- Email: [email protected]
Secant Varieties of Adjoint Varieties: Orbit Decomposition
Journal of Algebra 227, 26᎐44 Ž2000. doi:10.1006rjabr.1999.8223, available online at http:rrwww.idealibrary.com on
Secant Varieties of Adjoint Varieties: Orbit Decomposition Hajime Kaji Department of Mathematical Sciences, School of Science and Engineering, Waseda Uni¨ ersity, Tokyo 169-8555, Japan E-mail: [email protected]
and Osami Yasukura Department of Mathematics, Fukui Uni¨ ersity, Fukui 910-8507, Japan E-mail: [email protected] Communicated by Robert Steinberg Received April 9, 1999 DEDICATED TO PROFESSOR KIYOSI YAMAGUTI ON HIS
70TH BIRTHDAY
The orbit decomposition of secant varieties of adjoint varieties is given.
䊚 2000
Academic Press
Key Words: adjoint varieties; secant varieties; orbit decomposition.
INTRODUCTION As homogeneous projective varieties naturally obtained from complex simple Lie algebras ᒄ, we have the adjoint ¨ arieties, denoted by X Ž ᒄ ., that is, for each ᒄ, the Žunique. closed orbit of the action G 哭 ⺠#Ž ᒄ ., defined by the adjoint representation Ad: G哭 ᒄ, where G is the inner automorphism group Int ᒄ of ᒄ, and ⺠#Ž ᒄ . is the projectivization of ᒄ Žsee, for example, wBv, B1, B2, KOYx.. On the other hand, from each complex projective variety X : ⺠ N we obtain its secant ¨ ariety, denoted by Sec X, that is, the closure of the union of secant lines of X in ⺠ N. The secant varieties are quite fundamental in projective geometry, which are characterized by, for example, the following property for a smooth X : ⺠ N: the 26 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
27
SECANT VARIETIES
projection from a point P g ⺠ N to ⺠ Ny 1 gives an isomorphism of X onto its image if and only if P f Sec X Žsee, for example, wFR, LV, Zx.. If one considers the case of an adjoint variety X Ž ᒄ ., then the secant variety Sec X Ž ᒄ . allows the action of G since the action G 哭 ⺠#Ž ᒄ . is linear. The purpose of this article is to give an answer to the following Problem. Find the orbit decomposition of the secant varieties Sec X Ž ᒄ . of the adjoint varieties X Ž ᒄ . associated to complex simple Lie algebras ᒄ. Of course, the adjoint variety X Ž ᒄ . is an orbit in Sec X Ž ᒄ .. In fact, if one denotes by Omin the unique non-zero, minimal nilpotent orbit in ᒄ with respect to the closure ordering, then, as is well known, X Ž ᒄ . s Omin , where : ᒄ _ 04 ª ⺠#Ž ᒄ . is the canonical projection. On the other hand, it is known that the secant variety Sec X Ž ᒄ . has a dense orbit: In fact, if H, X, Y 4 is a Jacobson᎐Morozov standard triple corresponding to Omin , with neutral element H Žsee, for example, wCM, p. 33x., then, according to wKOY, Proposition 5.3x, we have Sec X Ž ᒄ . s G ⭈ H. Furthermore, according to wKY1x, we have the existence of a third orbit in Sec X Ž ᒄ . in case of rk ᒄ G 2, while it is easily verified that Sec X Ž ᒐ ᒉ 2 ⺓. consists of the two orbits above. Now our result is THEOREM. For the adjoint ¨ ariety X Ž ᒄ . : ⺠#Ž ᒄ . associated to a complex simple Lie algebra ᒄ, we ha¨ e: Ža. The secant ¨ ariety Sec X Ž ᒄ . is decomposed as a union of the dense orbit G ⭈ H and a finite number of projecti¨ izations of nilpotent orbits, including X Ž ᒄ .. Namely, there exist a finite number of nilpotent orbits, O0 [ Omin , O1 , . . . , Or : ᒄ such that r
Sec X Ž ᒄ . s G ⭈ H "
@ Oi . is0
Žb. The nilpotent orbits O0 , . . . , Or abo¨ e ha¨ e a unique maximal orbit with respect to the closure ordering F . Žc. If the maximal orbit in O0 , . . . , Or is denoted by Omax , then codim Ž Omax , Sec X Ž ᒄ . . s 1,
28
KAJI AND YASUKURA
and for any nilpotent orbit O : ᒄ, we ha¨ e
O : Sec X Ž ᒄ .
Žd.
m
O F Omax .
The maximal orbits Omax are gi¨ en as follows:
The nilpotent orbits in the table above are denoted by partition types for classical ᒄ and by the Bala᎐Carter label for exceptional ᒄ Žsee, for example, wSS, BC1, BC2, C, Chap. 13, CM, Chaps. 5 and 8x.. Note that we have isomorphisms ᒐ ᒌ 5 ⺓ , ᒐ ᒍ 4 ⺓ and ᒐ ᒌ 6 ⺓ , ᒐ ᒉ 4 ⺓. COROLLARY. The Hasse diagrams of nilpotent orbits whose projecti¨ izations are contained in Sec X Ž ᒄ . are gi¨ en as
SECANT VARIETIES
29
where the superscripts denote the dimension of the orbits. To prove the theorem, we have two basic results: one asserts the nilpotency of orbits in Sec X Ž ᒄ . not through H ŽProposition 1., and the other concerns the complement of the dense orbit through H in Sec X Ž ᒄ . ŽProposition 2.. In fact, by virtue of those results, the problem is reduced to proving Žb. and Žd. above. Then, as we will see below, the cases of ᒐ ᒍ 2 n⺓ and of exceptional types E6 , E7 , E8 , and G 2 are almost done. For the cases of ᒐ ᒉ n⺓ and ᒐ ᒌ n⺓, we need a simple observation on the rank of orbits ŽProposition 3.. The remaining case, of type F4 , is the most difficult, and we reduce the problem to finding the orbit decomposition of Int ᒄ 0 哭 ⺠#Ž ᒄ 1 . in case of type F4 ŽProposition 4., where ᒄ i denotes the graded piece of degree i of the graded decomposition ᒄ s [y2 F iF 2 ᒄ i of contact type Žsee Section 1.. It turns out that the realization of the exceptional simple Lie algebras given by Yamaguti wYx is essential. Finally we should mention that our result is closely related to the recent work of Kaneyuki wKn1, Kn2x Žsee, for details, Section 3..
30
KAJI AND YASUKURA
The authors thank Professor Stefan Helmke for his invaluable advice and stimulating conversation: In particular, Proposition 2 below came out of a conjecture posed by him, and our first proof was simplified by him. He also read the first draft very carefully and gave the authors helpful advice. The authors thank Professor Soji Kaneyuki, too. This work was started and has been encouraged by a question asked by him as follows: Is the number of orbits in the secant ¨ ariety of an adjoint ¨ ariety equal to rk ᒄ q 1?
1. PRELIMINARIES DEFINITION. For a complex simple Lie algebra ᒄ, the adjoint ¨ ariety associated to ᒄ, denoted by X Ž ᒄ ., is defined to be the Žunique. closed orbit of the action G 哭 ⺠#Ž ᒄ ., defined by the adjoint representation Ad: G 哭 ᒄ, where we set G [ Int ᒄ, the inner automorphism group of ᒄ, and ⺠#Ž ᒄ . [ Ž ᒄ _ 04.r⺓=. Take a Cartan subalgebra ᒅ and a base ⌬ of the root system R with respect to ᒅ, and fix an order on R defined by ⌬ Žsee, for example, wHm1x.. Let be the highest root, and take a highest root vector X g ᒄ. Then we have G ⭈ X s Omin . Let Xy be a lowest root vector of ᒄ, and set H [ w X , Xy x g ᒄ. Then, multiplying by suitable scalars, one obtains a standard triple H, X, Xy 4 , where H, X, Y 4 is called a ŽJacobson᎐Morozov. standard triple if these elements satisfy the relation of the standard base of ᒐ ᒉ 2 ⺓ as follows:
w H, X x s 2 X,
w H , Y x s y2Y ,
w X , Y x s H.
Those elements H, X, and Y are respectively called neutral, nil-positi¨ e, and nil-negati¨ e elements. Moreover we have an eigenspace decomposition of ᒄ with respect to ad H, ᒄs
[
y2FiF2
ᒄi,
ᒄ " 2 s ⺓ ⭈ X" ,
where ᒄ i [ Y g ᒄ N w H, Y x s iY 4 . This is called a graded decomposition of contact type in case of ᒄ 1 / 0, that is, rk ᒄ G 2. The neutral element H is called the characteristic element of the graded decomposition.
31
SECANT VARIETIES
DEFINITION. For a complex projective variety X : ⺠ N, the secant variety of X : ⺠ N, denoted by Sec X, is defined by Sec X [
D
x) y : ⺠ N,
x, ygX , x/y
where x ) y denotes the complex projective line in ⺠ N joining x and y. LEMMA 1. We ha¨ e Sec X Ž ᒄ . s G ⭈ ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Proof. Let us denote by Tan X Ž ᒄ . the tangent variety of X Ž ᒄ .; that is, set Tan X Ž ᒄ . [
D
Tx X Ž ᒄ . ,
xgX Ž ᒄ .
where Tx X Ž ᒄ . is the projective tangent space to X Ž ᒄ . at x in ⺠#Ž ᒄ .. Obviously it follows from the definition that Tan X Ž ᒄ . is a closed subset of ⺠#Ž ᒄ ., hence of Sec X Ž ᒄ .. On the other hand, we see that Tan X Ž ᒄ . contains a dense subset G ⭈ H of Sec X Ž ᒄ .. Indeed, we have Tan X Ž ᒄ . s G ⭈ T X X Ž ᒄ . and it follows from wKOY, Proposition 2.2x that T X X Ž ᒄ . s ⺠# Ž ⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 . with H g ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Therefore we see that Sec X Ž ᒄ . s Tan X Ž ᒄ . s G ⭈ ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Remark. The equality Sec X Ž ᒄ . s Tan X Ž ᒄ . above follows also from a corollary to the Fulton᎐Hansen connectedness theorem Žsee, for example, wZ, I, 1.4. Theoremx. and the fact that dim Sec X Ž ᒄ . s 2 dim X Ž ᒄ . Žsee wKOY, Theorem 5.1x.. LEMMA 2. We ha¨ e Ž H q ᒄ 1 q ᒄ 2 . : G ⭈ H. Proof. Take Y1 g ᒄ 1 and Y2 g ᒄ 2 , and set U [ Ž exp ⺓ ⭈ ad H . ⭈ Ž H q Y1 q Y2 . . It follows that
Ž exp t ad H . Ž H q Y1 q Y2 . s
1
Ý nG0
n!
n
Ž t ad H . Ž H q Y1 q Y2 .
s H q e t Y1 q e 2 t Y2 ,
32
KAJI AND YASUKURA
since w H, Yi x s iYi for i s 1, 2. Therefore we have U s H q sY1 q s 2 Y2 N s g ⺓= 4 . Taking the closure of the image U in ⺠#Ž ᒄ ., we obtain
U s s02 H q s0 s1Y1 q s12 Y2 N Ž s0 : s1 . g ⺠ 1 4 : ⺠# Ž ᒄ . . Then we find
Ž G ⭈ H . l U / ⭋. Indeed, U is of course dense in U, and Ž G ⭈ H . l U is a non-empty open subset of U since H g Ž G ⭈ H . l U and G ⭈ H is an open subset of Sec X Ž ᒄ . Žsee wKOY, Proposition 5.3x.. Therefore we have U : G ⭈ H since exp ⺓ ⭈ ad H : G; hence Ž H q Y1 q Y2 . g G ⭈ H. LEMMA 3. We ha¨ e Ž Y1 q ᒄ 2 . : G ⭈ Y1 for any non-zero Y1 g ᒄ 1. Proof. Consider the root space decomposition of ᒄ 1 , ᒄ1 s
[
␣gR
ᒄ␣ ,
where R [ ␣ g RqN y ␣ g R4 , R is the system of roots, and Rq is the set of positive roots. According to this decomposition, we have Y1 s
Ý
␣gR
Y␣ ,
Y␣ g ᒄ ␣ .
It follows that Y␥ / 0 for some ␥ g R since Y1 / 0. Set  [ y ␥ g R. Then  g R and one can choose Z g ᒄ  such that w Z, Y␥ x s X since w ᒄ  , ᒄ ␥ x s ᒄ 2 . Moreover we see that if ␣ / ␥ , then w Z, ᒄ ␣ x s 0: Indeed, if w Z, ᒄ ␣ x / 0, then  q ␣ s since w Z, ᒄ ␣ x : w ᒄ 1 , ᒄ 1 x s ᒄ 2 s ᒄ with Z g ᒄ  . Therefore, Žexp sZ .Y␥ s Y␥ q sX , and Žexp sZ .Y␣ s Y␣ for any ␣ g R with ␣ / ␥ . Thus
Ž exp sZ . Y1 s Y1 q sX , and we obtain the conclusion since exp sZ g G. PROPOSITION 1. Any orbit in Sec X Ž ᒄ . through neither H nor X intersects ⺠#Ž ᒄ 1 .. In particular, any orbit in Sec X Ž ᒄ . not through H is nilpotent. Proof. According to Lemma 1, any orbit in Sec X Ž ᒄ . intersects ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. On the other hand, it follows from Lemma 2 that any orbit
SECANT VARIETIES
33
through ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 . but not through H intersects ⺠#Ž ᒄ 1 [ ᒄ 2 ., and from Lemma 3 that any orbit through ⺠#Ž ᒄ 1 [ ᒄ 2 . but not through X intersects ⺠#Ž ᒄ 1 .. Therefore we obtain the result. A restriction of G 哭 ⺠#Ž ᒄ . yields a natural action, G 0 哭 ⺠# Ž ᒄ 1 . , where we set G 0 [ Int ᒄ 0 . Indeed, it follows from the Jacobi identity that w ᒄ i , ᒄ j x : ᒄ iqj . Since G 0 : G and ⺠#Ž ᒄ 1 . : Sec X Ž ᒄ ., we have a well-defined, natural map, ⌿ : orbits of G 0 哭 ⺠# Ž ᒄ 1 . 4 ª orbits of G 哭 Sec X Ž ᒄ . 4 . Then Proposition 1 yields that the image of ⌿ contains the set of orbits through neither X nor H. Remark. Ža. The image of ⌿ does not contain G ⭈ H, since ad H is not nilpotent. Žb. The image of ⌿ does not contain X Ž ᒄ . in general: In fact, according to wKY1x, ⺠#Ž ᒄ 1 . l X Ž ᒄ . s ⭋ if and only if ᒄ , ᒐ ᒉ 2 ⺓ or ᒄ , ᒐ ᒍ 2 n⺓ for some n. Žc. Using wKY1x, one can show that ⌿ is not injective if ᒄ , ᒐ ᒉ n⺓ for any n G 3. Moreover, it turns out that the converse is also true: In fact, we will give in a forthcoming paper wKY2x a complete description of the orbit decomposition of G 0 哭 ⺠#Ž ᒄ 1 .. PROPOSITION 2. For a semi-simple element E of a semi-simple Lie algebra ᒄ with G [ Int ᒄ, consider the projecti¨ ization G ⭈ E : ⺠#Ž ᒄ . of the adjoint orbit through E in ᒄ. Then the complement in its closure, G ⭈ E _ G ⭈ E, has pure codimension 1 in G ⭈ E. Proof. The orbit G ⭈ E is affine since it follow from wS, 4.4.5x that the orbit G ⭈ E is closed in the affine space ᒄ. On the other hand,
N G⭈E : G ⭈ E ª G ⭈ E is a quotient by CG Ž E .rCG Ž E ., where CG Ž E . and CG Ž E . respectively denote the stabilizers of E g ⺠#Ž ᒄ . and of E g ᒄ. Since E is semi-simple, CG Ž E .rCG Ž E . is a finite group Žsee, for example, wDS, Lemma 5.1x., so that N G⭈E is a finite morphism in the scheme-theoretic sense. Therefore, it follows from wHr1, II.1.5x that G ⭈ E is also affine. Now the claim follows from wHr1, II.3.1x.
34
KAJI AND YASUKURA
Remark. With the assumptions of Proposition 2, it follows from wHr1, II.6.2x that if dim G ⭈ E G 2, then the complement is connected.
2. PROOF OF THEOREM First of all, the statement Ža. of the theorem follows from Proposition 1 since the number of nilpotent orbits in a semi-simple ᒄ is finite, as is well known Žsee, for example, wCM, Theorem 3.5.4x.. In particular, it is obvious that there exist maximal elements of O0 , . . . , Or 4 . Next, for any nilpotent orbit O : ᒄ, we have that O : Sec X Ž ᒄ . if and only if there is a maximal element Oi of O0 , . . . , Or 4 such that O F Oi . On the other hand, Oi is a maximal element of O0 , . . . , Or 4 if and only if codim Ž Oi , Sec X Ž ᒄ . . s 1, since it follows from Proposition 2 that any irreducible component of Sec X Ž ᒄ . _ G ⭈ H has codimension 1 in Sec X Ž ᒄ . s G ⭈ H. Therefore, the statement Žc. is clear, once the uniqueness Žb. is established. Note that the condition above is equivalent to
Ž codim1.
dim Oi s 2 dim Omin y 2.
Indeed, it follows from wKOY, Theorem 5.1x that dim Sec X Ž ᒄ . s 2 dim X Ž ᒄ ., and we have in general dim O s dim O y 1 for a nilpotent orbit O : ᒄ. Thus, it suffices to show the statements Žb. and Žd.: we check the claims case by case. Let us start with C: For ᒄ s ᒐ ᒍ 2 n⺓, according to wCM, Theorem 5.1.3x, the nilpotent orbits correspond bijectively to the partitions of 2 n such that odd parts occur with even multiplicity, and the Hasse diagram for ᒐ ᒍ 2 n⺓ is as follows:
SECANT VARIETIES
35
We have Omin s Ow212 ny 2 x , and Ow2 2 12 ny 4 x is the only nilpotent orbit satisfying the condition Žcodim1. above since it follows from wCM, Corollary 6.1.4x that dim Ow2 2 12 ny 4 x s 4 n y 2 and dim Ow212 ny 2 x s 2 n. Therefore, Ow2 2 12 ny 4 x must be the unique maximal orbit. EG: It follows from wCM, Sect. 8.4x that OA 2 is the only nilpotent orbit satisfying Žcodim1. for the cases E6 , E7 and E8 , and OG 2 Ž a1 . for G 2 , while Omin s OA 1 for all exceptional ᒄ. Therefore, the orbit satisfying Žcodim1. must be the unique maximal orbit in each case. As we will see below, in the remaining cases, the nilpotent orbits satisfying Žcodim1. are not necessarily unique: For the classical cases, we need PROPOSITION 3. Assume that ᒄ is a subalgebra of ᒐ ᒉ N ⺓. Then for any orbit O : ᒄ if O : Sec X Ž ᒄ ., then rk O F 2 rk Omin , where rk O is defined to be the rank of any element in O as a matrix ¨ ia the embedding ᒄ : ᒐ ᒉ N ⺓. Proof. It suffices to show that rk A F 2 rk Omin for a general A g ᒄ such that A g Sec X Ž ᒄ .: Indeed, rk is a lower semi-continuous function on ⺠#Ž ᒄ .. Then we may assume that A g A 0 ) A1 for some A i g ᒄ with A i g X Ž ᒄ .. Replacing the A i with suitable scalar multiples Žif necessary ., we may assume moreover that A s A 0 q A1. Then we have rk A F rk A 0 q rk A1 s 2 rk Omin since A i g Omin . A: For ᒄ s ᒐ ᒉ n⺓, according to wCM, Theorem 5.1.1x, the nilpotent orbits correspond bijectively to the partitions of n, and the Hasse diagram for ᒐ ᒉ n⺓ is as follows:
We have Omin s Ow21ny 2 x. It follows from Proposition 3 that Ow2 31ny 6 x Sec X Ž ᒄ . since rk Ow2 31ny 6 x s 3 while rk Omin s 1. On the other hand, Ow31ny 3 x satisfies the condition Žcodim1. since it follows from wCM, Corol-
36
KAJI AND YASUKURA
lary 6.1.4x that dim Ow31ny 3 x s 4 n y 6 and dim Ow21ny 2 x s 2 n y 2. Therefore, for any n G 3, Ow31ny 3 x must be the unique maximal orbit. In case of n s 2 obviously Ow2x is the unique maximal orbit. BD: For ᒄ s ᒐ ᒌ n⺓, according to wCM, Theorems 5.1.2, 5.1.4x, the nilpotent orbits correspond bijectively to the partitions of n such that even parts occur with even multiplicity except that in case n is even, very even partitions d correspond to two orbits, denoted by OdI and OdII, where a partition d is said to be ¨ ery e¨ en if d has only even parts and each multiplicity is even. Therefore, the Hasse diagram for ᒐ ᒌ n⺓ is
where Ow2Ž I,4 1IIny. 8 x with n s 8 and Ow2Ž I,6 1IIny. 12 x with n s 12 are corresponding to the very even case, and decompose into two orbits. We have Omin s Ow2 2 1ny 4 x. It follows from Proposition 3 that neither Ow32 4 1ny 11 x nor Ow2 6 1ny 12 x are contained in Sec X Ž ᒄ . since rk Ow32 4 1ny 11 x s rk Ow2 6 1ny 12 x s 6 while rk Omin s 2. On the other hand, Ow3 2 1ny 6 x satisfies Žcodim1. since it follows from wCM, Corollary 6.1.4x that dim Ow3 2 1ny 6 x s 4 n y 14 and dim Ow2 2 1ny 4 x s 2 n y 6. Therefore if n G 6, then Ow3 2 1ny 6 x must be the unique maximal orbit. For the last case of type F4 we need the following notation: If ᒅ is a Cartan subalgebra of ᒄ and ⌬ s ␣ 1 , . . . , ␣ l 4 is a base of simple roots, then the weighted Dynkin diagram of H g ᒅ is defined to be the Dynkin diagram of ᒄ with ␣ i Ž H . inserted at the ith node. If H, X, Y 4 is a standard triple with H g ᒅ, then the weighted Dynkin diagram of the orbit through X is defined to be that of H.
SECANT VARIETIES
37
F: Assume ᒄ is of type F4 . It follows from wCM, Sect. 8.4x that OA 2 as well as OA˜2 satisfy Žcodim1. and their weighted Dynkin diagrams are
while OA 1 s Omin . It suffices to show that OA˜2 Sec X Ž F4 ., and the required result follows from Proposition 1 and PROPOSITION 4. Let ᒄ be the complex simple Lie algebra of type F4 with graded decomposition ᒄ s [y2 F iF 2 ᒄ i of contact type, and let ⌽i N 1 F i F 144 be the base of ᒄ 1 gi¨ en in wY, Appendixx. Then representati¨ es for the orbits of G 0 哭 ⺠#Ž ᒄ 1 . and the weighted Dynkin diagrams for nilpotent orbits in ᒄ through those representati¨ es are gi¨ en as follows:
Proof. Observe that if suffices to show that the nilpotent orbits in ᒄ through the four elements above have the required weighted Dynkin diagrams: Indeed, this implies that the four orbits in ᒄ 1 through those elements are distinct, while we have as a fact that the action G 0 哭 ⺠#Ž ᒄ 1 . has exactly four orbits. Actually, this fact would be best described in connection with the realization of Freudenthal’s magic square via the theory of Jordan algebras Žsee wFx; see also wM, Ž5.10., At, p. 22x.. A proof by means of the classification of Lie algebras is given as follows: We have a decomposition ᒄ 0 s ⺓ ⭈ H [ ᒄ SS 0 , w x where ᒄ SS 0 denotes the semi-simple part ᒄ 0 , ᒄ 0 of a reductive algebra ᒄ 0 . We see from the Dynkin diagram of ᒄ 0 that ᒄ SS 0 is a simple Lie algebra of type C3 Žsee, for example, wAs, p. 53x.. According to this decomposition, it
38
KAJI AND YASUKURA
follows from Weyl’s dimension formula Žsee, for example, wSK, Sect. 1x. that the action G 0 哭 ᒄ 1 turns out to be GL1⺓ = Sp6 ⺓ 哭 ⺓ m ⌫ Ž ⌳ . , where ⌫ Ž ⌳ . is the irreducible Sp6 ⺓-module of dimension 14 with highest weight ⌳ s 2 or ⌳ s 3 . Now it follows from wV, Proposition 2x that the number of orbits of G 0 哭 ᒄ 1 is finite. On the other hand, according to wSK, Proposition 20x, GL1⺓ = Sp6 ⺓ 哭 ⺓ m ⌫ Ž 2 . has infinitely many orbits. Thus ⌳ s 3 and the action in question is equivalent to GL1⺓ = Sp6 ⺓ 哭⺓ m ⌫ Ž 3 .. Now the claim follows from wI, Proposition 7x as well as wKm, Sect. 9x. This completes the proof of the fact. Now, we compute the weighted Dynkin diagrams for the orbits in ᒄ through the nilpotent elements above. Let ⌽i N 1 F i F 144 and H1 , . . . , H4 4 be, respectively, the bases of ᒄy1 and of a Cartan subalgebra ᒅ given in wY, Appendixx. Denote by 1 , . . . , 4 4 the dual of Hi ’s. Then it follows from wY, Appendixx that
1 q 2 ,
1 y 2 ,
3 q 4 ,
3 y 4
form a base of positive, strongly orthogonal roots, and we have four standard triples as follows: neutral
nil-positive
nil-negative
H1 q H 2 H1 y H 2 H3 q H4 H3 y H4
2 ⌽7 2 ⌽8 2 ⌽9 2 ⌽14
y2⌽ 1 y2 ⌽ 2 y2⌽ 3 y2 ⌽13
Recall that a base of roots for ᒅ* is strongly orthogonal if any sum or difference of two of its elements is never a root. From the standard triples above we obtain the following standard triples: neutral
nil-positive
nil-negative
H1 q H 2 2 H1 2 H1 q H3 q H4 2Ž H1 q H3 .
2 ⌽7 2Ž ⌽ 7 q ⌽ 8 . 2Ž ⌽ 7 q ⌽ 8 q ⌽ 9 . 2Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14.
y2 ⌽ 1 y2Ž ⌽ 1 q ⌽ 2 . y2Ž ⌽ 1 q ⌽ 2 q ⌽ 3 . y2Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13.
39
SECANT VARIETIES
Indeed, it follows from the strong orthogonality that, for example, 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . s 2 H1 q 2 H 3 , 2 Ž H1 q H3 . , 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . , 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . s H1 q H2 , 2 ⌽ 7 q H1 y H2 , 2 ⌽ 8 q H3 q H4 , 2 ⌽ 9 q H3 y H4 , 2 ⌽14 s 4⌽ 7 q 4⌽ 8 q 4⌽ 9 q 4⌽14 , 2 Ž H1 q H3 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s w H1 q H2 , y2⌽ 1 x q w H1 y H2 , y2⌽ 2 x q w H3 q H4 , y2⌽ 3 x q H3 y H4 , y2⌽13 s 4⌽ 1 q 4⌽ 2 q 4⌽ 3 q 4⌽13 . Now set
␣1 [ 2 y 3 , ␣4 [
␣ 2 [ 3 y 4 , 1 2
␣ 3 [ 4 ,
Ž 1 y 2 y 3 y 4 . .
Then, it follows from wY, Appendixx that ⌬ s ␣ 1 , . . . , ␣ 4 4 forms a base of simple roots, and the Dynkin diagram is as follows:
40
KAJI AND YASUKURA
Therefore the weighted Dynkin diagrams of H1 q H2 , 2 H1 , 2 H1 q H3 q H4 , 2 H1 q 2 H3 are as follows:
In particular, H1 q H2 and 2 H1 are dominant with respect to ⌬. Therefore, the weighted Dynkin diagrams of H1 q H2 and 2 H1 above respectively give those of nilpotent orbits through ⌽ 7 and ⌽ 7 q ⌽ 8 , as is required. On the other hand, we see that ␣ 2 ␣ 1Ž2 H1 q H3 q H4 . is dominant though 2 H1 q H3 q H4 is not, where ␣ denotes the reflection with respect to a simple root ␣ : Indeed, it follows from the recipe below that
Therefore, the last diagram corresponding to ␣ 2 ␣ 1Ž2 H1 q H3 q H4 . gives the weighted Dynkin diagram of the nilpotent orbit through ⌽ 7 q ⌽ 8 q ⌽ 9 , as is required. Similarly we see that ␣ 1Ž2 H1 q 2 H3 . is dominant since
and the last diagram gives the weighted Dynkin diagram of the nilpotent orbit through ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽ 14. Thus we obtain the required result. RECIPE. Let c be the coefficient of the ith node in the diagram for H abo¨ e. To compute ␣ iŽ H ., change c to yc at the ith node and add yc ⭈ 2Ž ␣ j , ␣ i .rŽ ␣ i , ␣ i . at e¨ ery jth node adjacent to the ith node. Ž Here Ž⭈, ⭈ . is any positi¨ e definite bilinear form which is in¨ ariant under the action of the Weyl group.. Obser¨ e that this is just c itself unless ␣ j is longer than ␣ i .
41
SECANT VARIETIES
This follows from the formula for a reflection: Indeed, we have ␣ j Ž ␣ iŽ H .. s ␣ Ž H . with ␣ [ ␣ iŽ ␣ j . s ␣ j y 2Ž ␣ j , ␣ i .rŽ ␣ i , ␣ i . ⭈ ␣ i . 3. FINAL REMARK Let ᒄ be a complex simple Lie algebra. For a graded decomposition ᒄs
[
y FiF
ᒄi
of order with characteristic element Z, consider the coset space, M [ GrCG Ž Z . , where we set G [ Int ᒄ and CG Ž Z . is the stabilizer of Z g ᒄ, as before. In wKn1, Sect. 3x Žsee also wKn3, 3.2x., Kaneyuki gave a G-equivariant ˜ Moreover in case of s 1, he compactification of M, denoted by M. ˜ and found that the number of obtained the orbit decomposition of M ˜ orbits in M is equal to rk M q 1, where rk M is the split rank of M as a simple reducible pseudo-hermitian symmetric space. In this context, our work in the case of rk ᒄ G 2 is closely related to the case of the graded decomposition of contact type: In fact, consider M corresponding to the graded decomposition of contact type. Then the secant variety Sec X Ž ᒄ . is a G-equivariant compactification of M CG Ž H . rCG Ž H .
,
where CG Ž H . is the stabilizer of H g ⺠#Ž ᒄ . and H is the characteristic element for the gradation of contact type. Indeed, we have GrCG Ž H . s G ⭈ H and it follows from wKOY, Proposition 5.3x that G ⭈ Hs Sec X Ž ᒄ ., which allows the action of G. For the case ᒄ s ᒐ ᒉ 2 ⺓, it is easily shown that X Ž ᒐ ᒉ 2 ⺓. is a conic in ⺠#Ž ᒐ ᒉ 2 ⺓. s ⺠ 2 with Sec X Ž ᒐ ᒉ 2 ⺓. s ⺠#Ž ᒐ ᒉ 2 ⺓., and GrCG Ž H . s G ⭈ H coincides with the complement of X Ž ᒐ ᒉ 2 ⺓. Žsee wKOY, Proposition 4.1x.. This corresponds to the basic example of Kaneyuki’s theory Žsee wKn1, p. 333x.. Note that CG Ž H .rCG Ž H . is a finite group since H is semi-simple. In fact, we have PROPOSITION 5. If H is the neutral element of a standard triple corresponding to Omin , gi¨ en in Section 1, then
Ž CG Ž H . : CG Ž H . . s 2.
42
KAJI AND YASUKURA
Proof. It suffices to show that
y1 H l G ⭈ H s H , yH 4 . The inclusion : is obtained by following the argument in the proof of wDS, Lemma 5.1x. Indeed, if H g G ⭈ H with g ⺓, then ad H and ad H have the same set of eigenvalues, which is equal to 0, " 24 in case of ᒄ s ᒐ ᒉ 2 ⺓, and 0, " 1, " 24 otherwise. Therefore, it follows that s "1. To show the converse, we first consider the case of ᒄ s ᒐ ᒉ 2 ⺓. We may assume that ᒄ is the set of traceless matrices of size 2 with G s PSL2 ⺓ and Hs
1 0
0 . y1
Then we have G ⭈ H 2 g ⭈ H s gHgy1 s yH g y1 H, where g g G is represented by g[
0 y1
1 g SL 2 ⺓. 0
Next, for arbitrary ᒄ, consider the embedding
: PSL2 ⺓ ¨ G associated to the composition of ᒐ ᒉ 2 ⺓ , ᒄy2 [ ⺓ ⭈ H [ ᒄ 2 : ᒄ . Then, using g g PSL2 ⺓ above, we have g ⭈ H s yH with g g G, and we obtain the result. Remark. For another proof of this result, see wKY1, Sect. 3, Corollaryx.
REFERENCES wAsx wAtx wBC1x wBC2x
H. Asano, Symplectic triple systems and simple Lie algebras Žin Japanese., RIMS Kokyuroku, Kyoto Uni¨ . 308 Ž1977., 41᎐54. K. Atsuyama, Freudenthal’s projective geometry Žin Japanese., in ‘‘Projective VarietiesrProjective Geometry of Algebraic Varieties 2 q Singularities,’’ Proc. Symp. Algebraic Geometry, pp. 1᎐62, Waseda Univ., Japan, 1997. P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups, I, Math. Proc. Cambridge Philos. Soc. 79 Ž1976., 401᎐425. P. Bala, R. W. Carter, Classes of unipotent elements in simple algebraic groups, II, Math. Proc. Cambridge Philos. Soc. 80 Ž1976., 1᎐17.
SECANT VARIETIES
wBvx
43
A. Beauville, Fano contact manifolds and nilpotent orbits, Comment. Math. Hel¨ . 73 Ž1998., 566᎐583. wB1x W. Boothby, Homogeneous complex contact manifolds, Amer. Math. Soc. Proc. Symp. Pure Math. 3 Ž1961., 144᎐154. wB2x W. Boothby, A note on homogeneous complex contact manifolds, Proc. Amer. Math. Soc. 13 Ž1962., 276᎐280. ´´ wBrx N. Bourbaki, ‘‘Elements de Mathematique, Groupes et Algebres de Lie,’’ Chapters ´ ` 4, 5, and 6, Hermann, Paris, 1968. wCx R. W. Carter, ‘‘Finite Groups of Lie TypeᎏConjugacy Classes and Complex Characters,’’ Wiley, Chichester, 1993. wCMx D. H. Collingwood and W. M. McGovern, ‘‘Nilpotent Orbits in Semisimple Lie Algebras,’’ Van Nostrand Reinhold, New York, 1993. wDSx A. Dancer and A. Swann, ‘‘Quaternionic Kahler Manifolds of Cohomogeneity One,’’ ¨ preprint, math.DGr9808097, 1998. wFx H. Freudenthal, Lie groups in the foundations of geometry, Ad¨ . Math. 1 Ž1964., 145᎐190. wFRx T. Fujita and J. Roberts, Varieties with small secant varieties: The extremal case, Amer. J. Math. 103 Ž1981., 953᎐976. wFHx W. Fulton, J. Harris, ‘‘Representation Theory: A First Course,’’ GTM, Vol. 129, Springer-Verlag, New York, 1991. wHr1x R. Hartshorne, ‘‘Ample Subvarieties of Algebraic Varieties,’’ Lecture Notes in Math., Vol. 156, Springer-Verlag, New York, 1970. wHr2x R. Hartshorne, ‘‘Algebraic Geometry,’’ GTM, Vol. 52, Springer-Verlag, New York, 1977. wHm1x J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ GTM, Vol. 9, Springer-Verlag, New York, 1972. wHm2x J. E. Humphreys, ‘‘Linear Algebraic Groups,’’ GTM, Vol. 21, Springer-Verlag, New York, 1975. wIx J.-i. Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 91 Ž1970., 997᎐1028. wKOYx H. Kaji, M. Ohno, and O. Yasukaru, Adjoint varieties and their secant varieties, Indag. Math. 10 Ž1999., 45᎐57. wKY1x H. Kaji, O. Yasukura, ‘‘Projective Geometry of Adjoint Varieties,’’ preprint, 1998. wKY2x H. Kaji and O. Yasukura, The orbit structure of simple graded Lie algebras of contact type, in preparation. wKn1x S. Kaneyuki, Homogeneous symplectic manifolds and dipolarizations in Lie algebras, Japan. J. Math. 13 Ž1987., 333᎐370. wKn2x S. Kaneyuki, Graded Lie algebras, related geometric structures and pseudo-Hermitian symmetric spaces, in ‘‘Analysis and Geometry on Complex Homogeneous Domains and Related Topics,’’ Progress in Math., Birkhauser, BaselrBoston, to ¨ appear. wKmx T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 Ž1982., 1᎐80. wLVx R. Lazarsfeld and A. Van de Ven, ‘‘Topics in the Geometry of Projective SpaceᎏRecent Work of F. L. Zak,’’ DMV Sem. 4, Birkhauser, BaselrBoston, 1984. ¨ wMx S. Mukai, Projective geometry of homogeneous spaces Žin Japanese., in ‘‘Projective VarietiesrProjective Geometry of Algebraic Varieties,’’ Proc. Symp. Algebraic Geometry, pp. 1᎐52, Waseda University, Japan, 1994.
44 wSKx wSx wSSx wVx wYx wZx
KAJI AND YASUKURA
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 Ž1977., 1᎐155. T. A. Springer, ‘‘Linear Algebraic Groups,’’ Progress in Math., Vol. 9, Birkhauser, ¨ BaselrBoston, 1981. T. Springer and R. Steinberg, Conjugacy classes, in ‘‘Seminar on Algebraic Groups and Related Finite Groups,’’ Lecture Notes in Math., Vol. 131, pp. 167᎐266, Springer-Verlag, New York, 1970. E. B. Vinberg, The Weyl group of a graded Lie algebra, Iz¨ . Akad. Nauk SSSR Ser. Mat. 40, 463᎐495. K. Yamaguti, On weak representations of a class of ternary systems, Mem. Fac. Gen. Ed. Kumamoto Uni¨ . Natur. Sci. 9 Ž1974., 1᎐8. F. L. Zak, ‘‘Tangents and Secants of Algebraic Varieties,’’ Translations of Math. Monographs, Vol. 127, Am. Math. Soc., Providence, 1993.
Secant Varieties of Adjoint Varieties: Orbit Decomposition Hajime Kaji Department of Mathematical Sciences, School of Science and Engineering, Waseda Uni¨ ersity, Tokyo 169-8555, Japan E-mail: [email protected]
and Osami Yasukura Department of Mathematics, Fukui Uni¨ ersity, Fukui 910-8507, Japan E-mail: [email protected] Communicated by Robert Steinberg Received April 9, 1999 DEDICATED TO PROFESSOR KIYOSI YAMAGUTI ON HIS
70TH BIRTHDAY
The orbit decomposition of secant varieties of adjoint varieties is given.
䊚 2000
Academic Press
Key Words: adjoint varieties; secant varieties; orbit decomposition.
INTRODUCTION As homogeneous projective varieties naturally obtained from complex simple Lie algebras ᒄ, we have the adjoint ¨ arieties, denoted by X Ž ᒄ ., that is, for each ᒄ, the Žunique. closed orbit of the action G 哭 ⺠#Ž ᒄ ., defined by the adjoint representation Ad: G哭 ᒄ, where G is the inner automorphism group Int ᒄ of ᒄ, and ⺠#Ž ᒄ . is the projectivization of ᒄ Žsee, for example, wBv, B1, B2, KOYx.. On the other hand, from each complex projective variety X : ⺠ N we obtain its secant ¨ ariety, denoted by Sec X, that is, the closure of the union of secant lines of X in ⺠ N. The secant varieties are quite fundamental in projective geometry, which are characterized by, for example, the following property for a smooth X : ⺠ N: the 26 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
27
SECANT VARIETIES
projection from a point P g ⺠ N to ⺠ Ny 1 gives an isomorphism of X onto its image if and only if P f Sec X Žsee, for example, wFR, LV, Zx.. If one considers the case of an adjoint variety X Ž ᒄ ., then the secant variety Sec X Ž ᒄ . allows the action of G since the action G 哭 ⺠#Ž ᒄ . is linear. The purpose of this article is to give an answer to the following Problem. Find the orbit decomposition of the secant varieties Sec X Ž ᒄ . of the adjoint varieties X Ž ᒄ . associated to complex simple Lie algebras ᒄ. Of course, the adjoint variety X Ž ᒄ . is an orbit in Sec X Ž ᒄ .. In fact, if one denotes by Omin the unique non-zero, minimal nilpotent orbit in ᒄ with respect to the closure ordering, then, as is well known, X Ž ᒄ . s Omin , where : ᒄ _ 04 ª ⺠#Ž ᒄ . is the canonical projection. On the other hand, it is known that the secant variety Sec X Ž ᒄ . has a dense orbit: In fact, if H, X, Y 4 is a Jacobson᎐Morozov standard triple corresponding to Omin , with neutral element H Žsee, for example, wCM, p. 33x., then, according to wKOY, Proposition 5.3x, we have Sec X Ž ᒄ . s G ⭈ H. Furthermore, according to wKY1x, we have the existence of a third orbit in Sec X Ž ᒄ . in case of rk ᒄ G 2, while it is easily verified that Sec X Ž ᒐ ᒉ 2 ⺓. consists of the two orbits above. Now our result is THEOREM. For the adjoint ¨ ariety X Ž ᒄ . : ⺠#Ž ᒄ . associated to a complex simple Lie algebra ᒄ, we ha¨ e: Ža. The secant ¨ ariety Sec X Ž ᒄ . is decomposed as a union of the dense orbit G ⭈ H and a finite number of projecti¨ izations of nilpotent orbits, including X Ž ᒄ .. Namely, there exist a finite number of nilpotent orbits, O0 [ Omin , O1 , . . . , Or : ᒄ such that r
Sec X Ž ᒄ . s G ⭈ H "
@ Oi . is0
Žb. The nilpotent orbits O0 , . . . , Or abo¨ e ha¨ e a unique maximal orbit with respect to the closure ordering F . Žc. If the maximal orbit in O0 , . . . , Or is denoted by Omax , then codim Ž Omax , Sec X Ž ᒄ . . s 1,
28
KAJI AND YASUKURA
and for any nilpotent orbit O : ᒄ, we ha¨ e
O : Sec X Ž ᒄ .
Žd.
m
O F Omax .
The maximal orbits Omax are gi¨ en as follows:
The nilpotent orbits in the table above are denoted by partition types for classical ᒄ and by the Bala᎐Carter label for exceptional ᒄ Žsee, for example, wSS, BC1, BC2, C, Chap. 13, CM, Chaps. 5 and 8x.. Note that we have isomorphisms ᒐ ᒌ 5 ⺓ , ᒐ ᒍ 4 ⺓ and ᒐ ᒌ 6 ⺓ , ᒐ ᒉ 4 ⺓. COROLLARY. The Hasse diagrams of nilpotent orbits whose projecti¨ izations are contained in Sec X Ž ᒄ . are gi¨ en as
SECANT VARIETIES
29
where the superscripts denote the dimension of the orbits. To prove the theorem, we have two basic results: one asserts the nilpotency of orbits in Sec X Ž ᒄ . not through H ŽProposition 1., and the other concerns the complement of the dense orbit through H in Sec X Ž ᒄ . ŽProposition 2.. In fact, by virtue of those results, the problem is reduced to proving Žb. and Žd. above. Then, as we will see below, the cases of ᒐ ᒍ 2 n⺓ and of exceptional types E6 , E7 , E8 , and G 2 are almost done. For the cases of ᒐ ᒉ n⺓ and ᒐ ᒌ n⺓, we need a simple observation on the rank of orbits ŽProposition 3.. The remaining case, of type F4 , is the most difficult, and we reduce the problem to finding the orbit decomposition of Int ᒄ 0 哭 ⺠#Ž ᒄ 1 . in case of type F4 ŽProposition 4., where ᒄ i denotes the graded piece of degree i of the graded decomposition ᒄ s [y2 F iF 2 ᒄ i of contact type Žsee Section 1.. It turns out that the realization of the exceptional simple Lie algebras given by Yamaguti wYx is essential. Finally we should mention that our result is closely related to the recent work of Kaneyuki wKn1, Kn2x Žsee, for details, Section 3..
30
KAJI AND YASUKURA
The authors thank Professor Stefan Helmke for his invaluable advice and stimulating conversation: In particular, Proposition 2 below came out of a conjecture posed by him, and our first proof was simplified by him. He also read the first draft very carefully and gave the authors helpful advice. The authors thank Professor Soji Kaneyuki, too. This work was started and has been encouraged by a question asked by him as follows: Is the number of orbits in the secant ¨ ariety of an adjoint ¨ ariety equal to rk ᒄ q 1?
1. PRELIMINARIES DEFINITION. For a complex simple Lie algebra ᒄ, the adjoint ¨ ariety associated to ᒄ, denoted by X Ž ᒄ ., is defined to be the Žunique. closed orbit of the action G 哭 ⺠#Ž ᒄ ., defined by the adjoint representation Ad: G 哭 ᒄ, where we set G [ Int ᒄ, the inner automorphism group of ᒄ, and ⺠#Ž ᒄ . [ Ž ᒄ _ 04.r⺓=. Take a Cartan subalgebra ᒅ and a base ⌬ of the root system R with respect to ᒅ, and fix an order on R defined by ⌬ Žsee, for example, wHm1x.. Let be the highest root, and take a highest root vector X g ᒄ. Then we have G ⭈ X s Omin . Let Xy be a lowest root vector of ᒄ, and set H [ w X , Xy x g ᒄ. Then, multiplying by suitable scalars, one obtains a standard triple H, X, Xy 4 , where H, X, Y 4 is called a ŽJacobson᎐Morozov. standard triple if these elements satisfy the relation of the standard base of ᒐ ᒉ 2 ⺓ as follows:
w H, X x s 2 X,
w H , Y x s y2Y ,
w X , Y x s H.
Those elements H, X, and Y are respectively called neutral, nil-positi¨ e, and nil-negati¨ e elements. Moreover we have an eigenspace decomposition of ᒄ with respect to ad H, ᒄs
[
y2FiF2
ᒄi,
ᒄ " 2 s ⺓ ⭈ X" ,
where ᒄ i [ Y g ᒄ N w H, Y x s iY 4 . This is called a graded decomposition of contact type in case of ᒄ 1 / 0, that is, rk ᒄ G 2. The neutral element H is called the characteristic element of the graded decomposition.
31
SECANT VARIETIES
DEFINITION. For a complex projective variety X : ⺠ N, the secant variety of X : ⺠ N, denoted by Sec X, is defined by Sec X [
D
x) y : ⺠ N,
x, ygX , x/y
where x ) y denotes the complex projective line in ⺠ N joining x and y. LEMMA 1. We ha¨ e Sec X Ž ᒄ . s G ⭈ ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Proof. Let us denote by Tan X Ž ᒄ . the tangent variety of X Ž ᒄ .; that is, set Tan X Ž ᒄ . [
D
Tx X Ž ᒄ . ,
xgX Ž ᒄ .
where Tx X Ž ᒄ . is the projective tangent space to X Ž ᒄ . at x in ⺠#Ž ᒄ .. Obviously it follows from the definition that Tan X Ž ᒄ . is a closed subset of ⺠#Ž ᒄ ., hence of Sec X Ž ᒄ .. On the other hand, we see that Tan X Ž ᒄ . contains a dense subset G ⭈ H of Sec X Ž ᒄ .. Indeed, we have Tan X Ž ᒄ . s G ⭈ T X X Ž ᒄ . and it follows from wKOY, Proposition 2.2x that T X X Ž ᒄ . s ⺠# Ž ⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 . with H g ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Therefore we see that Sec X Ž ᒄ . s Tan X Ž ᒄ . s G ⭈ ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. Remark. The equality Sec X Ž ᒄ . s Tan X Ž ᒄ . above follows also from a corollary to the Fulton᎐Hansen connectedness theorem Žsee, for example, wZ, I, 1.4. Theoremx. and the fact that dim Sec X Ž ᒄ . s 2 dim X Ž ᒄ . Žsee wKOY, Theorem 5.1x.. LEMMA 2. We ha¨ e Ž H q ᒄ 1 q ᒄ 2 . : G ⭈ H. Proof. Take Y1 g ᒄ 1 and Y2 g ᒄ 2 , and set U [ Ž exp ⺓ ⭈ ad H . ⭈ Ž H q Y1 q Y2 . . It follows that
Ž exp t ad H . Ž H q Y1 q Y2 . s
1
Ý nG0
n!
n
Ž t ad H . Ž H q Y1 q Y2 .
s H q e t Y1 q e 2 t Y2 ,
32
KAJI AND YASUKURA
since w H, Yi x s iYi for i s 1, 2. Therefore we have U s H q sY1 q s 2 Y2 N s g ⺓= 4 . Taking the closure of the image U in ⺠#Ž ᒄ ., we obtain
U s s02 H q s0 s1Y1 q s12 Y2 N Ž s0 : s1 . g ⺠ 1 4 : ⺠# Ž ᒄ . . Then we find
Ž G ⭈ H . l U / ⭋. Indeed, U is of course dense in U, and Ž G ⭈ H . l U is a non-empty open subset of U since H g Ž G ⭈ H . l U and G ⭈ H is an open subset of Sec X Ž ᒄ . Žsee wKOY, Proposition 5.3x.. Therefore we have U : G ⭈ H since exp ⺓ ⭈ ad H : G; hence Ž H q Y1 q Y2 . g G ⭈ H. LEMMA 3. We ha¨ e Ž Y1 q ᒄ 2 . : G ⭈ Y1 for any non-zero Y1 g ᒄ 1. Proof. Consider the root space decomposition of ᒄ 1 , ᒄ1 s
[
␣gR
ᒄ␣ ,
where R [ ␣ g RqN y ␣ g R4 , R is the system of roots, and Rq is the set of positive roots. According to this decomposition, we have Y1 s
Ý
␣gR
Y␣ ,
Y␣ g ᒄ ␣ .
It follows that Y␥ / 0 for some ␥ g R since Y1 / 0. Set  [ y ␥ g R. Then  g R and one can choose Z g ᒄ  such that w Z, Y␥ x s X since w ᒄ  , ᒄ ␥ x s ᒄ 2 . Moreover we see that if ␣ / ␥ , then w Z, ᒄ ␣ x s 0: Indeed, if w Z, ᒄ ␣ x / 0, then  q ␣ s since w Z, ᒄ ␣ x : w ᒄ 1 , ᒄ 1 x s ᒄ 2 s ᒄ with Z g ᒄ  . Therefore, Žexp sZ .Y␥ s Y␥ q sX , and Žexp sZ .Y␣ s Y␣ for any ␣ g R with ␣ / ␥ . Thus
Ž exp sZ . Y1 s Y1 q sX , and we obtain the conclusion since exp sZ g G. PROPOSITION 1. Any orbit in Sec X Ž ᒄ . through neither H nor X intersects ⺠#Ž ᒄ 1 .. In particular, any orbit in Sec X Ž ᒄ . not through H is nilpotent. Proof. According to Lemma 1, any orbit in Sec X Ž ᒄ . intersects ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 .. On the other hand, it follows from Lemma 2 that any orbit
SECANT VARIETIES
33
through ⺠#Ž⺓ ⭈ H [ ᒄ 1 [ ᒄ 2 . but not through H intersects ⺠#Ž ᒄ 1 [ ᒄ 2 ., and from Lemma 3 that any orbit through ⺠#Ž ᒄ 1 [ ᒄ 2 . but not through X intersects ⺠#Ž ᒄ 1 .. Therefore we obtain the result. A restriction of G 哭 ⺠#Ž ᒄ . yields a natural action, G 0 哭 ⺠# Ž ᒄ 1 . , where we set G 0 [ Int ᒄ 0 . Indeed, it follows from the Jacobi identity that w ᒄ i , ᒄ j x : ᒄ iqj . Since G 0 : G and ⺠#Ž ᒄ 1 . : Sec X Ž ᒄ ., we have a well-defined, natural map, ⌿ : orbits of G 0 哭 ⺠# Ž ᒄ 1 . 4 ª orbits of G 哭 Sec X Ž ᒄ . 4 . Then Proposition 1 yields that the image of ⌿ contains the set of orbits through neither X nor H. Remark. Ža. The image of ⌿ does not contain G ⭈ H, since ad H is not nilpotent. Žb. The image of ⌿ does not contain X Ž ᒄ . in general: In fact, according to wKY1x, ⺠#Ž ᒄ 1 . l X Ž ᒄ . s ⭋ if and only if ᒄ , ᒐ ᒉ 2 ⺓ or ᒄ , ᒐ ᒍ 2 n⺓ for some n. Žc. Using wKY1x, one can show that ⌿ is not injective if ᒄ , ᒐ ᒉ n⺓ for any n G 3. Moreover, it turns out that the converse is also true: In fact, we will give in a forthcoming paper wKY2x a complete description of the orbit decomposition of G 0 哭 ⺠#Ž ᒄ 1 .. PROPOSITION 2. For a semi-simple element E of a semi-simple Lie algebra ᒄ with G [ Int ᒄ, consider the projecti¨ ization G ⭈ E : ⺠#Ž ᒄ . of the adjoint orbit through E in ᒄ. Then the complement in its closure, G ⭈ E _ G ⭈ E, has pure codimension 1 in G ⭈ E. Proof. The orbit G ⭈ E is affine since it follow from wS, 4.4.5x that the orbit G ⭈ E is closed in the affine space ᒄ. On the other hand,
N G⭈E : G ⭈ E ª G ⭈ E is a quotient by CG Ž E .rCG Ž E ., where CG Ž E . and CG Ž E . respectively denote the stabilizers of E g ⺠#Ž ᒄ . and of E g ᒄ. Since E is semi-simple, CG Ž E .rCG Ž E . is a finite group Žsee, for example, wDS, Lemma 5.1x., so that N G⭈E is a finite morphism in the scheme-theoretic sense. Therefore, it follows from wHr1, II.1.5x that G ⭈ E is also affine. Now the claim follows from wHr1, II.3.1x.
34
KAJI AND YASUKURA
Remark. With the assumptions of Proposition 2, it follows from wHr1, II.6.2x that if dim G ⭈ E G 2, then the complement is connected.
2. PROOF OF THEOREM First of all, the statement Ža. of the theorem follows from Proposition 1 since the number of nilpotent orbits in a semi-simple ᒄ is finite, as is well known Žsee, for example, wCM, Theorem 3.5.4x.. In particular, it is obvious that there exist maximal elements of O0 , . . . , Or 4 . Next, for any nilpotent orbit O : ᒄ, we have that O : Sec X Ž ᒄ . if and only if there is a maximal element Oi of O0 , . . . , Or 4 such that O F Oi . On the other hand, Oi is a maximal element of O0 , . . . , Or 4 if and only if codim Ž Oi , Sec X Ž ᒄ . . s 1, since it follows from Proposition 2 that any irreducible component of Sec X Ž ᒄ . _ G ⭈ H has codimension 1 in Sec X Ž ᒄ . s G ⭈ H. Therefore, the statement Žc. is clear, once the uniqueness Žb. is established. Note that the condition above is equivalent to
Ž codim1.
dim Oi s 2 dim Omin y 2.
Indeed, it follows from wKOY, Theorem 5.1x that dim Sec X Ž ᒄ . s 2 dim X Ž ᒄ ., and we have in general dim O s dim O y 1 for a nilpotent orbit O : ᒄ. Thus, it suffices to show the statements Žb. and Žd.: we check the claims case by case. Let us start with C: For ᒄ s ᒐ ᒍ 2 n⺓, according to wCM, Theorem 5.1.3x, the nilpotent orbits correspond bijectively to the partitions of 2 n such that odd parts occur with even multiplicity, and the Hasse diagram for ᒐ ᒍ 2 n⺓ is as follows:
SECANT VARIETIES
35
We have Omin s Ow212 ny 2 x , and Ow2 2 12 ny 4 x is the only nilpotent orbit satisfying the condition Žcodim1. above since it follows from wCM, Corollary 6.1.4x that dim Ow2 2 12 ny 4 x s 4 n y 2 and dim Ow212 ny 2 x s 2 n. Therefore, Ow2 2 12 ny 4 x must be the unique maximal orbit. EG: It follows from wCM, Sect. 8.4x that OA 2 is the only nilpotent orbit satisfying Žcodim1. for the cases E6 , E7 and E8 , and OG 2 Ž a1 . for G 2 , while Omin s OA 1 for all exceptional ᒄ. Therefore, the orbit satisfying Žcodim1. must be the unique maximal orbit in each case. As we will see below, in the remaining cases, the nilpotent orbits satisfying Žcodim1. are not necessarily unique: For the classical cases, we need PROPOSITION 3. Assume that ᒄ is a subalgebra of ᒐ ᒉ N ⺓. Then for any orbit O : ᒄ if O : Sec X Ž ᒄ ., then rk O F 2 rk Omin , where rk O is defined to be the rank of any element in O as a matrix ¨ ia the embedding ᒄ : ᒐ ᒉ N ⺓. Proof. It suffices to show that rk A F 2 rk Omin for a general A g ᒄ such that A g Sec X Ž ᒄ .: Indeed, rk is a lower semi-continuous function on ⺠#Ž ᒄ .. Then we may assume that A g A 0 ) A1 for some A i g ᒄ with A i g X Ž ᒄ .. Replacing the A i with suitable scalar multiples Žif necessary ., we may assume moreover that A s A 0 q A1. Then we have rk A F rk A 0 q rk A1 s 2 rk Omin since A i g Omin . A: For ᒄ s ᒐ ᒉ n⺓, according to wCM, Theorem 5.1.1x, the nilpotent orbits correspond bijectively to the partitions of n, and the Hasse diagram for ᒐ ᒉ n⺓ is as follows:
We have Omin s Ow21ny 2 x. It follows from Proposition 3 that Ow2 31ny 6 x Sec X Ž ᒄ . since rk Ow2 31ny 6 x s 3 while rk Omin s 1. On the other hand, Ow31ny 3 x satisfies the condition Žcodim1. since it follows from wCM, Corol-
36
KAJI AND YASUKURA
lary 6.1.4x that dim Ow31ny 3 x s 4 n y 6 and dim Ow21ny 2 x s 2 n y 2. Therefore, for any n G 3, Ow31ny 3 x must be the unique maximal orbit. In case of n s 2 obviously Ow2x is the unique maximal orbit. BD: For ᒄ s ᒐ ᒌ n⺓, according to wCM, Theorems 5.1.2, 5.1.4x, the nilpotent orbits correspond bijectively to the partitions of n such that even parts occur with even multiplicity except that in case n is even, very even partitions d correspond to two orbits, denoted by OdI and OdII, where a partition d is said to be ¨ ery e¨ en if d has only even parts and each multiplicity is even. Therefore, the Hasse diagram for ᒐ ᒌ n⺓ is
where Ow2Ž I,4 1IIny. 8 x with n s 8 and Ow2Ž I,6 1IIny. 12 x with n s 12 are corresponding to the very even case, and decompose into two orbits. We have Omin s Ow2 2 1ny 4 x. It follows from Proposition 3 that neither Ow32 4 1ny 11 x nor Ow2 6 1ny 12 x are contained in Sec X Ž ᒄ . since rk Ow32 4 1ny 11 x s rk Ow2 6 1ny 12 x s 6 while rk Omin s 2. On the other hand, Ow3 2 1ny 6 x satisfies Žcodim1. since it follows from wCM, Corollary 6.1.4x that dim Ow3 2 1ny 6 x s 4 n y 14 and dim Ow2 2 1ny 4 x s 2 n y 6. Therefore if n G 6, then Ow3 2 1ny 6 x must be the unique maximal orbit. For the last case of type F4 we need the following notation: If ᒅ is a Cartan subalgebra of ᒄ and ⌬ s ␣ 1 , . . . , ␣ l 4 is a base of simple roots, then the weighted Dynkin diagram of H g ᒅ is defined to be the Dynkin diagram of ᒄ with ␣ i Ž H . inserted at the ith node. If H, X, Y 4 is a standard triple with H g ᒅ, then the weighted Dynkin diagram of the orbit through X is defined to be that of H.
SECANT VARIETIES
37
F: Assume ᒄ is of type F4 . It follows from wCM, Sect. 8.4x that OA 2 as well as OA˜2 satisfy Žcodim1. and their weighted Dynkin diagrams are
while OA 1 s Omin . It suffices to show that OA˜2 Sec X Ž F4 ., and the required result follows from Proposition 1 and PROPOSITION 4. Let ᒄ be the complex simple Lie algebra of type F4 with graded decomposition ᒄ s [y2 F iF 2 ᒄ i of contact type, and let ⌽i N 1 F i F 144 be the base of ᒄ 1 gi¨ en in wY, Appendixx. Then representati¨ es for the orbits of G 0 哭 ⺠#Ž ᒄ 1 . and the weighted Dynkin diagrams for nilpotent orbits in ᒄ through those representati¨ es are gi¨ en as follows:
Proof. Observe that if suffices to show that the nilpotent orbits in ᒄ through the four elements above have the required weighted Dynkin diagrams: Indeed, this implies that the four orbits in ᒄ 1 through those elements are distinct, while we have as a fact that the action G 0 哭 ⺠#Ž ᒄ 1 . has exactly four orbits. Actually, this fact would be best described in connection with the realization of Freudenthal’s magic square via the theory of Jordan algebras Žsee wFx; see also wM, Ž5.10., At, p. 22x.. A proof by means of the classification of Lie algebras is given as follows: We have a decomposition ᒄ 0 s ⺓ ⭈ H [ ᒄ SS 0 , w x where ᒄ SS 0 denotes the semi-simple part ᒄ 0 , ᒄ 0 of a reductive algebra ᒄ 0 . We see from the Dynkin diagram of ᒄ 0 that ᒄ SS 0 is a simple Lie algebra of type C3 Žsee, for example, wAs, p. 53x.. According to this decomposition, it
38
KAJI AND YASUKURA
follows from Weyl’s dimension formula Žsee, for example, wSK, Sect. 1x. that the action G 0 哭 ᒄ 1 turns out to be GL1⺓ = Sp6 ⺓ 哭 ⺓ m ⌫ Ž ⌳ . , where ⌫ Ž ⌳ . is the irreducible Sp6 ⺓-module of dimension 14 with highest weight ⌳ s 2 or ⌳ s 3 . Now it follows from wV, Proposition 2x that the number of orbits of G 0 哭 ᒄ 1 is finite. On the other hand, according to wSK, Proposition 20x, GL1⺓ = Sp6 ⺓ 哭 ⺓ m ⌫ Ž 2 . has infinitely many orbits. Thus ⌳ s 3 and the action in question is equivalent to GL1⺓ = Sp6 ⺓ 哭⺓ m ⌫ Ž 3 .. Now the claim follows from wI, Proposition 7x as well as wKm, Sect. 9x. This completes the proof of the fact. Now, we compute the weighted Dynkin diagrams for the orbits in ᒄ through the nilpotent elements above. Let ⌽i N 1 F i F 144 and H1 , . . . , H4 4 be, respectively, the bases of ᒄy1 and of a Cartan subalgebra ᒅ given in wY, Appendixx. Denote by 1 , . . . , 4 4 the dual of Hi ’s. Then it follows from wY, Appendixx that
1 q 2 ,
1 y 2 ,
3 q 4 ,
3 y 4
form a base of positive, strongly orthogonal roots, and we have four standard triples as follows: neutral
nil-positive
nil-negative
H1 q H 2 H1 y H 2 H3 q H4 H3 y H4
2 ⌽7 2 ⌽8 2 ⌽9 2 ⌽14
y2⌽ 1 y2 ⌽ 2 y2⌽ 3 y2 ⌽13
Recall that a base of roots for ᒅ* is strongly orthogonal if any sum or difference of two of its elements is never a root. From the standard triples above we obtain the following standard triples: neutral
nil-positive
nil-negative
H1 q H 2 2 H1 2 H1 q H3 q H4 2Ž H1 q H3 .
2 ⌽7 2Ž ⌽ 7 q ⌽ 8 . 2Ž ⌽ 7 q ⌽ 8 q ⌽ 9 . 2Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14.
y2 ⌽ 1 y2Ž ⌽ 1 q ⌽ 2 . y2Ž ⌽ 1 q ⌽ 2 q ⌽ 3 . y2Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13.
39
SECANT VARIETIES
Indeed, it follows from the strong orthogonality that, for example, 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . s 2 H1 q 2 H 3 , 2 Ž H1 q H3 . , 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . , 2 Ž ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽14 . s H1 q H2 , 2 ⌽ 7 q H1 y H2 , 2 ⌽ 8 q H3 q H4 , 2 ⌽ 9 q H3 y H4 , 2 ⌽14 s 4⌽ 7 q 4⌽ 8 q 4⌽ 9 q 4⌽14 , 2 Ž H1 q H3 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s Ž H1 q H2 . q Ž H1 y H2 . q Ž H3 q H4 . q Ž H3 y H4 . , y2 Ž ⌽ 1 q ⌽ 2 q ⌽ 3 q ⌽13 . s w H1 q H2 , y2⌽ 1 x q w H1 y H2 , y2⌽ 2 x q w H3 q H4 , y2⌽ 3 x q H3 y H4 , y2⌽13 s 4⌽ 1 q 4⌽ 2 q 4⌽ 3 q 4⌽13 . Now set
␣1 [ 2 y 3 , ␣4 [
␣ 2 [ 3 y 4 , 1 2
␣ 3 [ 4 ,
Ž 1 y 2 y 3 y 4 . .
Then, it follows from wY, Appendixx that ⌬ s ␣ 1 , . . . , ␣ 4 4 forms a base of simple roots, and the Dynkin diagram is as follows:
40
KAJI AND YASUKURA
Therefore the weighted Dynkin diagrams of H1 q H2 , 2 H1 , 2 H1 q H3 q H4 , 2 H1 q 2 H3 are as follows:
In particular, H1 q H2 and 2 H1 are dominant with respect to ⌬. Therefore, the weighted Dynkin diagrams of H1 q H2 and 2 H1 above respectively give those of nilpotent orbits through ⌽ 7 and ⌽ 7 q ⌽ 8 , as is required. On the other hand, we see that ␣ 2 ␣ 1Ž2 H1 q H3 q H4 . is dominant though 2 H1 q H3 q H4 is not, where ␣ denotes the reflection with respect to a simple root ␣ : Indeed, it follows from the recipe below that
Therefore, the last diagram corresponding to ␣ 2 ␣ 1Ž2 H1 q H3 q H4 . gives the weighted Dynkin diagram of the nilpotent orbit through ⌽ 7 q ⌽ 8 q ⌽ 9 , as is required. Similarly we see that ␣ 1Ž2 H1 q 2 H3 . is dominant since
and the last diagram gives the weighted Dynkin diagram of the nilpotent orbit through ⌽ 7 q ⌽ 8 q ⌽ 9 q ⌽ 14. Thus we obtain the required result. RECIPE. Let c be the coefficient of the ith node in the diagram for H abo¨ e. To compute ␣ iŽ H ., change c to yc at the ith node and add yc ⭈ 2Ž ␣ j , ␣ i .rŽ ␣ i , ␣ i . at e¨ ery jth node adjacent to the ith node. Ž Here Ž⭈, ⭈ . is any positi¨ e definite bilinear form which is in¨ ariant under the action of the Weyl group.. Obser¨ e that this is just c itself unless ␣ j is longer than ␣ i .
41
SECANT VARIETIES
This follows from the formula for a reflection: Indeed, we have ␣ j Ž ␣ iŽ H .. s ␣ Ž H . with ␣ [ ␣ iŽ ␣ j . s ␣ j y 2Ž ␣ j , ␣ i .rŽ ␣ i , ␣ i . ⭈ ␣ i . 3. FINAL REMARK Let ᒄ be a complex simple Lie algebra. For a graded decomposition ᒄs
[
y FiF
ᒄi
of order with characteristic element Z, consider the coset space, M [ GrCG Ž Z . , where we set G [ Int ᒄ and CG Ž Z . is the stabilizer of Z g ᒄ, as before. In wKn1, Sect. 3x Žsee also wKn3, 3.2x., Kaneyuki gave a G-equivariant ˜ Moreover in case of s 1, he compactification of M, denoted by M. ˜ and found that the number of obtained the orbit decomposition of M ˜ orbits in M is equal to rk M q 1, where rk M is the split rank of M as a simple reducible pseudo-hermitian symmetric space. In this context, our work in the case of rk ᒄ G 2 is closely related to the case of the graded decomposition of contact type: In fact, consider M corresponding to the graded decomposition of contact type. Then the secant variety Sec X Ž ᒄ . is a G-equivariant compactification of M CG Ž H . rCG Ž H .
,
where CG Ž H . is the stabilizer of H g ⺠#Ž ᒄ . and H is the characteristic element for the gradation of contact type. Indeed, we have GrCG Ž H . s G ⭈ H and it follows from wKOY, Proposition 5.3x that G ⭈ Hs Sec X Ž ᒄ ., which allows the action of G. For the case ᒄ s ᒐ ᒉ 2 ⺓, it is easily shown that X Ž ᒐ ᒉ 2 ⺓. is a conic in ⺠#Ž ᒐ ᒉ 2 ⺓. s ⺠ 2 with Sec X Ž ᒐ ᒉ 2 ⺓. s ⺠#Ž ᒐ ᒉ 2 ⺓., and GrCG Ž H . s G ⭈ H coincides with the complement of X Ž ᒐ ᒉ 2 ⺓. Žsee wKOY, Proposition 4.1x.. This corresponds to the basic example of Kaneyuki’s theory Žsee wKn1, p. 333x.. Note that CG Ž H .rCG Ž H . is a finite group since H is semi-simple. In fact, we have PROPOSITION 5. If H is the neutral element of a standard triple corresponding to Omin , gi¨ en in Section 1, then
Ž CG Ž H . : CG Ž H . . s 2.
42
KAJI AND YASUKURA
Proof. It suffices to show that
y1 H l G ⭈ H s H , yH 4 . The inclusion : is obtained by following the argument in the proof of wDS, Lemma 5.1x. Indeed, if H g G ⭈ H with g ⺓, then ad H and ad H have the same set of eigenvalues, which is equal to 0, " 24 in case of ᒄ s ᒐ ᒉ 2 ⺓, and 0, " 1, " 24 otherwise. Therefore, it follows that s "1. To show the converse, we first consider the case of ᒄ s ᒐ ᒉ 2 ⺓. We may assume that ᒄ is the set of traceless matrices of size 2 with G s PSL2 ⺓ and Hs
1 0
0 . y1
Then we have G ⭈ H 2 g ⭈ H s gHgy1 s yH g y1 H, where g g G is represented by g[
0 y1
1 g SL 2 ⺓. 0
Next, for arbitrary ᒄ, consider the embedding
: PSL2 ⺓ ¨ G associated to the composition of ᒐ ᒉ 2 ⺓ , ᒄy2 [ ⺓ ⭈ H [ ᒄ 2 : ᒄ . Then, using g g PSL2 ⺓ above, we have g ⭈ H s yH with g g G, and we obtain the result. Remark. For another proof of this result, see wKY1, Sect. 3, Corollaryx.
REFERENCES wAsx wAtx wBC1x wBC2x
H. Asano, Symplectic triple systems and simple Lie algebras Žin Japanese., RIMS Kokyuroku, Kyoto Uni¨ . 308 Ž1977., 41᎐54. K. Atsuyama, Freudenthal’s projective geometry Žin Japanese., in ‘‘Projective VarietiesrProjective Geometry of Algebraic Varieties 2 q Singularities,’’ Proc. Symp. Algebraic Geometry, pp. 1᎐62, Waseda Univ., Japan, 1997. P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups, I, Math. Proc. Cambridge Philos. Soc. 79 Ž1976., 401᎐425. P. Bala, R. W. Carter, Classes of unipotent elements in simple algebraic groups, II, Math. Proc. Cambridge Philos. Soc. 80 Ž1976., 1᎐17.
SECANT VARIETIES
wBvx
43
A. Beauville, Fano contact manifolds and nilpotent orbits, Comment. Math. Hel¨ . 73 Ž1998., 566᎐583. wB1x W. Boothby, Homogeneous complex contact manifolds, Amer. Math. Soc. Proc. Symp. Pure Math. 3 Ž1961., 144᎐154. wB2x W. Boothby, A note on homogeneous complex contact manifolds, Proc. Amer. Math. Soc. 13 Ž1962., 276᎐280. ´´ wBrx N. Bourbaki, ‘‘Elements de Mathematique, Groupes et Algebres de Lie,’’ Chapters ´ ` 4, 5, and 6, Hermann, Paris, 1968. wCx R. W. Carter, ‘‘Finite Groups of Lie TypeᎏConjugacy Classes and Complex Characters,’’ Wiley, Chichester, 1993. wCMx D. H. Collingwood and W. M. McGovern, ‘‘Nilpotent Orbits in Semisimple Lie Algebras,’’ Van Nostrand Reinhold, New York, 1993. wDSx A. Dancer and A. Swann, ‘‘Quaternionic Kahler Manifolds of Cohomogeneity One,’’ ¨ preprint, math.DGr9808097, 1998. wFx H. Freudenthal, Lie groups in the foundations of geometry, Ad¨ . Math. 1 Ž1964., 145᎐190. wFRx T. Fujita and J. Roberts, Varieties with small secant varieties: The extremal case, Amer. J. Math. 103 Ž1981., 953᎐976. wFHx W. Fulton, J. Harris, ‘‘Representation Theory: A First Course,’’ GTM, Vol. 129, Springer-Verlag, New York, 1991. wHr1x R. Hartshorne, ‘‘Ample Subvarieties of Algebraic Varieties,’’ Lecture Notes in Math., Vol. 156, Springer-Verlag, New York, 1970. wHr2x R. Hartshorne, ‘‘Algebraic Geometry,’’ GTM, Vol. 52, Springer-Verlag, New York, 1977. wHm1x J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ GTM, Vol. 9, Springer-Verlag, New York, 1972. wHm2x J. E. Humphreys, ‘‘Linear Algebraic Groups,’’ GTM, Vol. 21, Springer-Verlag, New York, 1975. wIx J.-i. Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 91 Ž1970., 997᎐1028. wKOYx H. Kaji, M. Ohno, and O. Yasukaru, Adjoint varieties and their secant varieties, Indag. Math. 10 Ž1999., 45᎐57. wKY1x H. Kaji, O. Yasukura, ‘‘Projective Geometry of Adjoint Varieties,’’ preprint, 1998. wKY2x H. Kaji and O. Yasukura, The orbit structure of simple graded Lie algebras of contact type, in preparation. wKn1x S. Kaneyuki, Homogeneous symplectic manifolds and dipolarizations in Lie algebras, Japan. J. Math. 13 Ž1987., 333᎐370. wKn2x S. Kaneyuki, Graded Lie algebras, related geometric structures and pseudo-Hermitian symmetric spaces, in ‘‘Analysis and Geometry on Complex Homogeneous Domains and Related Topics,’’ Progress in Math., Birkhauser, BaselrBoston, to ¨ appear. wKmx T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 Ž1982., 1᎐80. wLVx R. Lazarsfeld and A. Van de Ven, ‘‘Topics in the Geometry of Projective SpaceᎏRecent Work of F. L. Zak,’’ DMV Sem. 4, Birkhauser, BaselrBoston, 1984. ¨ wMx S. Mukai, Projective geometry of homogeneous spaces Žin Japanese., in ‘‘Projective VarietiesrProjective Geometry of Algebraic Varieties,’’ Proc. Symp. Algebraic Geometry, pp. 1᎐52, Waseda University, Japan, 1994.
44 wSKx wSx wSSx wVx wYx wZx
KAJI AND YASUKURA
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 Ž1977., 1᎐155. T. A. Springer, ‘‘Linear Algebraic Groups,’’ Progress in Math., Vol. 9, Birkhauser, ¨ BaselrBoston, 1981. T. Springer and R. Steinberg, Conjugacy classes, in ‘‘Seminar on Algebraic Groups and Related Finite Groups,’’ Lecture Notes in Math., Vol. 131, pp. 167᎐266, Springer-Verlag, New York, 1970. E. B. Vinberg, The Weyl group of a graded Lie algebra, Iz¨ . Akad. Nauk SSSR Ser. Mat. 40, 463᎐495. K. Yamaguti, On weak representations of a class of ternary systems, Mem. Fac. Gen. Ed. Kumamoto Uni¨ . Natur. Sci. 9 Ž1974., 1᎐8. F. L. Zak, ‘‘Tangents and Secants of Algebraic Varieties,’’ Translations of Math. Monographs, Vol. 127, Am. Math. Soc., Providence, 1993.