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On Invariant Convex Cones in Simple Lie Algebras

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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 construc- tion 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 decompo- sition. 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 posi- tive 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 nota- tion, 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 decomposi- tion ofg. Fix a maximal abelian subspace a of p.
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