The Determination of the Admissible Nilpotent Orbits in Real Classical Groups. by James 0. Schwartz SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1987 b Massachusetts Institute of Technology 1987 Signature of Author ,4 Department ofYMathematics May 1, Certified by Accepted " 1987 David A. Vogan rpýsis Supervisor by Sigurdur Helgason by~ Chairman, Departmental Graduate Committee AUG 22 1988 ARCHIVES The Determination of the Admissible Nilpotent Orbits in Real Classical Groups. by James O. Schwartz Submitted to the Department of Mathematics on May 1, 1987 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ABSTRACT This thesis classifies the admissible nilpotent orbits, in the sense of Duflo, in the following groups: SL(n,R), Sp(2n.R). O(pq), U(p,q), SO(p,q) and SU(p,q). The philosophy of coadjoint orbits suggests that the admissible orbits should be the ones to which one can attach representations. Thesis Supervisor: Title: David A. Vogan Professor of Mathematics, M.I.T. Dedication This thesis is dedicated to my parents with love and gratitude. Acknowledgments Interacting with my advisor David Vogan over the past four years has been a remarkable experience. He has It been tremendously generous with his time and ideas. of such to a person exposed has been a privilege to be deep commitment and excellence in his discipline. He is an inspiring example of clear thinking and intellectual honesty. I am very grateful to Jerry Orloff for the many hours he spent with me in the dungeon teaching me the basics of Lie theory, and later, puzzling over the meaning of D.V.'s remarks. Nesmith Ankeny got me started at M.I.T. and his presence and unique theories about the world have enriched this experience. I've spent many fruitful and not so fruitful hours discussing math and other quandaries with Jeff Adams, Jesper Bang-Jensen, Sam Evens, Bill Graham, Monty McGovern and Roger Zierau. Jamie Kiggen and Bill Schmitt have been a great source of support, humor and affection over the past five years. My friendships with my brother Tony, and sister Cassie, Jerry, Kathi, Steve, Peter B., Peter H., Annie I have and Jeff have meant an enormous amount to me. shared many good times with Martin whose obsession with movies and other forms of fun were a welcome relief from I thank John Rhodes, the colder beauty of mathematics. the playful Spartan, for his comradeship during the years we were tutors. Thanks to Phyllis Ruby for her helpfulness and Maggie Beucler for typing my thesis. Most of all I wish to express my love for Fluke, Flip, Wig, Bertha, Fatface, and the Hund; many names for the most important person, my wife, Ann Hochschild. Table of Contents Abs tract 0. Introduction and Preliminaries 1. Reduction to LnK 2. Complex Structure on 3. Classification of Nilpotent Orbits 4. Explicit Forms for 5. Admissibility in Sp(2n,R), O(p,q), U(p,q), SO(p,q) and SU(p,q) . 44 SL(n,R) 76 6. Admissibility in References g~/g E 0 0 L and LnK 85 -1INTRODUCTION This thesis is a study of the connection between representations of a semisimple Lie group and the partition of the dual of its Lie algebra under the coadjoint representation. g into orbits Kirillov first introduced this method in the study of simply connected nilpotent groups and it was generalized by Auslander-Kostant and Duflo to simply connected type I solvable groups. In these cases the orbit method has been successful in setting up a correspondence between equivalence classes of irreducible unitary representations and coadjoint orbits of G on go. The connection between the algebraic picture of the unitary dual and the more geometric orbit picture has provided useful insight and added coherence to the representation theory. To implement the correspondence discussed above, Duflo introduced a method for picking out a subset of orbits he calls admissible orbits in go . He describes a bijection between admissible orbits and unitary representations for nilpotent and Type I solvable groups. In the case of a real semisimple Lie group it is not at present understood which nilpotent coadjoint orbits should correspond to representations of the group. -2Moreover, there isn't a general technique for attaching representations to nilpotent coadjoint orbits -geometric quantization has not been applied successfully to nilpotent orbits except in special cases. The admissible nilpotent orbits are good candidates on which to attempt quantization. We have classified the nilpotent orbits and determined which of them are admissible in the following real classical groups: Sp(2n,IR) , O(p,q) and U(p,q) -- the first two are the groups preserving a symplectic and a symmetric form, respectively, on a real vector space, and the latter is the group preserving an Hermitian form on a complex vector space. In addition, we have determined the admissibility of the nilpotent orbits in the following semisimple groups: SU(p,q) and SO(p,q) SL(n,R) . Chapters 1, 2, and 4 describe the method used to determine whether an orbit is admissible. In Chapter 3 we recall how the real nilpotent orbits under each of the real classical groups above are parametrized by equivalence classes of representations of SL(2,IR) on a vector space preserving the appropriate non-degenerate sesquilinear symplectic or Hermitian form (Springer, Steinberg). In Chapter 6 we do the same in the case of SL(n,R) , except we look at representations of SL(2,IR) -3preserving a multilinear n-form. A nilpotent orbit under a complex group may break up into a disjoint union of several nilpotent orbits under a real form of the complex group, but it turns out, in the cases we have studied, that the issue of admissibility of a nilpotent orbit depends on its orbit under the complexification of G The final two Theorems in Chapter 5 show which complex nilpotent orbits are admissible in terms of the parametrization discussed above. In Chapter 6, we show all nilpotent orbits in are admissible. SL(n,R) PRELIMINARIES When G is semisimple, we can make use of the (or any nondegenerate G-invariant Killing form, B bilinear form on go), to identify this identifies adjoint orbits on orbits on We denote go . {Adg(X)jg E G} G {g E GAdg(X) = X} be called go go with go , and with coadjoint the orbit of X E g by 0 = (F E qo X, G X which fixes The Lie algebra of [X,F] = 0} . GX will The adjoint representation gives us a smooth, transitive action of on 0X ; thus the coset space OX = We will call the isotropy subgroup of X , the subgroup of GX go is a homogeneous space isomorphic to G/G G -4We now introduce an example which shows why integrality of an orbit is not a discriminating enough criterion to use to pick out orbits which should correspond to representations. integral An orbit OX is called if there exists a finite dimensional, irreducible unitary representation v GE of satisfying the following condition: iB(E,X)*I = dv e(X) If E is nilpotent, then Lemma 1). VX E gq E B(E, q) = 0 It follows immediately that for all nilpotent orbits -- take representation. v (Chapter 1, E is admissible to be the trivial But an orbit and the representation attached to it should match in dimension in the following sense. The representation attached to a nilpotent orbit should have Gelfand-Kirillov dimension equal the dimension of the orbit. shown that to one half Howe and Vogan [9] have there are no such representations associated to the minimal nilpotent orbit in Sp(2n,IR) . This example points out the need for an extended notion of integrality. Duflo's definition of admissibility is closely related to integrality, but it makes use of the symplectic structure of the orbits. We now describe this structure. OX is a symplectic manifold, i.e. there is a closed 2-form which is non-degenerate on each tangent space. -5E E 0X , we construct a skew-symmetric For a fixed bilinear form on * smooth mapping g E . given by -)) TE(OX) g -- Adg(E) is onto and has We define our bilinear form on go/go X ) OX Thus we have a canonical isomorphism: E TE(X) E(O G - : consider the as follows: d~le : Then the map kernel TE(OX) and carry it over to 0X by means of this E o q0 0 isomorphism. To begin, define the skew-symmetric mapping WE : R 90Xgo - The radical of as follows: wE E is go wE(V,W) = B(E,[V,W]) , and therefore wE makes sense when it is considered as a form on the quotient E space g /g go /go E Clearly, We will omit . GE The group E is non-degenerate on the verification that OX . a closed 2-form on wE wE gives us (See Guillemin-Sternberg). acts on the symplectic vector space via the adjoint representation and preserves the g /g form wE . For g E G E E E g.(Y+g o ) = Adg(Y) + g o and It Y + g E E Eo/go E is easy to check , we define that this E into , Sp(go/ao) vector space the group which preserves g /g o wE on the E o The symplectic group of any vector space V, Sp(V) has a well-known two-fold cover called the metaplectic cover, Mp(V) (Shale [6]). let T Let 7 : Mp(V) --+ Sp(V) and -6denote our mapping mp (GE , G E -- SP(go /o)0 E the "pullback" cover of . G (GE mp = {(x,y) 6 GE x Mp(go/o E)(x mapping p (GE) m p -G : E We define as follows: = ) r(y)} . The defined by projection on the first factor is a two-fold cover of GE MP(g o /E (GE)mp GEE E Sp(go /g) T We say a representation vmp of (GE )mP is genuine if it is not trivial on the kernel of the map pl : (GE)mp -, G E representation i-B(E,X)*I Finally. we say a genuine vmp is admissible if dumPle(X) = E E X Eg . The group G and the for all orbit are said to be admissible if admissible representation. that i*B(E,X) = 0 nilpotent. (GE ) z t Thus , and (GE)mp , G where kernel of the map For E E g admits an It is not difficult to see for every vmp (GE) m p X E g E if E is must be trivial on will be admissible if and only if z pl is the nontrivial element of the : (GE)mp -- GE nilpotent, the algebraic group be decomposed as follows: GE = L k U , where L GE is can -7reductive and U simply connected. to It follows that L and its two-fold cover Possibly after replacing we can embed E may (GE)mp 6 ' where take is isomorphic 9t(2) the Cartan involution of 0 (9I(2)) , or polar decomposition enables us Because as L LmP by another element of in a 6-stable copy of is = X E to write 0E inside g L = ZG(SL(2)) reductive subgroup in our decomposition of L , and Lmp x U , which reduces the question of admissibility to the subgroup go GE is unipotent, normal in GE . , as . Then we the The G = Kexp(p ) is 9-stable and algebraic, we can decompose L = (LnK)exp(lop o) , and, as above, the question of admissibility reduces to a question about the two-fold cover of by mapping LnK . LnK We recall that this cover is constructed into Sp(go/g) . Hence the question of admissibility of an orbit finally reduces to analyzing a representation of a compact group on a finite dimensional vector space. -8Chapter 1 We begin by showing how the question of admissibility for the nilpotent orbit 0E is equivalent to a topological question about the two-fold metaplectic cover (GE )mp GE C G . of the subgroup We begin with an easy Lemma. Lemma 1. Let Proof: . [go [E,X] = 0 this qf(2) B([goli, [o]j) = 0 s9(2)-theory tells us that E E C i< [Io 0 i Proposition 1. , where i . ; therefore A nilpotent orbit We decompose [go• i is i = j . unless 2 E o) 0E if and only if the kernel of the map is not in the identity component to embed the The go-invariance of E E [Co B(E, go implies that {E,F,H} . s~(2)-triple under i implies B(E,X) = 0 is the Killing form on H-eigenspace with eigenvalue B Then Use the Jacobson-Morosov Theorem in a standard = B We must show that B(E,X) = 0 0 goq be nilpotent. E X E g0 , where for all E E The and = 0 . C go r (GE mp : is admissible (GE)mp of (GE )mP GE -9Proof. Let (e,z} M = denote order two subgroup of GE clear that = 0 mp vmp Hence Vmp(z) = I Now assume z C (GE) mp an it is EGE (GE )mp then, by = I vmpI vm p and of . In is not genuine. (G E mp , and we construct a genuine z representation vmp of = I ; that mp r -- is not admissible, because if there were dvmPI particular, If (GE mP an admissible representation the Lemma, the kernel of (GE mP is vmp such that is admissible. (GE mp Let M = {z,e} component group denote the image of (GE)mP = (GE)mP/(GE)mp M . in the Then SC (GE)mp , an order 2 normal subgroup, is contained in the center of the group, hence K C Z((GE)mp) . Let a be the non-trivial one dimensional representation of M Let p denote the right regular representation on Ind(GEmp(a) . M p(z).f : Let For f Ind (E)mpa) we calculate M g E (GE mp , then p(z)f(g) = f(g z) = f(z g) = z.f(g) = -f(g) Thus (GE) m p p(z) X I . by setting We define a representation vmp(g) = p(g) where v of g E (GE) mp andI -10g is the image of vmp g (GE) m p in It is clear that is an admissible representation of GE We now show reductive subgroup admissible. E E go is admissible if and only if a L C GE Proposition 2. 2 . which we define below, is Use the Jacobson-Morosov Theorem to embed in a standard L = ZG(sX( )) GE , U 9s(2)-triple {EF,H} C g 0 is isomorphic to is L K U where the closed subgroup of corresponding to the Lie subalgebra U (GE )mp u 0 = g E GE n [E, 0 '0 o] is unipotent, connected and simply connected. Proof: Assume 9X(2)-triple u E U . is the nilpositive element in the (E,F,H} C go . g 6 GE element and E Fix First we show that every can be decomposed g E G {E.Adg-1 (F),Adg-1 (H)} . Then g = e E L -*u with {E,F,H} and are both 9s(2)-triples with the same nilpositive element. By Kostant (1959), all triples with the same nilpositive are conjugate by an element of U . Furthermore, the set of neutral elements for all triples with nilpositive where = [E, o] n g u 00 U --+ H+u o which sends E 0 E is the linear coset H+u 0 and there is a 1-1 onto mapping: u ---+ Adu(H) . Hence there is a -11unique u E U such that , Adu(Adg -1(H)) {E, Adu(Adg-1 (F)) Thus u*g -1 = L E ZG(9t(2)) g = Now we , and we have decomposed -1 (g.u-1 )u show that connected, simply connected, and u The exponential map: u -- the exponential map, exp nilpotent group dimension of dim n = 1 n . is unipotent is normal in GE U : is 1-1 and onto. n -- N To see , for a connected This statement is true for because the exponential map is onto for a n E N n has a nontrivial center with j , such that exp(X+Zl)-exp(-Z exp there exists exp(X+Z 1 ) = n-z E U U is a nilpotent Lie algebra. therefore we may assume Z2 , N , is onto we use induction on the connected abelian group. Thus for . E LU L n U = {e} We will show below that = {E,F,H} 2 exp Z 2 ) = n exp(X+Z -Z2) = n . z E Z n/g ---+ N/Z X+g E n/j , Z1 E = z j . is onto. such that Also we can find ; therefore and this gives To see basis for the matrices : n exp is one-to-one, we find a in which they are strictly upper triangular (Engel's Theorem). It is easy to see by -12direct computation that exp matrices. is homeomorphic to Euclidean Therefore space, i.e. Let U U is one-to-one on those is connected and simply connected. [go] k be the H-eigenspace with eigenvalue under the adjoint representation of sI(2) on go k As an easy consequence of the 9X(2)-theory, we get u [ o C Let . u C U and W E u such that k>O exp W = u If X E $ [go , k then Adu(X) = X+Y with kŽO Y E [go ; thus k U is unipotent. By the same k>O argument, if X C [go] 0 commutes with the and L n U = {e} thus GE = L enough to show X E u . Adg(X) E [E,W] = X , . XI(2) . U = u 0 go . 0 On the , because . U . is connected it Since [E,Adg(x)] U is normal for all g C GE = Adg[E,X] = 0 We know there is a W E go AdR This shows We now show in GE is and , hence such that so: Adg(X) = Adg([E,W]) hence action on Adg(X) C u We have E Adu(X) f [go] e E L , AdR(X) E [go other hand, for that then Adg(X) E [E,go] as desired. = [Adg(e),Adg(W)] = [E,Adg(W)] and we get We note that L Adg(X) C gE n [E,go is reductive, because -13the centralizer of a reductive algebraic subgroup of an algebraic group is reductive. Recall 0 the covering map GE have shown that (GE)m p -- : GE L x U . is isomorphic to We . Define _-1 Lmp = -1(L) and we prove Proposition 3. Let U be the analytic subgroup of Lie algebra u . Since an isomorphism of by L , U is Lmp x U isomorphic to Proof: (GE) m p The metaplectic cover U 7 is simply connected, U onto U is normalized by Since u with is is normalized o Lmp We have an injective map must show this map is onto. covering maps as (GE)mp i : Let (GE )m p L mPU - denote the 72 and V1 and in the diagram: LmP U (GE)mP - GE Choose x 1 E (G Emp mp show that either i(y 1 ) ; i(y 2 ) and {y, Y2 i(y 1 ) = x I or = -1 2 or 1 (X i(y 2 ) = x but they both maps to xl 1 ) We will 1 under 7T 1 -14Since r 1 (x 1 ) = x1 conclude either there exists a or equals 0 = -F' and subalgebra spanned by . OH' t' 9' on the V'E = -F as follows: e to a Cartan p' ' = [R(E-F) with 0 0 and p' RH e IR(E+F) = involution g = p" " $ 9") and 9" V' C o Helgason). . One can extend 9' on the whole Lie algebra to a Cartan go so that (the Cartan decomposition with respect to f" o and p' C p" o o (Mostow, p. 277 Since all Cartan decompositions are conjugate, we can find and {E' = = -H' {E,F,H} sT(2) = Then satisfies the This corresponds decomposition of . such that the sf(2)-triple We define a Cartan involution 8'H = -H 0 go a Cartan involution on g C G OE' xl1 be a standard F' = Adg(F), H' = Adg(H)] relations and i(Y 2 ) {E,F,H} C go Let sf(2)-triple and Proof: ,1 is a two-to-one map, we i(y 1 ) Proposition 4. Adg(E), and Adg(p") o = p decomposition on where g E G such that go= p0 0 O go with respect to Adg(t") = f o is 8. the Cartan We have -1 Adgog"oAdg and the triple {Adg(E),Adg(F),Adg(H)} satisfies the desired relations. O o = -15We recall the polar decompositions for a reductive G . group G = K exp(p o) We can write the Lie algebra decomposition of the 1-eigenspace and involution 8 for G C = t p is the(-1)-eigenspace for a Cartan go defined on compact subgroup of . po go which comes from , and K the maximal corresponding to the subalgebra The following two Lemmas give this decomposition L . Lemma 2. The centralizer of a O-stable subgroup H C G is O-stable. Proof: c g : It suffices G --- G be defined by We observe that c c(eg) = Goc g0 g Ad(Og) = OoAdgoO (x) = g*x*g because -1 . Let , x EG . c g(x) = = O(g)-x-O(g1 ) = O(g(Ox)g- ) = O(cg(Ox)) (Og).x*(Og)for to show that x C G . Differentiating this equation gives us what we want. 0 Lemma 3. Suppose IH/HOI = n Then H is a O-stable subgroup of H = (H n K)exp(po n § ) G and where §o = Lie(H) Proof: It is clear that and we want to show = (H n K)exp(po f H H11 = H . Assume ) CH for a moment we -16know n HO = (H on) K) 0 exp(po can write h = k exp X -l calculate (Oh)-1*h = exp2X E H get (exp 2 X)n Fix h E k , with E exp(po0n exp X o) ýo) X E po and k E H H' = (H n K) 0 exp(po n We IH/HnI = n . we and, therefore, o K So it remains to prove that o0) . . X E po Then = exp(2nX) E H 0 , and hence and Since exp(2nX) E H0 N exp po = exp(po 2nX E po h E H , then we So . O = (H f K) 0 exp(po §o) is an open set in H0 because it contains a neighborhood of the identity, therefore, assuming it's a group, it is both open and closed which implies group. H' = H . Now we show H' is a The following argument shows that it is sufficient to show for expX expY E exp(po n p,o X,Y E ~ o) . Let kl'expX 1 ,k2 *expX 2 E H' then (k 1 exp X 1 )-(k 2 exp X 2 ) = kl(k -1 2 k2 )*expX 1 *k 2 (expX 2 ) = kl'k2 expX 3 *expX 2 which is an element of H' expX 3 *expX 2 E exp(po n So we set o) O(t) = exptX-exptY = k(t)exp(Z(t)) , -17 k(t) E K , Z(t) C po t . We want to show and that - k(t), Z( a re analytic in Z(t) fn t) but it is enough to show for small P0 t for all t It is enough to show that the coefficients in the power series expansion of Z(t) belong (t) (00(t))-1 (e@(t)) p(t) to 40 . We have = exp(Z(t)) , hence = exp(2Z(t)), 1 Z(t) - 1 2 -1 log((eO(t)) 0(t)) log(exptY*exptX-exptX.exptY) By Campbell-Baker Hansdorf this last expression can be written as a power series in bracket in ý X and Y . t whose coefficients are Since X and , these coefficients do as well. Proposition 5. We have n L = (L Y belong to 0 K)exp(po n IO) where 10 0 = Lie L Proof: Follows directly from Lemmas 2 and 3. Proposition 6. Proof. The group LmP = O (L n K)mP exp(pO nlO) Apply the same argument used in Proposition 3. 0 Recall that an orbit is admissible if and only if -18the non-trivial element in the kernel of 7: z GE (GE)mp - the map is not in the identity component. be the non-trivial element in the kernel of Proposition 3, Therefore, (GE)mp = LmP U z E (GE) mp Proposition 6, LmP and U By is connected. if and only if = (LnK)mP.exp(ponl v . Let z E (LmP)0 ) and . By exp(p ol is connected. Thus, z E (LK)p We conclude that the admissibility of an . z E (LmP)O ) if and only if orbit can be decided by considering the compact subgroup LAK c G E GE We have seen that representation into Sp(2n) maps under the adjoint , where LK C G E The image of the compact subgroup compact subgroup of are conjugate in Sp(2n) . Sp(2n) 2n = dim (to/ E E must be a Since all maximal compacts , we can map LnK into Sp(2n) n O(2n) , a maximal compact subgroup of isomorphic to U(n) . Sp(2n) The following Proposition, which we will not prove, enables us to compute the metaplectic cover U(n) m p Proposition. of U(n) The group {(k,e i) explicitly. U(n)mp is isomorphic to E U(n) x S 1 I det(k) = e2i Define a character X on LnK . by composing the -19following maps: LnK Sp(2n)nO(2n) ---S We construct the pullback cover metaplectic cover Introduction. (LAK)mP Si from the as described in the The next Corollary follows directly from the description of Corollary. U(n)mp U(n) det U(n)mp given above. (LAK) m p The pullback cover is equal to 1 1 x(g) = z2 } {(g,z) E (LAK)xS The next Theorem shows that an orbit if and only if the character there is another character K is admissible \ is a square, of LAK that is, such that S2 Theorem. Let kernel the covering map z t of (LAK)0o Proof: <=> be the non-trivial element in the t (LnK) mop 0 7 the character First, assume be a path from z z ) = \2 the identity to Let {(g(O),e i) : (LAK)mp - y of LAK . LAK Then is a square. and we show there cannot z in (LAK)o E (LnK)xS11 0 < , that e is < 7} -20be a path from (e,1) , g(r) = e g(O) = e to (e,-1) , and " P(g(O)) = e iO Therefore \(g(O)) = \i(g(r)) = e P(e) i7n = 1 , = -1 (g(O)) = \ (g(O)) \(g(o)) = -e or p(g(O)) hence which (LnK) m0 p in = e ie 2 W e have = e2i 0 ie But Thus is impossibl e because g(7r) = e Conversely, assume a square. Define a character g E LnK , P(g) = {wj check this P(g) and we sh ow (g,w) E (LNK)o } we have as desired. k(g) = w It is easy to LnK ; therefore 0 In the next Chapter we develop the machinery necessary to compute X . x P by setting, f or is a well defined character of (g,w) E (LK)m k(g) = z e (LnK)mp For -21Chapter 2 In order to construct a map from LnK into U(n) go/go we will find an (LnK)-invariant hermitian form on considered as a complex vector space with respect to a complex structure J . The vector space go/g is endowed with the natural GE-invariant symplectic form WE . We seek an (LnK)-invariant inner product complex structure J Hermitian form on g /g such that E and is a , that is, the following relation must be satisfied by iT(x,Y) = T = b + iwE b T(Jx,y) J all and T : x,y E goa 0 0 . Equivalently, the compatibility condition can be expressed in terms of J, b , and b(x,y) = oE(Jxy) If b and T(x,y) wE all wE as: E x,y E go/o0 are (LnK)-invariant then so is = b(x,y) + iwE(x,y) = b(e.x,e.y) + = T(e.x,e.y) iwE(e.x,..y) T a -22Assume we have found an endomorphism J go/ E on following four properties: satisfying the following four properties: 1) J2 = -1 2) WE(Jxy) = -WE(xJy) 3) WE(Jx,x) 4) J x,y E go all E with equality if and only if x = 0 _0 commutes with the action of Then we define a symmetric bilinear form LfK . b go/ on 0 def b(x,y) = oE(Jx,y) by setting 0 and we make the following Claim. on = b + iwE T !o9 o b J The necessary compatibility among is built into the definitions. property 2.) as follows: wE(Jy,x) = b(y,x) . b b(x,x) = wE(Jx,x) > 0 (LOK)-invariant because e (LOK)-invariant hermitian form viewed as a complex vector space with complex structure given by Proof: is a E LnK we have: b J , wE , and is symmetric by b(x,y) = wE(Jx,y) = -WE(x,Jy) = is positive definite because if x X 0 . J Finally, T commutes with LQK is . For -23- T(x,y) = oE(Jx3y) + iWE(x'y) = WE(e-JxCy) + iWE(e.x.~.y) = WE(J(e.x).e.y) + iCE(e'x'e.y) = T(e.x,e.y) 0 . g /gq We now exhibit a complex structure on satisfies properties (l)-(4). isotypic subspaces under Decompose sT(2) o go into W1 ; go= which o where i=l W1 m.V Let [W1], Define an irreducible 9X(2)-module, dim Vi = i V , 1 be the H-eigenspace with eigenvalue J on Definition. [Wile for x E [W ]e For • -i-1 • e , define i-3 e as follows; J(x) = (Je.i) -1/ 2 .(adE(x)) where depending on e , and defined as follows. i and the weight spaces of constants j V j2 are one dimensional, such that for .i is a constant . Since there are x E [Vi]% [F.[E,x]] = je.i-x . We compute and F j9,i in the by looking at the action of el(2) representation on homogeneous polynomials of degree i-1. 0 1 operator r((O 0 ) ) operator x 8 ay E In this representation, the corresponds to the differential 10 00 ) r( 0 to , 8x ,0 y 8 , and 1 7r(o -1 0 ) to -24- a 8 So we have: Y -- ax i-l+e i-1- 2 defn.i-1+ 2 = i Ri ,i = J-e-2,i 2 y ) 2 i-1+e i-1-e 2 2 .. conclude that F.E.(x i-3-P i+1+e ri-i-el i-] .i.. 2 = y) e,i(x Proposition. i-1- 2 = i+1+ i-l-e 1 2 2 at and note that J , defined above, satisfies properties (1)-(4). Proof: 1.) We check J2 = -1 . It is enough to check this on a basis, and we choose a special basis made up of eigenvectors spanning each isotypic component x E [Wi Pick J 2 x = J(( (J = -1/2 ,i )*O(adE(x)) -1/2 = (J,,i )J(O[E,x]) -1/2 ,i )-J(O[E,x]) ) 8[E,[ O E, O x]] -1 = (-J~ i)-[F.[E,x]] X C q ; -1/2 -1/2 S(J,i )-(J--2,i -- W since ,i since O[E,x]E[W and 1 -- E = -F 2 -25- 2.) Check w E(Jx,y) = -WE(xJy) We check this equality on a basis of eigenvectors chosen as in 1.). note that W E(E o]i,[go]j] = B([E,[go O equals unless j = -i-2 we conclu de that for x,y -W E(xJy) = 0 WE(JXY) . i]l[¾o]j) 0 Since We which J[[go]i] = [o]-i-2 in different eigenspaces . Now take x,y E [Wi]e Then . WE(JXy) = B([E,Jx],y) = -B(Jx,[E,y]) = -1/2 -1/2 B(O[E,x],[E,y]) = -(Ji .~ -(Je,i ).B([E,x],6[E,y]) = -WE(X Jy) 3.) -WE(X, Check wE(Jx,x) > 0 0 x -1/2 which is alway s positive and non-zeroi 0([E,x],[E,x]) non-zero on 4.) to fixed by 0 0 go/g Finally we must see that LnK C GE Certainly se e why 8 conclude f or WE(JX,x) = Jx) = -B([E,x],(jei )*-[E,x]) -1/2 want if and 8 adE J coimmutes with commutes with commutes with LfK . 0oAd(g) = Ad(Og)oO for LnK , so we Since g E G k E LfK , BoAd(k) = Ad(k)oe K is we as desired. n Now we decompose g = ( [go]i into a direct sum i=n of eigenspaces under representation. the H action of Because the 9s(2) L = ZG(s~( 2 )) , L preserves o -26[go] the eigenspaces i for all i Thus for e E L we have: detR(Ade) = II de t AdR[go]i] i=l For g E GE we let Adg Adg on the uotient space Ad(LAK) as the complex onto pace E g /g0 Ad(LOK) . Proposition. Proof: . Let Then V = [go , we have go 01o-1 We will h detC(hlV) transformations of J J maps i , hence ). rrr _detc[A-"•l~,] )*detd l[•oCg]i m Eol) i0-1 = ±1 o]_-i-2 = i[ consider The complex structure (+1 go preserves = detR h 0o# if h E J([oi [go ] i igo] o h E LAK , V = [g ].i[goi ] detC(hlV) ([goli)e • Since J . This is possible because for all [go]-i-2 det C(Ad- )=det C(Ad-- i X -1 E go/ group of complex linear commutes wi [go ] denote the action induced by ) (LnK)o ) = and commutes with E R . But LAK is compact, therefore its image under the determinant map must be ±l (and +1 on the identity component). 0 -27By the discussion leading up to the Proposition and the Proposition we can conclude: detC(Adh) = ± detC[Adh • [o]_] Our goal is to calculate this determinant. We begin this task by finding a parametrization for the nilpotent in g0 (under certain groups). this problem. The next chapter addresses -28Chapter 3 Classification of Nilpotents Let G algebra go be a real classical Lie group with real Lie We describe a correspondence between Lie algebra maps from st(2) into go and nilpotent orbits for which we need the following Definition. A Lie algebra map equivalent to another such map s~(2,R) -- : *' : go sl(2,R) -- is go if -1 there exists g E G such that O(X) = go'(X)g- Vx E s9(2,[R) Proposition 1 (Kostant). There is a bijection between equivalence classes of maps from nilpotent orbits under Proof: into go and G . By the Jacobson-Morozov Theorem we can embed any nilpotent 1(2,IR) sT(2) . E E go in a subalgebra of go isomorphic to Thus there is at least one equivalence class of maps corresponding to each orbit. To see there is only one, we use the fact that any two copies of 91(2,R) C go with the same nilpositive element are conjugate by an element of the subgroup U C GE C G -29(Kostant, 1959), where subgroup of = [E,g ] U is the closed, connected corresponding to the Lie subalgebra GE E this establishes the bijection. 0to u0 0 Thus, we conclude that the classification of nilpotent orbits under (for the moment) an arbitrary rea 1 group G is equivalent to the classification of equivalence classes of maps from Now we assume sT(2,R) into is a real linear Lie group which G preserves a non-degenerate sesquilinear form vector space G = G(wo) in M n(F) defined over a field V0 Then . go is a real (for some n) sT(2,R) - ~ from , --- G orbits under maps from G on a We denote wO lift . Because Lie to Lie group map s the classification of nilpotent is equivalent to the classification of SL(2,R) -* element of F. wO matrix algebra containe d preserving algebra maps from SL(2,IR) g G(wo) up to conjugacy by an G(wO) We introduce a notion of equivalence among pairs (o,V) w where a vector space is a non-degenerate sequilinear form on V/F . Definition. We define an F-linear isomorphism W 1 (VW) = w2(Tv,Tw) (wlV1) 1 T : for all V1 - 2,V2) V2 v,w E V 1 if there exists such that -30- This isomorphism is unique up to left multiplication by an element of G(w 2 ) element of . T : G(wl) V 1 -- * V 2 isomorphism OT : T or right multiplication by an Let (wlV 1 )~(w2,V 2 ) and implement the equivalence. Then the induces an isomorphism of groups for defined as follows: G(w 2 ) --- G(wo) -1 g G(w 2 ) let OT(g) = T ogoT E G(w) Fix an equivalence class all maps SL(2,R) -- G(wi) , and let (wo 0 ,V) (WOVO) . be a representative of . (w0 ,VO) We consider the set of such that (wi,Vi) E (w0 ,Vo) and define an equivalence relation on this set as follows. Definition. 0 : Let T SL(2IR) ---- G(wl) : SL(2,R) --- G(w 2 ) . there is an isomorphism forms w1 and w2 and We say T T : -1 V 1 7 ~ S ---- V 2 and if and only if preserving oj(x)oT = T(x) the for all x 6 SL(2) We observe that the set of equivalence classes of mappings of SL(2) -- G(i) , i E I , is in bijection with the set of equivalence classes of mappings from SL(2) --- G(w 0 ) for the fixed group G(wO) equivalence is defined by conjugacy within where G(O) . the -31Now consider the set of all t riples form w and on SL(2) a representation of {(7r,o,V) 7r is leaving invariant the V Define an equivalence (w,V) E (o0 ,VO)} relation on this set as follows: Definition. (lW isomorphism T 1, V1)) : V V2 - 1 if there is an 2,( 2,V2) which action and preserves the forms intertwines the and w1 SL(2) in the sense w2 already explained. Clearly, the following two sets are just descriptions of the same thing: {(r,w,V)Ir is a representation of preserving o and (W,V) E (o 0 ,VO)} (wo,Vo) SL(2) V SL( 2) {f : --- equivalence classes of and G(wi)I(J i,Vi As we've noted, the . , i E I} on set of the same as the latter set is set of equivalence classes (under conjugation in ({ : SL(2) -- G(wO)} . of nilpotent orbits under Therefore, (w,V) E (WO,VO)} SL(2) G(O)), classifi cation the reduces to understa:nding G the partition of the set of triples representation of the on V {(7r,w,V) • preserving w is a a nd into equivalence classes. First we choose a standard model for an irre ducible es(2) representation over R , (ri,Vi) , where Vi the vector space of homogeneous polynomials of degree is -32i-i /R in two variables (irreducible fe(2)-representations are determined by their dimens ion). of SL(2) (aX+cY) (1bXV+ (0 -1) E = (O 0) and A A to the operators respectively. form on Vi nil is given by substitution the L i al a Xe Also, we fix a n ( sf(2) a 3y -y . and correspond a ax y --3 i-1 ,Y determine) = 1 .triples i) (up to equivalence) representation of (7i ,wii1,V nn V• SL(2) (wi,Vi) H = 2)-invariant bilinear It i- c d 'oVO)an d we wil 1 determine all 0 Now we use wi(X of n) y'* (a b)X i j gebra the elements 0 ( F = ax The action a b tcd ) E SL(2) , or a x by setting We fix form and X'YJ on a monomial of variables; tnat is I, j i where VV , triples is a an SL(2)-invariant E (00' the semisimplicity of a representation (r0 ,V0 ) 91(2) to break up into its irreducible components. Lemma. V 0 = @ Vi H i i IR and = H over Hom( 2 as 9l(2)-modules (V i is as above) , where )(V',V) H i is a vector IF Proof: Define a map r : $ Vi®H i -- V by i defn. T(e i (v.Ti)) It is clear T(vi) = that i T is where v.®T 1-1 and since i e V O®H dim V = space -33dim(G Vi H i ) , T i SVi OHi i is an isomorphism. We consider as an 9X(2)-module by making 9s(2) first factor alone, and then it is clear that intertwining map. act on the T is an 0 We want to find all s1(2)-invariant forms on V0 ViOH i We fix our canonical sI(2)-invariant form . i nondegenerate sesquilinear form Definition. Let S wieh h i on H i be a nondegenerate i * sesquilinear form on Vi H i given by i i 1 where V0 i v.®T ,w.@S 1 The form $ 1 1 Vi H iO.hi on $ Vi H i o as follows: i w(T(9 v.iT T(e W.Si)) ) i = (70'9 gives a form 0VO) and *i. i ( *h i T i , $ iw0Si ) hi , $ ViH i) (r, i . are equivalent i triples in the sense defined previously. A choice of on -34nondegenerate sequilinear form i , h fixes an e1(2)-invariant form on Proposition shows this gives all on e ViOH i Vi H for each 0 Vi H i . i The next sf(2)-invariant forms i Proposition. on on Fix the canonical sr(2)-invariant form * Then the SL(2)-invariant forms on Vi H i i all of the form @ o ®h i sequilinear form on Proof: hi (2)Q ViH is an arbitrary H SL(2) invariant forms on correspondence Hom where 0 Vi H i i are in 1-1 with intertwining operators, i s((2) i Hom 9 (2)( (G Vj@HJ) ) . j ViHiV), $((Vj We have ) *())) ii ,J ( Homs(2)(V i ,(V ) (V i (Vi) ) ) Hom(HiH) ij 0 Hom o( HomF(H. (Hi) ) ) i W3 are -35- But we have fixed an Hom identifies i sT(2) form (V ,(V2 )) )with sI(2) i o on V i which IF , and we conclude -36Chapter 4 We have seen that nilpotent orbits of G -- the group which preserves a nondegenerate sesquilinear form on a vector space V -- on its Lie algebra are in one to one correspondence with equivalence classes of representations of sf(2) on a vector space with an 9s(2)-invariant sesquilinear form. representative We fix a (v, $ (Vi@Hi), $ Wi@hi) i (same notation as in previous section), where we choose hi to be Hermitian or skew-Hermitian so that the form the vector space $ Vi H i $ w ihi i on is Hermitian or symplectic. i This gives us the nilpotent orbits in o(p,q) or u(p,q) if V sp(2n,[) and is real or complex, respectively. Definition. We say that a basis {(Ti is a l<jidim H standard basis for form Hi 3.) hi on Hi complex, for 2.) h Hi if for with respect to the sesquilinear 1.) for hi hi skew-Hermitian and symplectic and Hermitian, the matrix (H)j k equals 1.) P -iI q 2.) 0O -I or 3.) or 3.) Hi real, or (hi(Ti ,Tk) h ()) [ -Iq -37- In order find a representative of the orbit corresponding (v,@Vi H i the equivalence class i@h i @ Vi H i i we choose a standard basis on the linear transformation respect to this basis. of 0E Then . Let r(E) g = (g)ij E G Adg((E)ij) and and we realize {(e = n 1 j=1 (E)i j with is a representative g -1 =(g is realized by writing respect to the basis i i as a matrix (E)i j E go 0E -G )ij 'r(E) EG with g..e.} 31 3 l in We now show how to construct a Cartan involution 0 , and a O-stable copy of sT(2,R) C go whose nilpositive element is a representative of We define an inner product 0 b 1lb2 Q = @ 1 i E @ Vi Hi on 2 i which satisfies the following two properties. 1.) respect There is a standard basis of to the G-invariant form orthonormal with respect to @ @ Vi®Hi i o ®hi 1 with which is Q = @ b l®b i 2.) Q(v(E)v,w) = Q(v,v(F)w) Assume these map L . v,w E V two conditions are satisfied. be the matrix corresponding to (E)ij E go for all v(E) Let in this basis; The adjoint with respect to Q is given by conjugate transpose when written in an orthonormal basis; hence (E)i j of a linear L is (E)ij ij == (E) Ei j -38But condition 2.) = r(F) . e((E)ij) = -(F)i {(E)ij,(F)ij,(H)ij} C go Q . We now define Vi®Hi T b1 ( i-l-sy )X s i and an inner product b2 Let Then the basis i @hi ( i- in spanned Hi on V (i-1t on We define an inner product Claim I.) 91(2) SO<s<i-ll<k<dim H H , b I i by setting ) xi-l-ty t ) = st st) i H b (T 1 T 1 ) = 2( m n v@T,w@S E Vi@Hi or, is e-stable. i-1) 1/2 Xi-l-sys bl( (i-1s and the j r(E) Choose the following basis for Define an inner product for (E)j i = (F)i j Hence we conclude that other words, by gives us the following equation, b Ob by: 6 mn on b (vOT,w@S) ViHi as follows: b (v,w)b (T,S) be a symplectic form on Vi@Hi 1/2 xi--sOT O<s<i-1 l<k<dim H up to the sign and order of the elements is a Darboux and orthonormal with basis with respect to w @hi respect to bl@b 2 on II.) Let hi Vi@H i be an Hermitian form on ViHi w1i h1 be an Hermitian form on V OH1 II.) Let -39a) If i is odd then (up to order) 1i ) /2)(X s O<s<i-l, 1(-syS l<k<dim Hi i with regard to respect to b) If i i Tk w Oh @ i-l-s i Tk)} X sY is a standard basis i and orthonormal with b i@b 1 2 Hi is even and real, then (up to order) i-i s/2 O<s<i-1, i-1-s sT i k l<k<m where 1i Xi-1-s@ T i k+m dim Hi = 2m a standard basis with respect to c) If i is even and Hi i Ohi b and orthonormal with respect to is 1 1 b2 2 complex then (up to order) {(i s)/2 (X i - i k- i -XSyi-l-ss 1 Y @ i *Tk ± Xs i 1 is a standard basis with respect to and orthonormal with respect to Proof: Obvious. b 1 Ob 2 0 Hence we have shown that (1), O@hi Q satisfies condition and we now show that it satisfies property (2). suffices to check this on a basis for Vi@H i . For It -40k X e and m A n both Q(E.(k+) i- 1 1/2(xi-l(k+l)yk+l i- 1 1/2 Q((k+ i-l 1/2(Xi-l-eiyTi )) i m n (xi-l-(k+l)yk+l Ti)F.i-i (X Tm e equal zero. Thus, it suffices Q((i- (X 1/2(Xi-k+ k+ +1'lT),F.( (i-i IkT , k) i i- F lOwT) i-1-eR (X @ and Ti ) ) n)) Y to check: Q(E.(i-1)1/2(Xi-(k+1)k+T Q(E(k+1 ) 1/2 ( i-l-kyk Ti 1/2 {X 1/2 ))= i-1-kykTi Y (X)) i-1 1/2 k+l (k+l) The left hand side equals 1-1 hand side is i- and the right i-i1) 1/2 1/2 1/2 It is easy to check k+1 1/2 1/2 (k+1) /2(i--k)1 / 2 these are both equal to We now describe an explicit realization of matrices. Let V = $ Vi H i L as be the decomposition of V i under a e-stable copy of nilpositive element. linear isomorphism sT(2) Any g of V . Hence, L . sr(2) g as its gives a Furthermore, since commutes with intertwines the action of lemma this implies that E g E L = ZG(zl(2)) V --* V linear transformation which has 9l(2) V . on the , g By Schur's preserves isotypic components can be characterized abstractly as the group of intertwining operators T : V -- V where -41T V . preserves the bilinear form on We realize these intertwining operators as matrices with respect to the basis chosen on V 9s(2) This gives in go to determine the 8-stable copy of L as a 9-stable matrix group. Proposition. L a Sp(2ml)xO(p 2,q a.) 2 )xSp(2m 3 )x*** if G preserves a symplectic form on a real vector space where 2m i = dim Hi and p.+q. = dim H L a O(plql)xSp(2m 2 )xO(p ,q 3 b.) 3 j )x.*. if G preserves a Hermitian form on a real vector space 2m. = dim H i where L r c) and V V p.+q. = dim Hi. U(pl.,q)xU(p 2,q 2 )x*** if G Hermitian form on a complex vector space preserves a V , where = dim H p.+q Proof: We know form ciOhih 0S Vi H i V on * Vi®Hi . with G-invariant bilinear As we have observed g E L preserves isotypic components, therefore we can write g = where g g : ViH i -- . VH Recall that Hi i can be thought of as linear transformation of each map. But (V i V) • Hom i gi g E L of H i g defines a by acting on the range preserves the sesquilinear form -42on V and therefore form on H i . E Sp(2ml)x**.. g = 2 (Id)@g i g On the other hand, an element specifies a linear 0 Vi H i i of = (g g v.OT i E V iHi i . where N let with i(vi®g (Ti)) i 1 (g ,***,g transformation [as follows: = ,gN)(@ v.®Ti) (g ,' must preserve the sesquilinear )k dim H i g 1 (T) and gkj T k k=l d intertwine g Clearly, preserves the form on V . invariant form on g E ZG(sT( 2 )) Hi , therefore g preserves the These two observations show that O We ve seen that the question of admissibility comes down to the subgroup LflK . We now prove the following Lemma. a.) LAK E (Sp(2ml)NO(2ml))x(O(P2 ,q 2 )nO(p 2 +q 2 )x * * * (V symplectic) b.) LNK E (O(Plql)NO(Pl+ql))x( )(Sp(2m2)O(2m2))x (V real hermitian) c.) LflK (U(Pl'ql)AU(pl+q1 )) x(U(p 2 , (V complex Hermitian) 2 )U(p 2 +q 2 ** )) x *.. *** ) -43Proof: GnK is the linear group which preserves the sesquilinear form i w = i w$ @h on V as well as the i inner product Q = b b . The proofs of all three statements are the same, therefore we only do a.). must show that (gl'--.,gN) E (Sp(2ml)xO(2ml))x*** determines an element of product Q . So we L which preserves the inner But this is clear because {T } ljidim Hi ii Hi for is an orthonormal basis of b. 2 On the other hand an element element (gl,' e E LnK ',gN) E Sp(2ml)xO(plql)x... preserve the form i b2 on H i for each i . determines an which must Hence (gl,°,gN)E(Sp(2m)nO(2ml))x(O(P 2 q 2 )nO(p2 +q 2 ))x ** . -44Chapter 5 We now consider the case where or Hermitian (with signature (p,q)) will show is a matrix which preserves the standard symplectic C Mn () algebra go form on Fn We . is isomorphic in these two cases to the go space of symmetric or antisymmetric homogeneous polynomials in of degree two. V We begin with the following: Definition. Let be a vector space over V define V to be equal to we let C act on c E C , set HomF(V,V) , where c-v for is v C V V . V@V * V IF V@V IF as G modules and go is a vector space over F V nondegenerate G-invariant sesquilinear form from Proof: V and the modules where inherits the Then we F = C as sets, but when as follows: c.v = c-v multiplication in Lemma. V V F . with a w , and G-module structure and invariant form V w V . First, observe that the G-invariant form w V - V gives a G-equivariant. F-linear, mapping on -45, v v,w E V for defined as follows; --- v where def. v (w) = .(v,w) b V@V V0V --- Now define a map v,w E V . . O(v@w) = v@w by setting, for The adjoint actions on V. give rise to the contragredient actions on V If we take the representation on the tensor products arising V from these actions on that * is a (or go) equivariant isomorphism. G Finally, define a map follows for [T(v@f)](w) it is easy to check Vn and VOV T . It Hom(VV) as let v,w E V , f E V = f(w)-v --- is easy to check that this map defines an isomorphism which intertwines the adjoint action of the group G as well as the adjoint action of End V the whole algebra o . We now want to consider the symmetric or antisymmetric elements in V@V , by which we mean the real span of elements of the form , <v®w-w®v> because V respectively, in and V VOV . <v~w+w@v> or This makes sense are identical as sets. We note that such elements are not closed under scalar multiplication by elements of the following notation. field F = C . We introduce the -46- Definition. 2 A2 (V) Let and 2 S2(V) be the set of antisymmetric and symmetric elements in Proposition 1. Let preserving go C Hom(V,V) V®V . be a Lie algebra Under the identification Hom(V,V) 2 V@V described above we have: a.) if w is skew-Hermitian, then go corresponds to S2 (V) C VV. b.) if is Hermitian, then w go corresponds to Ta to be the unique A2(V) C V®V . Proof: T E VOV Let and we define element of VO. Vx,y E V . We want to find such that T = Write I Ow i w(Tx,y) = -w(x,Tay) (T E V@VIT = Ta} and we compute Ta i y,) = w(I W(wix)viy) i W(i v.iwi(x) = w(wix)w*(viyx) 1 W(=(wisvi(Y),X) i = e6.(x. w. v. (y)) i where e -= Ta = (-e) I i ( 1 if -1 if w.Ov. 1 1 w a is Hermitian is skew-Hermitian . Thus and the Proposition follows. D -47- We need the following additional notation: Definition. Let S(V,W) be the symmetric elements in Let A(V,W) be the antisymmetric elements V®W $ WOV . in VOW $ WOV We note that isomorphic to inside V®W on V submodules. Suppose (r,V) V = V 1V and are both is a representation of the direct sum of 2 sT(2) Then S2(V) = S2(V b) A2(V) = A 2(V) Obvious. Corollary. A(V,W) that is of interest to us. a) Proof: and , but it is the natural way each sits (V$W) 0 (V$W) Proposiition 2. es(2) S(V,W) Let 1) @ S(V 1 ,V 2 ) $ S2(V 2 ) $ A(V 1 ,V2 ) C A2(V 2 ) 0 V = @ Vi®H i , Vi the standard i irreducible 9s(2)-module over I with canonical st(2)-invariant form o i , H i a vector space with the standard Hermitian or skew-Hermitian form with respect to a hi defined -48basis {T} Sl(idim S2 a) H1 Then . dim H' =2(a (Vi@Hi) i = i (v'@F*T 1 )) qj j=l @ S 2 (Vi@F*Tk i,k , VJ OF*Te) $ S(V'iF-T k i>j k,e i i * (S(V @F.Tk i1 S2 A 2(9ViHi) = *** replace b) V i i FT)) by A2 and A by S , Now we use the Clebsch-Gordon formula to decompose Vi S2 , S 2 (V i ) , and 2 A (V i) . Proposition 3. where a) Vi@V j = M(i+j-1)@M(i+j-3)@**-OM(i-j+1) b) S2(V c) A2(V ) = N(2i-3)$N(2i-7)$--**N(p) where q = 1 q = 3 i ) = P(21-1)$P(2i-5).***@P(q) and and p = 3 p = 1 if if i i odd even and M(r) is an irreducible sf(2)-module C Vi Vj with dimension r , P(r) an irreducible 9X(2)-module C S2(V and N(r) i) of dimension r an irreducible ol(2)-module C A2(Vi) of dimension r. -49- We let {T1} be the basis for Hi described in Chapter 4 and we introduce the following additional notation: 2 A (V OWT ) 2 ii ii ii P= (2i-1)Pkk(2i-5)@,**@Pkk(q ) S 2 (V i®RT) k A(ViI-Tk kk kk , VJ OR-T (for kk )=Nkl(i+j-l)eNkf(i+j-3)E--..Nk(i-j+l) , k.T V OR*T) S(Vi ii ii ii = Nkk(2i-3)@Nkk(2i-7)$...@Nkk(P =Pkl(i+j-l)@Pkf(i+j-3)@**-@Pk(i-j+l) ifj or kXe) k (Vi O.Tk)@(VJOR-T) R First, assume ke (1+j-1)@*..@Mk(i-j+1) kes··M, = Mk i V = OVi H is a vector space over Then, using the Corollary to Proposition 2 above, we IR . get: S2(v) = S (V) = ii ii D [Pkk(2i-1)@-***Pkk(q i,k a [Pk (i+j-1)@**-@P )] (i-j+l)] a i>j k,.e ( k>e and [P"k(2i-1)@D* .@Pk ( (1)] -50A 2 (V) = [Nkk(2i-3)@*..*DN(p)] $ i,k ( [N i>j (i+j-1)..***N (i-j+l)] a k,e C [Nke(2i-1)C*** Nk (1)] 1 with q and p N V = $ V @H i=l Now let i.e. Hi u(p,q) of V as in Proposition 3. is complex. be a complex vector space, as an SL(2)-module. We define a real form as the real span of a subset of Definition. V= VR A2 (V) We want to decompose <Xi-1-s s V V : Ti k 1<i<N i l<k<dim H Proposition 4. Proof: and vOw - A2(V) = A2(VI) v®w-w@v E A 2 (V) Let w = w 1 +iw 2 wOv = with (v 1 +iv = V W + 1 2 i[v2 w . We can write v 1 ,V2 ,W 1 )1(wl+iw - + iA 2(V ) 2) - - V 1 OW 2 - E A2(V ) + i(S2(VR)) . Then (wl+iw 2 )1(vl+iv Wl@V1 + v 2Ow 2 1 E V 2 v = v 1 +iv - w 2 ®v . w 2v 1 o 2 2 + W1 V 2 ] ) 2 -51Thus for complex, we have V A 2 (V) = [Nkk(2i-3)@...Nkk(p)] i,k i>j k,e [Nk(2i-1)@***@Nk (1)] 0 a 1 k>i e i'[Pkk(2i-1)@** @PkkP (q)] i,k 0 i*[P ( (i +j-l)@**.- @P (i-j+l) i>j k,e 0 i-[Pke(2i-1)@..-@Pkf(1)] k>R with p and q as in Proposition 3. Proposition 5. sI(2) - (r,V,w) V = @W i and Proof: where w(Wi,Wj) = 0 Let [V]k be w . w(I[V]k,[V]_) v [W = 0 j_1 W is isomorphic to for i w copies of . the eigenspace with eigenvalue unless and w E [W] i > j Assume k = e . c-w(F.E.v,w) = -c.-(E.v,F.w) = 0 because m. By the st(2)-invariance, under the H-action. let be a representation of be with an st(2)-invariant sesquilinear form Then V Let 1 . Then for some is a lowest weight vector in w(v,w) = c E IR W and k -52- Proposition 6. B : The Killing form S2(V) x S2(V) -- R or B considered as a map A2 (V) x A2(V) - B : IR is given in terms of the canonical G-invariant sesquilinear form w on V as follows: for v@w,v'@w' E VOV B(v@w,v'@w') Proof: = co(v,v')*-(w,w') (c E IR) Up to a mul tiple, there is a unique nondegenerate G-invariant sesquil inear form on a simple Lie algebra. We recall that we want to fineI the determinant of the linear map induced by the adjojint representation of (LnK)O on the quotient space results, for E go / E By our earlier e E (LnK)o , it suffiices to restrict our attention to the (-1)-eigenspace ol det,(Adjg /go a ) = detC(Ade I[o_ 1) We now describe an explicit basis for identification of above. go with S 2(V) or [go]A2(V) using the explained a -53The only irreducible submodules in which contribute to the (-1) and S(Vi@Hi i 4 j 1.) If V is real, = (Vi@Hi)@(VJ@HJ) (t) those , VJOHj) or A(Vi®Hi,VJ HJ) k,P l<s<j MiJ M1 namely with i > j (mod 2) Definitions. Mij A2(V) eigenspace (under the H-act ion) are those of even dimension; compo sing or S2(V) = $ then we set: ij Mk ( i +j+l-2s) Mkf(t) k,e Mij = kR Mke(i+j+1-2s) 1l<s<j 2.) If V is complex, then we set: Mij = (Vi@Hi)@(Vjo HJ) [M -= (i+j+l-2s)@i-*M (i+j+1-2s)] k,e Mi(t) = k(t)][Mke(t k,e MiJ ke l<slj ) @ i-Mk ij( i + lj + [M ( i+j+l-2s)Di.M'kJ i~j~l-s)$ kR For the remaining calculations in will make use of : VOW -- S(V, W) A(V, ] this Chapter we the isomorphisms {a : VOW -- -2 s ) W) -54defined by: a(v@w) = v~w + w®v P(vOw) = vOw - w®v We get: j OH j ) (V'@H')@ (V VJ@H i -(V-i@H [A(Vi®H i Vj@H j ) j ) and (r) P M M(r Nke(r We will often suppr ess the distinction be tween ) (Vi@Hi)@(VOHj j S(Vi@Hi, VH A(Vi®H i , VJOHj ) anLd or ) Claim 1. WE(M i a.) ,Mi ) = 0 unless i=i' and j=j' b.) WE(Mij(t),Mij(t')) = 0 unless Proof: We have V = C Vi®H i t=t' with invariant sesquilinear i form w unless . Proposition 5 implies i = j B([E,MiJ],M WE(Mij,M i 'j i'j . We observe ) . ) = 0 Since that that [E,M ij] if and only if w(Vi®Hi,VJ®HJ) WE(M i,M C M ij i 'j ) we conclude B(MiJ,Mij) = 0 But, B(MiJ,M j) = B((ViOHi) (V®HJ),(Vi =cc(V'iH i , V'i Hi ®HJ)) Hi')®(V ).w(V O®H.V Hj ) 0 -55and this last expression is j = j' Fix i V Vi@H V = unless i = i' and O by Proposition 6. Claim 2. 0 and i > j, and let i lk,k'<m i If , and le,e'<m. C S 2 (V) Mk , then J for i even, j odd, m. = dim Hi and 2m. = dim H a) 1 kM ej B(Mk'~ X) 0 <=> k'=k e'=e and for i odd, j even, 2m. = dim H i and m. = dim H J b) 1 B(Me If j ,PM M e) C A 2 (V) and M V J 0 <=> k'=k real, e'=R and then for i odd, j even, m. = dim Hi and 2m. = dim H J c) i 1 k'' M C A 2 (V) ke dim H e'=e and for i even, j odd, 2m. = dim Hi and m. = dim H J d) If 0 <=> k'=k Mk,e+m.) B(Mk' J = m. e) Proof: B(M k+m.,e 1 V complex, dim H , then k e,M k) X 0 <=> k' 0 looks on i m. 1 = k and and e' = e O Use Proposition 6. In order to compute know how j J S2 S (V) on and and [•o]_1 A2 (V) (V) we will need to -56Proposition 7. X E go S : The Cartan involution 2 is given on S (V) A2 and X --4 -Vt for by: A (V) for (Xsyi-1-sOTk)O(X i (Xj-1-ty t t T ) E Mij 0((Xsy i -l - s tT Ti )(Xj-1-t = V j-1-t (-1) s+t+l hi s 0k i -l - s V. . where ) @ (X V hJ(TJ,T m P ) = 5e Pm hi (T1 ,T ) =km are the appropriate forms on H and and H and respectively. Proof: The Cartan involution conjugate transpose on . su(p,q) We use the A 2 (V) . and X E S2(V) basis on (or V hais bais{( go = sp(2n,R) V = $ViH H i (under Q) X. and set: or Q = @ bi b 2 1 i to compute then X = 0 on transformation t , hence OX = -X* We take the usual orthonormal k' ,, ,,. O<s<i-1 Q so(p,q) with respect to an orthonormal (i-1 ) 1/2.i-l-s Y~ sT OT i } X s L , If we write a linear A2(V)) So we now compute is given by negative inner product (defined earlier) on S2(V) 0 er WII i , mn, ULIU~I -57s and L = v4 v3 i is, . j i X 1,2 and = <v 3 v Lv. = 0 . , Q(v if i ,L v) j A 3,4 = 0 = 0 ,L v 2 ) and Q(Lv 4 ,v 1 ) if Therefore, . for i x 3,4 . if and only if Q(Lv 3,v 2 ) = ,L v 1 ) . We have 4 and Q(v 4 ,L*vl) = Q(Lv 4 ,vl) = Q(v Also, j X 2 = Q(v 4 ,L v ) . Finally, Q(Lv 3 ,v 2 ) = Q(v 3 ,L*v 2 ) two cases: go preserves a symplectic form: Q(Lv3', 2 ) = (-l)S(-l)i+lhi(Tk' T 3 = 0 if and only if we break the verification that Q(v 1 Thus it remains to show that (-)tQ(v4v4) , hence I.) for all i X 3,4 , and since the image of Q(Lv 4 ,v 1 ) = (-l)tQ(vl,V1) into L v. Q(Lv 4 vk) = Q(v 4 ,L Vk) = 0 Likewise, 3 for j) = Q(v 3 ,L v) Q(Lv 3 Q(v Q(Lvi,vj) = Q(vi.,Lvj) To begin, observe that Q(Lvi,v.) = 0 k A 1 we want 1- and L = (-1 t that , k j--tdemonstrate that We 2 Ti 1/2 xi-1-s i-1 1 ,L v 2 ) = (-)-1j1 )Q(v2 v2 ) and (TT)Q(v3v 3 ) -58V. Since hi(Tk conclude Q(Lv 3 ,v Q(v 3 2 ) = -(-1)s(- ,L v 2 ) k 1 ,Tk)Q(v2 ,v 2 ) ) i+lhi(T . V . 1. ,Tk) = (-1) 3 and ,L v 2 ) We know that 0 h(T k k Wk A (V) tells us, Mij(k) C W k . J = cOoadE : J : Our . L thus o and we're done. preserves the isotypic parts of g 9s(2) C qo ; i.e. 0 formula for in addition, that [-1 [go] -1 -- [Mij(k)]_l on 0 S 2 (V) and preserves , and thus we get (of course k must be [MiJ(k)]_l = {0}). Our description of 2 A2 (V) = (1)+land ,T) We have already observed that [Mij(k)]_ even or Now . under the action of a G-stable : and V . = -(-l)t+s(-l)t(-l)J+lh(TJ,T 3 ) . Q(Lv3',v2 ) = Q( 8 , we preserves an Hermitian form: . hi(T h (Tk'Tk) = (-) and Q(Lv 3 ,v 2 ) = Q(v3,L v2 ) go II.) = (-1) ,T) shows that commutes with L the action of 91(2) , L the (-l)-eigenspace i Mi. preserves preserves [M i(k)]_l . L on and Moreover, since M i(k) and also Putting all of this together we get the following conclusion: e E (LnK)O S2(V) for -59- detC(Ade) detC(Ad• [MiJ] i>j i j(2) detC (Ad- I i>j [Mij(t)] _-1 itj(2) t We want an explicit b asis for [Mij(t)]_l We begin by finding a basis f or the irreducible submodule MkU(i+j+l-2s) C M i j We define a basis on Mk(i+j+l-2s) isomorphic to the standard basis for an which is st(2) representation on homogene ous polynomials of degree i+j-2s (= Si+j-2s(X,Y)) operator, : i+j-2s(X, Y) -- Xk (i+j-2s) = O(Xi+ basis on Let . j- 2 s ) * be an intertwining MiJ(i+j+l-2s) . We and this fixes a canonical Mk(i+j+l-2s) kP6 (i+j-2s-2m) = ,(Xi+j- X Definition. 2 s-mm) 0 < m < i+j-2s . Lemma 1. The highest weight vector Mk(i+j+1-2s) S p=l1 ()p-l s-1 Xk(i+j-2s) is a multiple of i-p p-I1 p- l)(X 'P k )O(Xj-l-s+PyS -POT -J9 in set -60Xk We know Proof: must be a linear (i+j-2s) combination of eigenvectors under the H-action with eigenvalue i+j-2s ; that is, a sum of the following vectors in M (i+j+1-2s) {(Xi-pYp-@T' ) (X j - 1- s + p y s-p T )l<p<s s Xl k (i+j-2s) = Set where 2 p= c (Xi-p-1Ti)(Xj-1s+PYs-P®T) P c are some constants. Use the fact that p (i+j-2s)) = 0 to solve for c in terms of P adE(X cl 0 = adE(XUk(i+j-2s)) s-1 Ss- [(s-p)c- i j-l-s+p+l s-p@Te)@(x Y lTe) [(s-p)c pi(X p= 1 s-p-1 Oj Yp-10T 1 )i(Xj-l-s+p+l + pcp+(X Therefore (s-p)*c implies that c p + p-Cp+ (-1)P-1 p=l l)(XjPOT p=l s- 1 = 0 = (- 1 )-1(s-) p-1 We normalize so that s 1 i-pp-1 ~ )(X'-T X , which 1 < p < s-1 as desired. a (i+j-2s) = (j--s+p s- j -61X Lemma 2. (-i-j+2s) canonical basis on s (-i) ( P-1 s. I p=l Proof: Xiyj to 0 1 .(Xi-p 0 Mk the lowest weight vector in our (i+j-1-2s) p- p)[(X S-1 p-lk 1Y i- ).(X Y p (-1)i.jYi. -1T T equals i) @ T J) -e in . SL(2) It So (X-l-s+ )yS T (X s - p Y j - l - s + p (10 1 ) Consider the element sends -I , TJ) goes to goes to (-1)i+j-1-s*(Xp-lyi-POT )@(XS-p j-1-s+P@T ) , and (-1)i+j-1-s = (-1) s because i t j We will also need a basis for submodule Set X . (mod 2) 0 the irreducible iM k(i+j+l-2s) C i-S(Vi®IRTk , V)*IR.T ) (i+j-2s) = S(-1)P-lP-)[(Xi-p pp=l i.T k)@(Xj- -s+ps-p@T )] The same reasoning used in Lemmas 1 and 2 shows that XJ(i+j-2s) is a highest weight vector in kS i'M k (i+j+l-2s) s (-1)s2 (-1)Pl and that - s X(-i-j+2s) = l -1)[(XPlyi-P@iT )(i XS-pyj-l-s+p Oj)] p=l is a lowest weight, where we define a canonical basis s {Xke i+j-2s-2m)}0m(i+j2 Lemma 3. Fix V = $ VP®H as in the previous case. , i > j , $i@h p form on V , and let lk(<m 1 and l<e<m. j i canonical -62- If Mij C S2 (V) a) , then: for i even, j odd, m. = dim Hi ,I 1 O(Xk(i+j-2s)) = -h (T 9mi = dlim T 3 = • il |1 e+m (-1-j+2s)) J = dim Hi , m. = dim H J kTk)(X k for O(X If i odd, j even, 2m. k M i j C A 2 (V) c) is real, then: c~ , ((i+j-2s)) = -hi (TkTk)(X , k k k ((i+j-2s)) = dim H J is ~ complex, zm = aim n- +m.(-i-j+2s)) k kR~m hJ(TJ,T J)(Xlm v M i j C A 2 (V) , V (e) V for i even, j odd, 2m.1 = dim Hi O(X If and SV for i odd, j even, m. = dim Hi 1 O(X d) hJ (TJ,TJ)(Xlj (i+j-2s)) Lv i , mj = dim Hi •r 11 = dim Hi and then: e(X i(i+j-2s)) = -i.hJ O(X (T ,T )h'(Tk,Tk) (X k'k)(ke(-i-j+2s)) ee (i+j-2s)) = i-hj (T ,T )h(T',Tk)(Xke(-i-j+2s)) Proof: By Lemma 2 we have Xk(-i-j+2s) = (-)s I (-1)p-1( s - ) [ (X p - Y i - p OT p=l Compare this expression with: ) (X s - p J- 1- s + p T ) -63O(X k (-1) (i+j-2s)) = j+s+1 ) [ (XS-PYj-+P@TJ)®(Xp - 1y i- p® - (-1) i) p=1 Lemma 4. a) b) c) (Same setup as Lemma 3) J(Xk (-1)) J(XkU(-1)) J(Xi(-1)) (-1)) i+j+1-2s 2 =(-1) i+j-1-2s 2 =(-1) hi(Tk,Tk)(Xk' h (T ,T )(Xjm i+j+1-2s 2 T k'Tk) iT ij +m (Xk =(-1) i+j-1-2s 2 d) J(Xk e) J(X k(-)) = =(-1) +m.J (-1)) th I- (-1)) to,10 k ' (TJ,TJ)(Xlj i+j+1-2s (-1) 2 2 i-h (TR,TR)hi (T ki ,T ki ) (X i (-1)) J(X ke(-1)) = i+j-1-2s 2 S (-1) i-hJ(TJ ,T)h i kT .Ljj i ijk ,Tk',T)(Xke(-1)) -64Proof: ij (for Case a) only) J(Xk(-l)) = O(X ij -1/2 -1/2.eoadE(X k = (-1) i (1)) (-l)) , because [E,Xk(-l)] = jl/2Xk (1) i+j-1-2s = 6(co(adF) 2 where cO (Xke(i+j-2s)) i+j+l-2s 2 = (i+j-2s)! i+j-l-2s i+j-l-2s (-1) 2 2 c(adE) O(Xk(i+j-2s)) Apply Lemma 3-a.): i+j-l-2s i+j-l-2s -h (Tk, i+j+l-2s 2 (X In each of +.(-i-j+2s)) i 0 k,) Xk, e+m. (-1) Lemma 5. 2 co(adE) 2 = (-1) in Lemma 3, the cases is a complex orthogonal basis for {XkeC-1)}k,e [Mij(i+j+1- 2 s)]_l and all the vectors in this basis have the same length with respect to the inner product Proof: b on g /g 0 0 The orthogonality follows from Claim 2. The basis elements all have the same length because we constructed them the same way on each submodule M kP(i+j+l-2s) . O -65We recall the following identification described in a previous section: If then LK If then (Sp(2ml)nO(2m 1 ))x((O(p 2 )xO(q L C O(p,q) LNK If then L C Sp(V) 2 ))x... , (O(pl)xO(ql))x(Sp((2m 2 )nO(2m 2 ))x..L C U(p,q) LNK , (U(pl)xU(ql))x(U(p 2)xU(q The product of 2 ))x--. two elements on the left side of the isomorphisms above is the usual product on a Cartesian product of groups. e E LAK Therefore, if corresponds to (gl,---,gN) E (Sp(2ml)nO(2m 1 ))x(O(p 2 )x(O(q 2 ))x we conclude det,(AdP) = 7 detC(Adg ) j and we can comp'ute the determinant of any element in if we know Adg det C(Adgj) for all invariant we must compute j Mij Since detC(Adg Mij M i Z j mod (2) . ) LnK is for all -66Proposition 8. I. Assume i > j Symplectic Group a) BA (where be given by the matrix B are dimensional space HJ . M 1 det(A+iB)even power if m. even 1 g = [ BA E Sp(2m i ) n with the same conventions as above, then detC(Adg -, c) if m. odd ) = b) For i odd, j even, O(2m.) 2m. Then [det(A+iB)Odd power i and A m.xm. matrices) with respect to the standard symplectic basis on the detC(Adgj n O(2mj) gj E Sp(2mj) For i even, j odd, let ij) For i even, j odd, let = 1 gi C [O(pi )xO(q i)] O then det(Ad-gi d) ij) For i odd, j even, let then = 1 gj . E [O(pj)xO(qi)] 0 ' -67- detC(Adgi I II. = Group preserving a symmetric 1 form on a real vector space. a) For i even, j odd, gi i- A C Sp(2m i ) dim H J = m. 3 , Mij) M la b) = det(A+iB) odd power i ). dim Hi = 2m. " For i odd, j even, de t(Adgi For i even, j odd, if m. is odd even power if m. is even For i odd, j even, g detC(Adg•j d) O(2m then det(A+iB) det,(Adg. n = [_B Mij) E Sp(2m )nO(2m ) = 1 gi E [O(Pi)xO(qi)] 0 ij) g detC(Ad-gj I ij = 1 E [O(p )xO(q j)]0 = 1. ' -68III. Group preserving an Hermitian form on a complex vector space. a) For i even, j odd gi = A E U(pi)(or U(q.)) , dim H i P+q. dim H J = m. I det [(detA)Odd power (detA)even power i) (Adg i b) dim H I mij ) (or U(qj)), dim H = p. = m. 1 (detA)odd power odd S(detA)even even For i odd, j even, detc(Ad-g d) is even For i even, j odd gj = A E U(pj) det,(Adg j is odd For i odd, j even, detC(Ad-gi power gj E [U(p )xU(q )10 Mi) = 1 . gi E [U(pi)xU(qi)] Mij) = 1 0 -69We will only prove the statements for Proof: the Symplectic group because the proofs for the other groups are very similar. !g . hence gj L preserves isotypic submodules of ij M (i+j+l-2s) preserves detC(Adgj [Mi(i+j+1-2s)]_) detC(Adg Mij(i+j+l-2s) ) . which equals " m. Case (a): We have g (T ) = 2 m. a m=1 1 < e < m. T j me m + m=I (-bm )Tm+mM. m=1 i Therefore m. A We compute a 2 ke(-1)) m. Xim(-1) + -1 (-b)Xi m=1 me km m=1 m. 3 ii m. 3 mD)Xk,m+m.3 (-1) ) k.J(Xj(-1 2 a Xij(-1) + 7,(-bme) m8Fk~s km(-')) m=1 where m=1 i+j+l-2s 2 ak,s = (-1) m. = m=1 i i hi(TkTk ) (- (ame-ik sbme)Xj 1) Thus j det C Adg j2m =Mk(i+j+1-2s)]-1 R=1 and = I det(A-ia s=1 B) ) -70m. k=1 j s=1 det(A-iek, k=l s=l 1 det Since j Adgjj I C C [Mij]_l is odd and ek ,s B) 5 s we get: k,s+l m. detC(Ad-gj Case (b) : T det(A-iek k=1 Mij) Same reasoning as part "I B) (a) leads to the equation, m. det (Adg. I ) = case J I TI det(A-ie OsB) e=1 s=l [Mij]_ In this J is even (es = -e,s+) ,therefore m IT e=1 det(Adgi I . Case (c): Set U= Lemma 2-c. ) we get Adg i (U) C U and Sg = 1 m. m. 0 $ (D [MUk(i+j+1-2s)]_ s=1 k=l e=1 [Mij ]-1 Ado (JU TT ''J-J = U@J(U) C JU Since we conclude By 1 -71- detC(Adgi [Mij In fact det[(Adgi U) at the image in R ) = detR(Adg-i U ) E R equals because we are looking of a connected, compact set containing the identity and Case (d): 1 det(Ad(e)) = 1 Use the same reasoning as in part (c). O Theorem. (I) Let G be the symplectic group with same notation as above, then if j is even 1 odd power det((Adg ) = if j is odd and there are an odd number of even dimensional irreducibles of dimension > j in ( Vi H i det(A+iB)even power otherwise -72(II) Let be a group preserving an Hermitian form on G a real vector space, then 1 if j is odd let(A+iB)odd power det (Adgj) = if j is even and there are an odd number of odd dimensional irreducibles of dimension < j let(A+iB)even power III.) Let G in @ ViOH i i otherwise be a group preserving an Hermitian form on a complex vector space, then odd power power (det A) if j is odd and there are an odd number of even dimensional irreducibles of dimension >j in @ Vi H i i detC(Adgj) = d d power (det A)o if j is even and there are an odd number of odd dimensional irreducibles of dimension < j in @ Vi@H i i ev e (det A) n power otherwise -73- Corollary. G . Let E E go Fix 0 and g = (gl'--gN) E LnK N U det((Adg) = the orbit of E under . Then n. (detCgi) and 1 is admissible <=> n. E i=l is even for all Proof: i The proof follows directly from the Theorem above and the last Theorem in Chapter 1. Theorem. of E E Fix a nilpotent under under SU(p,q) U(p,q) with H J 0 {0} D E E su(p,q) . Then the orbit is admissible <=> 1.)the orbit of is admissible, or 2.) If for all odd j the number of even dimensional irreducibles of dim ) j in 9 Vi H i is odd and for all even j with H i $ {0} the number of odd dimensional irreducibles of dimension < j in * Vi®H i is even. i Proof: of this element under under EE su(p,q) . Fix a nilpotent element SU(p,q) U(p,q) The orbit is the same as the orbit , and, as we've seen, this orbit N corresponds to a decomposition of C ' * i ViH i=1 under the action of N H i=1 (U(Pi)xU(qi))} 9s(2) . Let the set be the subgroup {(gl-'-.gN) N LnK C U(p,q) -74determined by E (see Chapter 4). LnK C SU(p,q) determined by E Then the subgroup is N {(gl' ' . ' gN) . i=l I (U(P.)xU(q.)) N (det(gi))i = 1} . i=l the preceding Corol lary we know that ther e is By sequence of integers (n 1 ,'- S,nN) which determine the determinant character: n. detC(Adg) = U i=l (det gi) where g = (gl, '°°gN g = (gl,'.-'gN) E LnK C SU( p,q) For , 9 (det gi = 1 i therefore on this subgroup the c haracter has the following form: n.+ki detC(Adg) = I (de t gi) i where k is an arbitrary integer Thus E C su(p,q) is admissible if and only if this character is a square, which is true if and only if is even for all j or even for all j . Theorem. of E under Proof: Fix under O(p,q) The lemma because and n2j+1 n2j is either odd for all j o a nilpotent SO(p,q) . E E so(p,q) . Then the orbit is admissible <=> the orbit of E is admissible. SO(p,q) case follows directly from the next SO(p,q) 0 = O(p,q) 0o . -75Lemma. The orbit of a nilpotent for if and only if it is admissible for G Proof of Lemma: z i (GE mp E is admissible for -0 GE Lie(GE) and only if G O0 if and only if (GE = Lie(G) 0 , it p= (GE) 0 z f (GE)p . mp )E E G above that G is admissible We have the following coverings of groups: . (GE )mp D ( Since E E go . follows from the inclusions Therefore z t (GE)mp 00 if -76Chapter 6 SL(n,R) In Chapter III we analyzed nilpotent orbits under a group which preserves a nondegenerate symplectic or orthogonal bilinear form on a vector space V . We now consider nilpotent orbits under a group which preserves a nondegenerate, multilinear, alternating top-dimensional form on a vector space V0 ; that is, we are considered nilpotent orbits in sl(n,R) is the dimension of V0 under SL(n,IR) where n ' As before, Kostant's 1959 results give us a bijection between nilpotent orbits in se(n,IR) under SL(n,[R) and equivalence classes of mappings from SL(2,J) into SL(n,IR) (where two maps are equivalent if they are conjugate by an element of SL(n,IR)). We have a notion of equivalence among vector spaces with non-zero top dimensional multi-linear alternating forms. (,wV) Let the pair be a vector space with such a form. Definition. ((1 .V1 ) isomorphism T : W2 (TV1' n) " V1 where (w2 .V2 ) V2 if there exists a linear such that {v 1 '.0*Tv n } 1 (vl.' is a basis .. V n) = for V1 -77The isomorphism T is unique up to right or left multiplication by an element of SL(n,R) , and the map induces an isomorphism 2) G(wi) T : G(w G(wl) is the group preserving the form (We note that be a representative of to SL(2,IR) n . -- where w. , i = 1,2 1 G(wl) = G(w 2 ) = SL(n,R)). Fix an equivalence class equal --- T (wO,VO) (wo,VO) and let (o0 ,VO) with dimension of V0 We consider the set of all maps G(wi) where (wi,Vi) E (wo,VO) . Define an equivalence relation on this set in the same way we did for the orthogonal and symplectic groups III). (see Chapter The same reasoning used in the earlier cases shows that the classification of nilpotent orbits in under SL(n,[R) st(n,R) reduces to understanding the partition of the set of triples {(7,w,V) 7r is a representation of SL(2,R) on V which preserves a (dimension of V)-nondegenerate, multilinear alternating form, and (G,V) E (wo,VO)} into equivalence classes (where the notion of equivalence is analogous to the equivalence in Chapter III). Let (r,V) We recall that notation). be a representation of V @ Vi®H i i et(2,R) (see Chapter III for on V -78- Proposition. a.) If Hi 4 {0} for some odd i, then there is one (up to equivalence) non-zero SL(2,R)-invariant multilinear alternating form on b.) Hi = {0} If ( Vi@Hi i for all odd i, then there are two equivalence classes of s uch forms. V = $ Vi@Hi i Let Proof: intertwining operator T : V --- ) C iso typic i T(W ) C Wi and [Wip HJ s First, assume {v =X j-1 OT Any preserves V Therefore components and eigenspaces. T([W i] i W i = Vi@H and 1 . (0} w(Tv 1 ' .,TVn) V . take - 1OT v. * v.=Yj j 1' j+l v2 =X j-2 YTTJ 1' to be a basis for We If is an n-form on L = (det T)w(v 1 '9 ,v } V then Also by the nv) . observations above det T = i det(TI W i IT = i Let WO(v 1 , ' .Vn) ) _i-1 ) be two n-forms on O,0 1 = 1 and arbitrary constant (v,V,O0)~(-,V,Vl) . c . o(v 1 ''..,v) det(Tj i-i1 i V [Wi defined by putting = c , for an We want to show that To do this we define a linear -79V --4 V : T isomorphism as follows: 1 T(ve) = elc njIv for 1 • e • j T(vk) = Vk for j+1 It is clear that calculate: ~0 (V 1 , Thus we've shown . .,V) for all odd i. wO(v 1 .' 0,v 1 and = c > O n) 0(v'I (v,V, 10 )~(vV,Wl) , o 0 (v 1 So we assume , n) = 1 e. c < 0 , and we derive a contradiction. n= W l(V1---,Vn Assume that ) = An argument (W,V,•0 )~(V,V,Wl) similar to the one above shows that where . = (det T)l 1 (V1'900v n) Hi = {0} Now assume < n is an intertwining operator, and we T wl(Tv 1l,**,Tv) ec-11c = 1 = < k a = sign of c and T : V 1 ---- V 2 intertwines the s9(2)-representations and preserves the forms. T is an intertwining operator, depend on the eigenvalue det(T 2e) W p . det(T ) Because does not Thus, = I det(T 2e ) p [W ] )2e = (det TI > 0 , for all [W2e] Since (r,V.w 0 )~(r,V,Gl) we must have 1 = W00(v1.•-v ) = W 1 (Tv1,---Tv) (det T)l( 1(vl* .,vn) = = (det T)-c = [U (det TI R e)]*c 2 W -80But this is impossible because and c is negative. T(det TI e is positive 2 We conclude there are two equivalence classes of n-forms on all odd i. WC) V if H i = {0} for o N. Fix (v,V= @ VigH ,w) (notation as above) and, as i=l we've shown, this triple corresponds to a nilpotent orbit 0E in sT(V) We construct a nondegenerate bilinear form T = q i ®i@h i on V 9s(2)-invariant , where W i is our i canonical 9s(2)-invariant form on inner product on {T i} ,T Hi Vi and defined as follows: be a basis for Hi hi is an let and set dim H 6 hi(Tj ,Tk ) = The form V -- + . isomorphism jk T gives us an SL(2)-equivariant mapping This map gives us an SL(2)-equivariant V@V -- Now we decompose SL(2,IR) have: . V-@V gq(V) • V®V under the action of Using the notation introduced in Chapter 4, we -81- (V'O'@Vj~ i,j ij (V0 ®VM~ i,j = 8 i,j We want to compute order Proposition. g((V) We write denotes the center of sl(V) eigenspace under [we (v),go(v)] 0 , we conclude sits inside the for p X 0 . S[g(V),(V),(V)] qg(V) for . j Clearly both summands are as fo Slows: 13]p = (p(v) (V) [0(v)] (v) and = 0 = [q for So we may now restrict our attention to [(V)]_l , where Theref ore we may decompose the p th adH 19(V)]p In given above. qg(V) = g ad(gX(V))-invariant. (V) ]_1 ) . detC((LnK)O)[1 [g~(V)1p = [sI(V)]p Proof: p the to do this we mus t see how decomposition of Since M (i+j+l-2s) e,k s=1 In order to compute when p 0. [qI(V)]1 detC(Adh l[s(V)]_l h E LnK , we construct a basis for ([sI(V)]_I)c 0 -82To do this we want to know how for [ The calculation of 6 our canonical basis o -1/2 J = J-I *oadE -1 looks on ]-1 made in Chapter 4 goes through here, and we have: e((Xsy i-l-sOT i ®Wk)@(XJ -1-tY t TJ twJ) = (-1) s+t+l(Xtyj-1-t@T )@(Xi-l-sS sTk i) We will need the following: Lemma. O(XUk(i+j-2s) = (-1) Proof: O(Xlk(i+j-2s) Xk(-i-j+2s) ij S =( 2 - 1 (-1)p- p=l (1) i + s + l = (- )(X p-l p=1 (-1)s-l1 (-1) (-1) i+s+l i-Pyp-1T i)@(Xj-I-s+Pys-P®Ti)) s-i (-1)p-1 (Xp-Ij-p®T )O(XS-Pyi-s+POT i p=l (-l)i X 1(-i-j+2s) Wk Proposition. T )(XP-lyi-PT i p-1 )(XS-Py -1-s+P J(Xlj(-1)> 13 e 3i+j-1-2s 2 o (-1) k Sji k( -1) -83Proof: = ( J(Xk(-l)) = -1 /2 .oadE(Xk (-1)) (-1))i O(xk•(-1)) [E,X because (-1)] S(i+j-2s)1/2 1 X(1) i+j-1-2s = 0 cO(adF) where c 0 i+j-1-2s 2 Sc =(-1) i+j-1-2s 2 Sc =(-1) X'( 2 i+j-2s) (i+j+1-2s 2 = ij-2s i+j-2s O(adE) O(adE) i+j-1-2s 2 (-X) (i+ji+j-1-2s 2 s)) 2 (-1) Xek(-i-j+2s) 2i+j-1-2s 2 =(-1)} *ekX(-1) Corollary. Proof: j([MJl- Clear. 1) = [Mji D An argument analogous t o the one given in Proposition 4.1 shows GL(1)x-**xGL(N) subgroup of Theorem. L and that is isomorphic to a subgroup of LnK is isomorphic to a O(1)xo**xO(N) The determinant character is trivial on (LnK)O , therefore all nilpotent orbits are admissible. -84Proof: We know ([Mij 1_l) Fix [Mi]_-1e[Mi]_-1 preserves J [Mij ]-1 [Mij ]-1eJ([MiJl -1 E 0 (j) . [Mji 1 _l Because and Adg commutes with LnK we conclude: det(Adgj If and g = ij ([M' gj E (LnK)o , then ) = det (Adgjl I ) CM )CIR. 1- = 1 . det,(Adg I[([MiJ_ 1) -85References 1. M. Duflo, "Construction de representations unitairies d'un groupe de Lie," in Harmonic Analysis and Group Representations, C.I.M.E., 1982. 2. V. Guillemin and S. Sternberg, Geometric American Mathematical Asymptotics, Math Surveys 14. Society, Providence, Rhode Island (1978). 3. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. 4. B. Kostant, "The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group," The American Journal of Mathematics 81 (1959), 973-1032. 5. A.A. Kirillov, "Unitary Representations of Nilpotent Lie Groups," Uspehi Mat. Nauk 17 (1962), 57-110. 6. D. Shale, "Linear Symmetrics of Free Bozon Fields", American Mathematical Society Transactions 103 (1962), 149-167. 7. T.A. Springer and R. Steinberg, "Conjugacy Classes", in Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics 131, SpringerVerlag, Berlin/Heidelberg/New York, 1981. S. D. Vogan, "Representations of Reductive Lie Groups", to appear in Proceedings of International Conference of Mathematicians (1986). 9. D. Vogan, "Singular Unitary Representations", Non-Commutative Harmonic Analysis and Lie Groups, Lecture Notes in Mathematics 880, 506-536.