akhil Ranjat~

A Morse-theoretic proof of this (intinitesimat) version can be found in [4 ; theorem 1, Ch. 1; see also 2]. Let now U be any connected compact group with Lie algebra u. Let h be a Caftan subalgebra in u. Let tV denote the Weyl group of (u, h). Let u, = [u, uJ denote the semisimple part of u. Let z be thecentre ofu. Then h, = h fl u, is a Cartan subalgebra of u, and we have h = z@h,, and IV(u,, h,) = VC(u,h). We now apply the above result of Kostant to g = u, + iu~, and a = ih, (i = ~/----i). It is easily verified that IV, is canonically isomorphic to IV. Hence we deduce that if p~ denotes the projection of u, onto h, with respect to the Killing form then p~ (Ad U,(H)) is the convex-hull of {wH: we IV} for any Heh,. If H~h and u~U, then we write H=Z+H,,~ez, H, eh, and Adu=AdzAdu, with z in the centre of U and u, in the semisimple part of U. Then we see that Adu (H) = Z + Adu, (H0) and hence p~(Ad U(lt))is the convex-hull of{wH ,~ w ~ W} in the general case too.

Proof of Lemma 2 : Since /d is compact, and C is closed in h, Ad/tT(C0 is elosedink. Also, since Cis acone, forany t >0, keKand2/EC, t-Adk (1t) = Adk(ttt). I-Ience it is enough to show that for any X, YeAd/~(C) we have Jff + YeAd/~(C). Choose k E/Cso that Ad k (J~ + Y) eh. This is possible since h is maximal abelian in k. If X= Adk~H1, Y=Adk2H~ for some kz, k~eE and H~, H.,eC, then Adk(,~' + Y) = H for some Heh implies that Ad k/ca H~ + Ad k ka//2 = hr. Sio.ce by what we have seen abovepa (Ad kk~ H,)) convex-huH of {w H, ; w e Ivr} and since C is a Iv~-invariant convex cone in h, it follows that Zr + YeAd/t~(C). This proves the lemma. Let pa also denote the projection of g onto h via the Killing form.

Corollary 3 : The following are equivalent for any two Ad (K)-invztriantcones Q1 and Q, in k: (i) Two Ad (/~)-invariant (closed, convex) cones in k are equal: Q1 -~ Qz. (ii) Q~Nh--Q,f~h. (iii) Q, n h -- p~ (Q~) - p, (Q,) =. Q2 f1 h.

Proof : (i) :~ (ii) :This is trivial. (ii) => (i), Since Qa N h is a ~K-invariant convex cone in h, Q,D Ad/~(Q~ N h), the latter being a cone in k by lemma 2. If X e Q~, then Ad k X~E h for some k e/r and since Q, is Ad K-invariant, Ad k Ar E Q~ N h. I-Ience AreAdk-1 (Q~ n h). Thus Q~_Adld(Q~ f'i It). Therefore QI = Ad/a(Qa f'l h) - Ad K (Q2 rl h) = Q,. (ii) ~- (iii). Enough to show that forany Ad (K)-invariant convex cone Q- k, we have Q13h =p,(Q). ClearlyQNh_=p~(Q). If Ar~Q, then for some k E K, H=Ad k 2d e Q n h. Hencep~ (J~) = ph (Ad k-X H). By Kostavt's result pn (Ad L--x H) ~ Q t~ h.

We shall prove an analogous result for Ad (G)-invariant coI~es in g. Before doing this, we shall describe the cone C=s. (g) Iq h. Let C7~,~o (g) (resp. C*~i~ (g)) ¬e the interior of C**,, (g)(resp, C~l, (g)).

Propositio~t 4 C~==o (g) N h = {H e h : - i~ (H) > 0 for ~ ~ p~}. Simple Lie algebras 171

Proof: We remark that since ~ is purely imaginary on h, -ia (H)e R for any tIeh, and hence the condition above is meaningful. To prove the proposition, wc mal~ use of th~ following Theorem (Vinberg). LetJf~C~ Then Jdisconjugate to an element in k+ = (yck:J -~ ad YI~ is positive definite}. Here p is enclowe4 with the positive-dell.ire metric + B (,). To go bacl0 to the proof of the proposition, let H~h. Then ttEC~ N h if and only if d-x ad H [, > O. Now with the earlier notation, we remark that {Jffm, Y~, : a ~ P,} is a basis for p and we can easily see that J-L ad H on p with respect to the above basis can be represented by the matrix

1 0 7 ~ (H)

7 o

o (g) etc.. l where a~ runs through P~ and i = ~f-Z-l. Hence d-1 ad H is positive-definite on p if and only if 1/ia(t0 > O, that is, if and only if - i~(It) > 0 for all a eP~.

Remark : Let C denote the closure of C~ N h in h. Thatis C -- {H~h : - ia (/-/)>10 for all a ~ P~}. Then it is easily seen that C -- C~,I (g) N h.

Proposition 5 : We have p~ (Cm~ (g)) = Cm,~(g) N h = C. Proof : Since C = C~u (g) fh h G h, we have C _ p~ (C~u (g)). Th~ we need only to show that p~(C~(g))c_ C. Let ~feC=,~(g). Let ~ = A + B be the Cartan decomposition of X~. Then for some keK, we have AdkA =H~h. Let us write AdkX=H+ Y, with yep. Since XE Cm,s (g), which is Ad (G)-invariant, we have Ad k. XrE C~,~ (g). By the definition of C=~=(g), we have - B(Adk(J0,Aflexp z. (k0))~>0 for any z~p. (*) We now make a special choice of z~ p. We take z = t~a foI a~P~andt~R. Then the above condition leads : - B (Ad exp ( - t X~a) (H), ko) - B (Y, Ad exp J~a (/co)) ~> 0. Since k and p are orthogonal to each other with respect to B (,) and in the first term above the vector ko ~ k to compute - B (Ad exp (- tJ~a) (H), /co)), it is enottgh to calculate the k-component in the Caftan decomposition of the vector Adexp (- tX~a) (H). Since X~p, HE k, [k,p] ~_ p, and since Adexp (- tXa) (H) is

o0 (ad (n) (exp Ad (- tJff~)) (H) = ~_~ nl ' 0 172 S Kumaresan and Akhil Ranjan the k-component is given by

(:~) : 0 This is easily calculated, by observing

[H, ard = i1 a(x). Y, and t/t, Y,] = I a(H). X.

to be equal to

= H + ~(~ (eosh 2s - 1) Ha, where s = - t,

= H+ ~(H) sin h 2 tH,.

A similar reasoning tells us that it is enough to calculate (ad X) ~"+x (ko). Since [.~,, k0] = - J~, and (ad X~)' (k,) = [X~ [;r - JX~]] = [~, [~ Y,]], = 2 [1~,, iHa] = -- 2 [ilia, ~a] ---- -2.2Y,= -2 s. Ya, ete. We see that the p-component of Ad exp tXa (ko) is

-- (2n +1)! Y'~ = -2sinh 2t. Yo-

Thus (*)can be rewritten as -B(H, ko) + ia(H)sinhZtB(iH~, ko) sinh2t B(Y, Ya)>~O. Dividing throughout by sinh~t and using the fact that =sinht. cosh t, we rewrite this as 1 sinh 2 t B ( H, ko) + ia ( H) B ( iH~, ko) >~ - coth t B (Y, Y~).

Letting t ~ + oo, we get -- ia (H) ( - B (ill,, ko)) >1 ~ B (Y, Y~). Since - B (ilia, k0) > 0, we see that - i~ (H) >1 0. This completes the proof of the proposition. Let Te = ilia e h. Since - B (,) restricted to h is positive deil~_ite, we identify h' with h via - B (,), under this identification ta = Ta/[ + B (a, a)] corresponds to - ia. Now from Iemma 1 we see that the dual C* of C _ h can be described as follows : C*-{ 2~ a~ta ; a~>10} ={ ~ baT~ : bs>~O}. mtepm aePm We now want to prove p~ (C~ (g)) = C*. For this we need to set up some notation. Simple Lie algebras 173

We say that two roots ~ and a' in P, are strongly orthogonal if neither a + a nor a - a' is in A. A subset S _ P. is said to be strongly orthogonal if and only if any two roots in S are strongly orthogonal. It is well-lmown that there exists a (maximal) strongly orthogonal subset S --- P. such that the space

= I R (Z~ + Z~) is a maximal abelian subspace of p [3 ; see also 5, Prop. VIII. 7.4 Cot. 7.6].

Lemma 6: Let Z= Z' tT(Z~+~-~)ea, with t,reR. 'lteS Let H e h. Then we have fad exp Z)f10 =8 +Zs ? (8)sinh t~eosi t,r(Z_.r - Z, r) +~ssin h2 t,r(? (I0. H,r).

Proof : For Zas above, we can write exp Z = exp Yexp A exp =~ whele Y -- Zs tan t,r Z-.r, ~ -- Y~s tan t-r Z-r and A = ~s log (eosh t,r) [Z~, Z--r] [5~ Lemma VIII. 7.11]. We then easily see that Ad exp Xr (8) = H - E 7(8) tanh t~Z 7 (Ad exp AoAd exp Jff) (10 = 8 - Xr (8) tanh t-t.cosh* t.rZ.r, = H - ~: ? (H) sinh t 7 cosh t,rZ,r, (Ad exp YAfl exp A Ad exp ~Y) (H) = H + Z ?(~H)tanh t,rZ-v - Z ? (H)sinh t~eosh t,: (Z~ - tanh tTH.r - tanh z t,rZ-,r), = 8 + Z ? (H)sirda t-tcosh t,r (Z_.r - Zv) + Zsinh z t-t (7 (//) H~,). This completes ~e proof of the lemma. We let p, denote the projection of g onto k wi~ respect to the Killing form -B (,). We note that for k e K, and ~ g, we have (p, o Adk) (2#) =Ad k (p, (X)). Also, we have 7 (k0) = i for ? E P.. Corollary 7 : With the notation of the lemma 6, we have p~ (Ad exp Z(H)) = H + sinh 2 t7 (7 (H). n~). = Ph (Ad exp Z (H)). In pmticular for H = ko, we have Pi (Ad exp Z (ko)) =/Co + Z sinh 2 t.v (i8,~) = ko + Z sinh ~ t,~ T,r = Ph (Ad exp Z (ko)).

Moreover for any Z ~ a, we have p~ (A4 ~x-p Z (k.)) ~ C*.

Proof : The first statement follows from the abo~e lemma, since 7 (H) is purely imaginar% ? (H) (Z-~ - ZT) ~p. The last statement follows from the expression for pa (Ad exp Z (ko)) and the d~scription of C* given before the statement of lc~.ma 6, 174 S Kumaresan and Akhil Ranjan

Proposition 8 : PA (Cmxn (g)) = C*. Proof : Since Cmto(g) = the closed convex cone generated by Ad (exp X) (ka) for .Irep, it is enough to show thatph (Ad exp i~ (k0) e C*. By the corollary 7, if ~ea, then we are through. We now want to exploit the fact that p = I3 Adk(a) to prove the general case. ke/r We first prove the following claim : p, (C,,, (g)) = Ad/d (C*). For this it is enough to cheek that p, (Ad exp X. ko) = Aft/g(C*) for x ep. .~'ep can be written as Ad k (H) for some k e Kand He a. Thus we have only to show that p, (Ad (exp (Ad k (H))) ko) e Ad KfC*) for all k e K and H E a. Now the elem.~nt under inspection is p, (Ad k Ad exp H. /Co). Since p, o Ad k= Ad k op k, we have to cheek whether Pt (Ad k Ad exp Hko) =Ad kpi (Ad exp H. k 0) cAd/d(C*). By the corollary 7, we have p, (Ad exp Hko) = p, (Ad exp Hko) e C* and hence Ad kp~ (Ad o.p Hko) e Ad k (C*) _c Ad/d(C*). Thus we have proved p, (C~ln (g)) ~ Ad/d (C*). To prove the other inclusion, it is enough to show that C* ~ Cm,n (g). We note that C* is a W~invariant convex cone in h. Let He C*. Since Cm~ (g) = (C~,z (g))*, the dual of C,.:n (g) with respect to - B (,), we have only to check that - B (H, Rr) i> 0 for any ~ Cm= (g). Since HeC* ~ h, - B(//, ~') = - B(H,p,(~). Since Ph (ilr)e C by proposition 4 ar.d C* is the dual of C, we deduce that -B(H, phiV)>~O. This completes the proof of the claim. We now see that P/~ (Cmi. (g)) m_ p/~ (p, (Cml n (g))), = p, (Ad/r (by the claim above), = C* (by corollary 3). This completes the proof of the proposition.

Remark : Note that in view of corollary 3, we have shown in the course of the aboVe proof, that p, (C,~, (g)) = C* = Cm,, (g) n h. In fact, for any invariant convex c one Q c g, the proof of the claim can be adapted to show p,(Q) = Q N h. We treated here the special case when Q = C, ao(g) to bring out the salient features of the proof. The general case is formulated in a more precise way as theorem 11.

Corollary 10 : C* is the closed convex cone generated by {T.~ :r eS, we 1r Simple Lie algebras 175

Proof: This follows from corollary 7 and proposition 8. Note that /co being the average of T,'s already lies in the cone generated by IVK}. We cart now describe all invariant convex cones in g. We first observe that if X is any subset of h we denote by C (~, h) the conve~ cope generated by {w 3~; wE Bt~r, Jge Y~}in h. Note in everycase for anyX the element ~'-w ~Y w is an element of h that is left fixed by every element of BeK. Hence it has to be a scalar multiple of k 0. lqow, since C (X, h) is a Bt~invariant convex cone in h, by lemma 2, Ad K(C(~, h)) is an Ad(/0-invariant cone in k such that p, (AdE(C(F,,h))) = Ad K(C(F,,h)) = C(Y.,h). Let QC~) denote the closed convex Ad (G)-invariant cone in g generated by 5'. _ h. We can now put together all the pieces to state and pr ore our main result. Theorem 11 : Let Co be a Wa-invariant convex cone in h such that C* _ C O __G C. Let Q (Co) denote the Ad (O)-invariant convex cone in g generated by Co" We then have p, (Q (Co)) = Q n h = Co. Conversely given an Ad (G)-invariant convex cone Q in g, such that k o r Q, Q is generated by Q N h =p,(Q).

Proof : The proof is exactly similar to that of proposition 8 and uses lemmas 6 and 2 (or rather corollary 3). To prove the first statement, since obviously Co ~ p~ (Q (Co)), it is enough to prove that p, (Ad g (H)) ~ Co for H ~ Co, g ~ G. This follows from the following claim:

Claim 1 : IfHeC~ interior of C, and goeG, thenph (Ad go(Ho))econvex- hull of { Wx" Ho} + C*. To prove this claim we need a lemma.

Lemma : Let Ja e C7~, (g). Then either Are k + or there exists an element Y in the orbit of X such that (i) p~ (YO ~ convex-hull of {W'K. p, (lO} + C*, (ii) Ip, CI')I < Ip,( l where p, denotes the projection on the p-campo~.ent of g=k @p and Ip, (,V) ] is the norm of p~ (70 in the Killing norm.

Proof : Let ~ r k +. By conjugating by an element of Ad E, we may assume that X = H + Z' (t,X,, + t_,, Ya), with H ~ ~. aePn ta ) By taking Z=i Z(a-~t-a X,, ---(H) Ya we see that 176 S Kumaresan and Akhil Ranjan

= - Z + + s nePii ts t ~ . eh• aeP~ Put

~ [ t~_~ tg ]iH~. Notke that VaC*. v = : - i (H) § - IlePm Hence for t eR, we can write Adexp tZ(JO ffi H+ (1 - t) ~ (t~X~ + t_aYa) - V+ tS+ ~(t2), where ~(t 2) is the sum of terms invobing t ~. Her:ce for t > 0 and suflioently small we have H - Pi (Ad exp t Z(X~ = pi iX9 - Ph (Ad exp t Z(X)) e C*. We take Y = Ad oxp t Z(X) with t as above. Then I Pp (Y) J is dominated by (t - t)lPg(~')J and hence [p~(Y)J < Ip,(,V)l. The general case follows by the convexity theorem of Kostant: If X0 is the given element and ifAd k(~o).= 3~ is as above, then we have P* (~a) = Ph (Ad/r162 (.Xo) ) = p~ (p~ (Ad k --1Ad k (Xo))), = p. (Ad -sp. (Ad k (XJ)), = Ph (Ad k -1 p~ (Ad k (3fro))),since Ph (~) = Pk (3~). Thus the lcmma is completely proved. Proof of the claim : Let 3~= AdgQ(Ho) and p~(~) = HeC ~ Consider the orbit Ox={Adg~:geG). Since ~ is semi-simple, Ox is closed by a well-known result of Borel-Harish-Chandra [10]. In this orbit consider the set Ox+ = {Ye Ox : Heconvex-huU of {WK. p~ (Y)} + C* and IP,(:Y) I < Ip,(g) l}. We claim that Ox+ is compact. Since it is obviously closed, only its bounded- heSS to be proved, which we relegate to a lemma below. We now consider the function YI-* [ Pp (IT) [ on Ox* to o~. Since Ox* is compact, it attains its mini- mum say at Y. We contend that pp (Y) = 0, since otherwise, the lemma above will give us an element Y' such that Y'e Ox+ and I P, (Y')I < [PP (Y)I. Hence pp(Y)= 0. Thus Yek +. If we take //1 eC ~ to be such that Adk Y = H~, it then follows from what we have proved that//1 and Ho lie in C O _~ h, are conju- gate and are such that Hoe convex-hull of {FFr Hi} + C*, But then convex-hull of {WIt /-/1} = convex-hull of {Wx" Ho}. I-Icr.ccthe claim is completely proved modulos the assertion that O~x is compact. That O~ is bounded follows from the following: Lemma: LctHekandCaconstant >OandgeG. If]p~(Adg(H))p< C, then l Adg(H)l 2 < 2C + I Hp where I Xl is with respect to - B (~,0Jt) 1/*, Simple Lie algebras 177

Proof: Put g = kexp(X)~k~, ~'ep. Then Ip,(Adg(H))l = tp, Ad exp(l#)(H)l. Since ~'ep, we can choose an orthonormal basis e~,...,e, of k andre,...,f, of p such that Ad ~(e,) = ;t,f ,~ Ad ~(f~) ffi 2~eJ 1 ~ i ~ minfr, s).

All other basis elements, if any, go to zero under Ad ~. Let H = ~, a~e~. For simplicity we assume r = s. Hence we have Adexp ~(H) = Z a~ (cosh 2~, + sigh ~ft) and therefore [ Adexp (X)(H)i ~ --- Z aS (cosh z ~., + sird~~ ~3, _- I HiS + 2Za~ sinh~ A,. < I HI ~ + 2C, since [pp (Ad exp ~(H)) l* = Z a~ sinh~ A,. This completes the proof of the lemma and hence the claim. To prove the second part of the theorem let Q be the given invariant convex cone containing ko. Let ~eQO, the interior of Q. Since Q_ Crux(g), by Vinbcrg's theorem [9; theorem 5] there exists g e G such that Adg X e k+. But then by a suitable Ad (k)-conjugation we conclude that Ad k g (X) e h. Since Q is Ad (G)-invariant, we have Ad kg (X) ~ Q t'l h. This means that Ad kg (X) lies in the Ad (G)-invariant cone, generated by Q A h and hence itself lies in the invariant cone generated Q n h. By the first part of the theorem we have p~ (M) E Q n h. This completes the proof of the second part. The foUowing corollary is immediate from theorem 11 : Corollary : Let g be such that Cram (g) ~ Cm.- (g). (This is the same as saying C*~ C, in our notation.) Then there exists a continuum of invariant convex cones in g. We note that this corollary answers Vinbcrg's problem 1 [9], in the negative. We can also answer Vinberg's question 4 [9] in the a~rmative: Theorem 11 "describes "all the invariant convex cones in a simple Lie algebra in terms of the intersection with the compact Cartan subal~bra h. The second part of the problem 4 asl~s whether it is true Q* o h -- (Q o h)*. The answer to this question is yes. This also follows trivially from theorem 11 as shown below : Cones Q and Q* are determined by Q N h and Q* N h by theorem 11. Note that we have trivially Q* N h ~_ (Q N h)* sinceif ..~Q* N h, Y~Q N h, then we haVe - B GV, Y) >~ 0. Now let He (Q f3 h)*, 3YE Q. We need only show that

- B(H, X)>~O. But thep - BfH,~) = - B(H, r,(Yl)). Now by theorem 11, p,(~)EQ A h and so - B(H,p,()~F))>~O since H~(Q f3 h)*. This implies -B(H,M)>~O for alI~EQ. Thatis to say that H~Q*. This gives the complete answer to Vinberg's problem 4. This we formulate as 178 S Kumaresan and Akhil Ranjan

Proposition 12 : For any Ad (tT)-invariant convex cone Q _ g, we have (Q t-I h)* = Q* tq h. Hence Q is self-dual if and only if Q tq h is a self-dual cone in h.

Remark : Using our description of C and C* in h, one can easily verify that C = C* when and only when g = sp (n, R). In this ease, if h is identified with R', then C = C* = the ' quadrant' {x e ,~ : x~ ) 0}. Hence the only case for which C,~t, (g) = C,,x (g) is true occurs when g = sp (n, R). This is noticed by Vinberg [9], but his ploof is not clear to us.

Remark : One may now ask : Are there self-dual cones in g ? If they exist, are there only ttnitely many self-dual cones in g ? The answer to the former is yes. This can be easily seen by our explicit description of C and C*. All one has to observe is that one can squeeze a lYr-stable self-dual cone Co such that C* _~ Co ~ C. The reader may easily verify this claim. But the answer to the second question is lgo. To show this we consider the case of SO* (6) (which is locally SU(3, 1), but to identify h with R a, SO* (6) realization is easier to handle). With the standard notation of the root system /)3 (~ As) we may take as the unique noneompaet simple root e2 + e, (The simple roots are e2 + es, ea - ez, ez - ea). Then we have P, ={e~ +e~,~. + es, e~ +e3}. Then identifying h with R 8 with the standard inner product, we see that C =(x,y,z)eR s : x + y>~O, y + z>~O and x + z>~O and C* = {(x,y,z)e R3 : y + z>~x,z + x>~yandx + y>~z}. The Weyl group Wx can be identified with Ss which acts on R s in the usual manrer. In the appendix we construct for any n 1> 1, a family of cones C," such that one member of this family is self-dual. For n ~ m, C~ ~ C," for any r, s. This constrkletion shows that SO* (6) has an infinite number of self-dual invariant convex cones.

Remark : We want to record some observations we have made on the cones in h. In the followin by a cone in h or g we mean its interior. It is easily seen that if a l~a.-stable convex cone C _ h is homogeneous in h (for definition, (see [8]), then Q (C), the Ad (604nvariant cone generated by C in g is also homogeneous. If we assume further that C is self-dual too, then Q (C) is a homo- gelaeous self-dual cone. Notice that Q ((7) is a priori self-dual with respect to - B. But as is easily seen Q ({7) is self-dual with respect to-B0 where - B0 (X, Y) = - B (,~, 0 IT), i.e., Q ((7) is a self-dual homogeneous cone in a ' Euclidean ' space (g, - Bo). Then by a known result [1, 7], Q(C) is a Riemannian symmetric space with respect to the canonical Riemannian structure on it (for the definition of the canonical Riemannian structure see [7, 8 ]). It would be interesting to know whieh of the self-dual Wr-invariant cones C are homogeneous and find all the Riemannian symmetric spaces associated to those cones Q (C) in g. For example, in the ease when (7 = sp (n, ~) we have seen that C~ (g) = C,.,x (g) = Q. If one interprets sp (n, ,~) as quadratic forms on ~', then Q may be identified with those form that are positive definite with respect to the sympletie form [which defines sp (n, ~)]. Thus in this ease, the corresponding symmetric space is ,~+ x SL (2n, ,~)socs,}. Simple Lie algebras i79

APl~udix

In this appendix we discuss some very special type of convex cones in a finite dimensional Euclidean space and describe their duals. Then we shall further specialize to discuss a class of cones, their duals and the conditions for their self-duality. Let V denote a finite dimensional vector space over ~ and Y' its real dual. By a convex cone in V we mean a closed convex cone which has interior and which contains no straightlineentirely. We assume that V is endowed with an inner product (,). If C is a cone in IF, its dual, denoted by C*, is defined as follows : C*={v'eV':v'(v)>t0 for all vin C}. Then it is weU-hnown that (C*)* = C and that for two cones Ca - C~, we have C~ D C~. Using the inner product (,) we can and do identify V' with V.

Lemmal : Let {vx,...,vs} be a set of vectors in Vsuch that C={veV: (v,v~)l>0 for l~i~N) is aconein V. Then we hayer C*={ueV:u= 27 a,v,, a, l> 0.

Proof : It is easily seen that C' = {u e V : u = S a,v,, ai >~ O} is indeed a cone. Let C* be the dual of C. Then it is clear that C' ~ C*. Taking duals, we see that C C (C')*. Hence to show that C' ~ C*, it is enough to show that (C')* c_ C. Now if u e V is such that (u, Y~ a~v~) >~ 0 for any Z a~v~ e C', then it foltows that (u, v~) I> 0 for all i. That is, u e C. We now discuss some very special type of cone in ,~a with the usual inner product. We call a cone C a regular polygonal right cone (in short RPRC) if there exists a plane P not passing through the origin O such that (i) P N C is a regular polygon called the base, (ii) the line joipJng the centroid 6/ of P tq C to O is orthogonal to P. We shall caU t/ais line the axis of the cone (figure 1). We shall describe the duals of such cones. For simplicity we ~.all assume that P is the plane z = 1 in ,~a = {(x, y, z): x, y, z e R} and it is the plane of the paper. Suppose now that P N C is a regular (n + 1)-gon, its centroid is the point (0,0, t) and that one of its vertices v0 lies on the lir.e y = 0, z = 1. Letr be the distance of the vertices {v~}~ from O. (R.ecall that the vertices are equi- distant from G). With these normalizations, which do not result in any loss of ~.~-rality, we can write ( 2~k 2~k ) v~= rCOSn-~--~,rsin-- 1 O<~k<~n. - n+l' ' Let C, = {($ a, v~, a~ >/0}. Then C is an KPKC. By Lemma l, we have C~ = {(x,y,z)e~ s : ((x,y,z), v,)>~0; 0

Pnr

FIL~ 1

$1gure 2

Thus the bounding hyperplanes of C* are 2xk 2~tk rcos~-l x + rsin ~--~ y + z ffiO, O< k<~ n.

Intersection of these planes with the plane P : {z = 1} gives rise to an (n + 1)- gon whose sicks arc given by the equations 2~k 21tk I,l : r COS n-(.-T, x + r Sin n-~.--~ y + l = O, O <~ k <<.n.

L0 and /.1 are depicted in the diagram (figure 2). Note also that the equation for Lo is rx + I = 0 or x ffi - l/r and that the slope of L~ is - cot 2~tk/n + I which is independent of r. We now discuss two cases, of which the first one is what is needed in the main part of the paper. Simple Lie algebras 181

Case 1 : n+ lis odd, say2m+ 1. In this case the sides of the base of C* are parallel to that of the base of C. I-Ienee the dual (7,* of any C,~ {C,} again lies in the family, Le., C~ = Co for some s (ligure 3). In fact, s can be computed explicitly : s = (l'~/reos (:r/2m + 1), r \nj

01"

s = 1 (see (~/2m + 1)).

That is, C g "= C:t, (see nl2m + 1). In particular if ro -- (see 7r]2m + 1)1/~, then C~, = C~o, i.e., C,o is self-dual.

# 9 1.1 I ll'# ~.'1"7 :.. / /t,.o.,I I LoLl-" ,,

Figure 3

~/12m,,IF

' IVlli r

vm,I

ivm G i = r

Iv~Gi 9,*

l~ar~ 4 182 S Kumaresan and Akhil Ranjan

Figure $

Case 2 : n + 1 is even, say, equal to 2m. In this case, it can be checked that the dual C~, of C, has as its base a regular 2m-gon whose sides are not parallel to the bases of cones it~ C, (figure 4). In particular the families {(7,'} and {(2,} are distinct ai, d hence no C, can be self-dual. We now let Ss the symmetric group on {1,2,3} act on ,~a in the usual way, i.e., if(xa, x2, xs) ~ R s then for Ss o (xl, xz xs) = (x,~t, x,,t~ x,,ia~). Then it is clear that the cones C,st~"+t) constructed above are Sa-stable (the superscript denotes the number of sides of the base polygon).

References

[1] Dorfmeistcr J and Koccher M Regulate Kegel 1979 3". Deutsch Math. Verein 81 109-151 [2] Duistermaat J J, Kolk J A C and Varadarajan V S 1981 Preprint No. 200, University Utrecht [31 Harish-Chandra 1956 Am. J. Math. 78 564-628 [4] Heckman G J 1980 Projeotion of orbits and asymptotic behaviour of multiplicities for compact Lie groups. Thesis Rijksuniversiteit Leiden [5] Helgason S 1978 Differential geometry, Lie groups and symmetric spaces (New York ; Academic Press) [6] Kostant B 1973 Ann. Sci. Ec. Norm. Sup. 6 413-455 [7] Rothaus O S 1960 Ann. Math. 83 358-376 [8] Vinberg E B 1963 Trans. Moscow Math. See. 12 340--403 [9] Vinberg E B 1980 Invariant convex cones and orderings in Lie groups, functional analysis and its applications Eng. Trans. Vol. 14, 1-10 [10] Warner G I972 Harmonic analysis on ~emisimple Lie groups Vol. I (Berlin : Srrinser Wrta~