- Email: [email protected]
Quotients of Some Prehomogeneous Vector Spaces
192, 61]73 Ž1997. JA966979
JOURNAL OF ALGEBRA ARTICLE NO.
Quotients of Some Prehomogeneous Vector Spaces Hiroyuki Ochiai* Department of Mathematics, Rikkyo Uni¨ ersity, Nishi-Ikebukuro, Tokyo 171, Japan Communicated by Robert Steinberg Received December 7, 1995
We give quotients VrrG s SpecŽCw V xG . for some prehomogeneous vector spaces Ž G = H, V .. Q 1997 Academic Press
1. INTRODUCTION 1.1. Let C be an algebraically closed field of characteristic zero. For a prehomogeneous vector space Ž G, r , V ., the quotient VrrG s SpecŽCw V xG . in the sense of geometric invariant theory w12x reduces to one point. Nothing interesting happens Žat least over an algebraically closed field.. Let N be a normal subgroup of G, and consider the quotient VrrN s SpecŽCw V x N .. It can be a nontrivial variety with an action of the quotient group GrN. Now we consider a special situation and change the notation slightly. Let us consider a prehomogeneous vector space of the form Ž G = H, r , V .. The quotient VrrG s SpecŽCw V xG . is an H-variety. A representation V of G is called coregular if the ring Cw V xG of invariants is isomorphic to a polynomial ring. If Ž G, r , V . is coregular, then it is easy to see ŽProposition 6.1. that the action of H on the affine space VrrG is linear. That is, we know the existence of the quotient prehomogeneous ¨ ector space Ž H, r , VrrG. in such a case. Several problems for Ž G = H, r , V . are expected to be reduced to that for the quotient Ž H, r , VrrG .. For example, we can construct a relative invariant of a prehomogeneous vector space Ž G = H, r , V . using a relative invariant of the quotient Ž H, r , VrrG .. * E-mail: [email protected]. 61 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
62
HIROYUKI OCHIAI
In the 1970s, Sato and Kimura w15x completed the classifications of irreducible reduced prehomogeneous vector spaces. At the same time, the relative invariants are constructed for almost all of these spaces. However, for some complicated prehomogeneous vector spaces, numbers Ž10., Ž21., and Ž24., the construction of relative invariants had not been settled. In 1990, Gyoja w4x constructed relative invariants explicitly for these spaces Ž10.: Ž SLŽ5. = GLŽ3., L 2 m L 1 ., Ž21.: Ž SpinŽ10. = GLŽ3., half-spin m L 1 ., and Ž24.: Ž GSpinŽ14., half-spin. by some complicated calculations. In this paper, we construct relative invariants for the spaces Ž10. and Ž21. using the quotient VrrG. Although the ingredients of the argument have already appeared in w3x or w4x, it seems new to stress the role of the quotient. Moreover, we can also construct several basic relative invariants for some related prehomogeneous vector spaces ŽProposition 5.1.. This construction is expected to be useful in obtaining an explicit formula of the functional equations of the associated zeta functions w13x. It is also helpful for orbit decompositions over a number field w14x. Observe that this method is not effective for the space Ž24. because GSpinŽ14. has only one simple factor. 1.2. Let Ž G = H, r , V . be a prehomogeneous vector space such that the quotient W [ VrrG is also a vector space. Then Ž H, r , W . is a prehomogeneous vector space such that the projection to the second factor of the generic isotropy of G = H at V coincides with the generic isotropy of H at W. The quotient map c : V ª W is surjective and G = H-equivariant. Conversely ŽTheorem 2.2., if two prehomogeneous vector spaces Ž G = H, r , V . and Ž H, r , W . are connected by a surjective equivariant morphism c : V ª W and they satisfy the condition above on the generic isotropy subgroups, then Ž H, r , W . is the quotient of Ž G = H, r , V .. 1.3. We explain the organization of the paper. In Section 2, we give a sufficient condition ŽTheorem 2.2. to obtain the quotient. We construct the equivariant surjective map c for the spaces Ž10., Ž20., and Ž21. in Sections 3 and 4. In Section 5, we give an application of the description of the quotient and Section 6 is devoted to proving the existence of an equivariant quotient. Notation. We denote the mth symmetric Žresp. skew symmetric. tensor representation of C n by S m ŽC n . Žresp. Lm ŽC n ... We often identify S 2 ŽC n . with the set SymŽ n. s X g M Ž n. < t X s X 4 of symmetric matrices, and L2 ŽC n . with the set Alt Ž n. of skew symmetric matrices.
PREHOMOGENEOUS VECTOR SPACES
63
A vector space or a linear algebraic group is defined over C. Note that, for example, the symbol SO Ž m. means a complex special orthogonal group SO Ž m, C., instead of a Žusual. real special orthogonal group SO Ž m, R..
2. MAIN THEOREM First we recall the notion of equivariant quotient of an affine variety. DEFINITION 2.1. Let G, H be linear algebraic groups and V, W be affine algebraic varieties. Suppose V is a G = H-variety and W is an H-variety. If c : V ª W induces an isomorphism ;
c *: C w W x ª C w V x
G
as C-algebras and H-modules, then the affine variety W Žor, more pre; cisely, the induced morphism c : VrrG ª W . is called the H-equi¨ ariant quotient of V by G. We introduce further standard terminology on prehomogeneous vector spaces. For a prehomogeneous vector space Ž G, r , V ., the open G-orbit in V is denoted by V s V V , and its complement by S s SV [ V y V V . A point ¨ in the open orbit V is called a generic point and the Žisomorphism class of. G¨ is called a generic isotropy. The following theorem is the main result of the abstract part of this paper. Let Ž G = H, r , V . and Ž H, r , W . be prehomogeneous
THEOREM 2.2.
¨ ector spaces and c : V ª W be a surjecti¨ e polynomial map such that
c Ž r Ž g , h. ¨ . s r Ž h. Ž c Ž ¨ . .
for all g g G, h g H , ¨ g V .
Let p H : G = H ª H be the second projection, p H ŽŽ g, h.. s h, and suppose the condition
Ž a.
p H Ž Ž G = H . ¨ . s Hc Ž ¨ .
for a ¨ g V V .
;
Then c : VrrG ª W is an H-equi¨ ariant quotient. Proof. It is enough to show the surjectivity of c *. First of all, we have V W s H ? c Ž ¨ . ( HrHc Ž ¨ . , V VrrG s Ž G = H . ? ¨ rrG ( Ž G = H . r Ž G = H . ¨ rrG s Hrp H Ž Ž G = H . ¨ . . By condition Ža., V W ( V VrrG.
64
HIROYUKI OCHIAI
Take some f g Cw V xG . Since Cw V xG ; Cw V V xG ( Cw V W x, there exists an f˜g Cw V W x such that f s f˜( c . We want to prove f˜g Cw W x. Let SW s S1 j ??? j Sl j S9 be the decomposition of SW such that the Si ’s are the irreducible components of codimension one and S9 is the union of components of codimension greater than one. Note that Si s w g W < pi Ž w . s 04 with an appropriate basic relative invariant pi . l Let p s Ł is1 pi . Then there is an integer N G 0 such that p N ? f˜ is regular outside of S9. Since S9 has codimension at least two, p N ? f˜g Cw W x. Suppose N G 1. We set g [ p N ? f˜g Cw W x, then g ( c s Ž p( c . N ? f. Since c is surjective, for a w g W, there is a ¨ g V such that w s c Ž ¨ .. Then g Ž w . s g Ž c Ž ¨ .. s pŽ c Ž ¨ .. N f Ž ¨ . s pŽ w . N f Ž ¨ .. Then g vanishes on S1 j ??? j Sl s w g W < pŽ w . s 04 . Again grp is regular outside of S9, g can be divided by p in Cw W x. That is, p Ny 1 ? f˜g Cw W x. By an induction on N, we have f˜g Cw W x. Condition Ža. in Theorem 2.2 may be stated in the following form: LEMMA 2.3. conditions:
Let ¨ g V V be a generic point and consider the following
Ža9. p H ŽŽ G = H . ¨ . > Hc Ž ¨ . . Ža1. Each connected component of Hc Ž ¨ . meets p H ŽŽ G = H . ¨ .. Ža2. dim p H ŽŽ g [ h . ¨ . s dim h c Ž ¨ . . Ža29. dim G y dim G¨ s dim V y dim W. Ži. Žii. Žiii.
Condition Ža. is equi¨ alent to Ža9.. Conditions Ža1. and Ža2. are equi¨ alent to Ža.. Condition Ža2. is equi¨ alent to Ža29..
Proof. Ži. It is easy to see
p H Ž Ž G = H . ¨ . ; Hc Ž ¨ .
for ¨ g V V .
Žii. Clear. Žiii. The left-hand side of Ža2. is dim Ž g [ h . ¨ y dim Ž g ¨ . s dim Ž G = H . y dim V y dim G¨ . The right-hand side of Ža2. is dim H y dim W. We give an example of Theorem 2.2. PROPOSITION 2.4.
For m ) n,
M Ž m, n . rrSO Ž m . 2 X ¬t XX g S 2 Ž C n . is a GLŽ n.-equi¨ ariant isomorphism.
PREHOMOGENEOUS VECTOR SPACES
65
The result is, of course, well known. This is easily derived from the classical invariant theory w5, 18x. However, we will give a proof using Theorem 2.2 in order to illustrate the examples in Sections 3 and 4 below. Proof. It is enough to check condition Ža. in Theorem 2.2 for G s SO Ž m., H s GLŽ n., V s M Ž m, n., W s SymŽ n., and c : V ª W defined by c Ž X . st XX. It is easily verified that c is surjective and that condition Ža2. holds. We prove condition Ža1.. Take a generic point ¨ s Ž I0n . g V. Then c Ž ¨ . s In g W and Hc Ž ¨ . s O Ž n., which has two connected components. It is enough to prove that diagŽy1, 1, . . . , 1. g O Ž n. can be obtained by p H ŽŽ G = H . ¨ .. In fact, we have
Ž diag Ž y1, 1, . . . , 1, y1. , diag Ž y1, 1, . . . , 1. . g Ž SO Ž m . = GL Ž n . . ¨ . 3. THE SPACE Ž10.: Ž SLŽ5. = GLŽ3., M Ž10, 3.. In this section, we will construct the map c for the prehomogeneous vector space Ž SLŽ5. = GLŽ3., L 2 m L 1 , M Ž10, 3... 3.1. In this section, x, y will be elements of Alt Ž5. ( L2 ŽC 5 . ( C 10 . First define an SLŽ5.-equivariant symmetric bilinear function
b : Alt Ž 5 . = Alt Ž 5 . ª Ž C 5 . * by
bi Ž x, y . [ x iq1 iq2 yiq3 iq4 y x iq1 iq3 yiq2 iq4 q x iq1 iq4 yiq2 iq3 q yiq1 iq2 x iq3 iq4 y yiq1iq3 x iq2 iq4 q yiq1 iq4 x iq2 iq3 s
1 4
Ý Ž sgn s . x iq s Ž1., iq s Ž2. yiq s Ž3., iq s Ž4.
sgS 4
for i s 1, . . . , 5. Here indices are numbered modulo 5. In w4x, bi is written as g i . As is known, ŽC 5 .* ( L4 ŽC 5 . appears in L2 ŽC 2 . m L2 ŽC 2 . with multiplicity one. The projection to this irreducible component is nothing but b . We can also give b in terms of the 4 by 4 principal minor Pfaffian w8x. For an x g Alt Ž5. and for i s 1, . . . , 5, define x Ž i. g Alt Ž4. by the matrix obtained by deleting the ith row and ith column from x. Then we have Žy1. iy1bi Ž x, x . s 2 PfŽ x Ž i. . and Žy1. iy1bi Ž x, y . s PfŽŽ x q y .Ž i. . y PfŽ x Ž i. .
66
HIROYUKI OCHIAI
y PfŽ y Ž i. .. That is, bi is a symmetric bilinear form corresponding to the quadratic form 2 PfŽ?Ž i. .. 3.2. Next we define an SLŽ5.-equivariant trilinear mapping ² ? < ? < ? : : Ž C 5 . * = Alt Ž 5 . = Ž C 5 . * ª C by Ž j , u, h . ¬ ² j < u
¡Ž a.
Ž b. Ž c. ~ Ž d. Ž e. Ž f. Ž g.
¢
²h < u < j : s y² j < u
Here x, y, z, u g Alt Ž5., j , h g ŽC 5 .*. Proof. Although these are proved in w4, Sect. 2x, we repeat the proof briefly. Ža. and Žb. follow from u g Alt Ž5.. Žd. is the differential of Žc. in the direction x y . To be more precise, substitute y q « x in y of Žc., and look at the coefficient of « 1. If we set h s b Ž x, y . in Žd., we have Že. with the help of Žb.. Žf. is ‘‘z x Že.’’ and Žg. is ‘‘z y Žd..’’ Now we prove Žc.. ² b Ž y, y . < x < j s
5
Ý bi Ž y, y . x i j s 2 Pf Ž A . , is1
where A s Ž a p q . g Alt Ž6. is defined by a p q s y p q , a p6 s x p j , a6 q s x jq if p, q F 5, and a66 s 0. In particular, Ž² b Ž y, y .< y < j . 2 s PfŽ A. 2 s detŽ A. s 0. 3.3. Now we define a map c : M Ž10, 3. ª SymŽ3.. We often identify an element of M Ž10, 3. with a triplet Ž x, y, z . of three alternative matrices in Alt Ž5.. c Ž x, y, z . [
ž
2² b Ž x, x . < z < b Ž x, y . : ² b Ž x, x . < z < b Ž y, y . : ² b Ž z, z . < y < b Ž x, x . : ² b Ž x, x . < z < b Ž y, y . : 2² b Ž y, y . < x < b Ž y, z . : ² b Ž y, y . < x < b Ž z, z . : . ² b Ž z, z . < y < b Ž x, x . : ² b Ž y, y . < x < b Ž z, z . : 2² b Ž z, z . < y < b Ž z, x . :
/
PREHOMOGENEOUS VECTOR SPACES
67
For e¨ ery g g SLŽ5., h g GLŽ3., Ž x, y, z . g M Ž10, 3., we
LEMMA 3.2. ha¨ e
c Ž gx, gy, gz . s c Ž x, y, z . , t
t
c Ž Ž x, y, z . h . s det Ž h . h c Ž x, y, z . h.
Ž 3.1.
Proof. We may prove the equivariance Ž3.1. for h’s of one of the 1 « 0
a 0 0
following three forms: h s 0 1 0 , 0 1 0 , or permutation matrices. 0 0 1 0 0 1 Checking Ž3.1. for diagonal or permutation matrices is easy. For h s 1 « 0 0 1 0 , we use relations Ža. ] Žg. listed in Section 3.2.
ž / ž /
ž /Ž 0 0 1
The 1, 1.-entry of c ŽŽ x, y, z . t h. s c Ž x q « , y, z . is calculated as 2² b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . < z < b Ž x, y . : q 2 « ² b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . < z < b Ž y, y . : Ž b.
s 2² b Ž x, x . < z < b Ž x, y . : q 2 « ² b Ž x, x . < z < b Ž y, y . : q Ž 2 « 2 y 4« 2 . ² b Ž y, y . < z < b Ž x, y . : s 2² b Ž x, x . < z < b Ž x, y . : q 2 « ² b Ž x, x . < z < b Ž y, y . : q 2 « 2 ² b Ž y, y . < x < b Ž y, z . : by using Ža. and Žf.. The Ž1, 3.-entry is ² b Ž z, z . < y < b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . : Žc.
s ² b Ž z, z . < y < b Ž x, x . : q 2 « ² b Ž z, z . < y < b Ž x, y . : s ² b Ž z, z . < y < b Ž x, x . : q « ² b Ž y, y . < x < b Ž z, z . : by Žd. and Ža.. Similarly, the Ž1, 2.-entry is checked by Žb., the Ž2, 2.-entry by Žc., the Ž2, 3.-entry by Žc., and the Ž3, 3.-entry by Že.. LEMMA 3.3. The map c is surjecti¨ e. Proof. Let ¨ i g C 5 Ž i s 1, . . . , 5. be the standard unit vector, let Ei j g Ž M 5. be the standard matrix unit, and define Yi j s Ei j y Eji g Alt Ž5. for a moment. We set x s Y12 q Y34 , y s Y23 q Y45 , z s Y13 q Y25 , z9 s Y15 , and z0 s Y13 . Then we have b Ž x, x . s 2 ¨ 5 , b Ž y, y . s 2 ¨ 1 , b Ž z, z . s 2 ¨ 4 , b Ž x, y . s ¨ 3 , b Ž x, z . s ¨ 1 , b Ž y, z . s ¨ 2 , b Ž x, z9. s ¨ 2 , b Ž y, z9. s ¨ 4 , b Ž y, z0 . s ¨ 2 , b Ž x, z0 . s 0, b Ž z9, z9. s 0, and b Ž z0, z0 . s 0. By a direct calculation, ² b Ž y, y .< x < s 2 ¨ 2 , ² b Ž x, x .< z < s y2 ¨ 2 , ² b Ž z, z .< y < s 2 ¨ 5 , ² b Ž x, x .< z 9 < s y2 ¨ 1 , ² b Ž x, x .< z 0 < s 0, ² b Ž z 9, z 9.< y < s 0, and ² b Ž z0, z0 .< y < s 0.
68
HIROYUKI OCHIAI
Finally, we
c Ž x, y, z0 . s
get
c Ž x, y, z . s
ž
0 0 4 0 4 0 4 0 0
/,
c Ž x, y, z 9. s
ž
0 y4 0 y4 0 0 0 0 0
/,
0 0 0 0 4 0 0 0 0
c Ž x, y, 0. s 0. Since the map c is GLŽ3.equivariant, the image is a GLŽ3.-stable subset of SymŽ3. containing the four elements above. It must be SymŽ3. itself.
ž / , and
THEOREM 3.4. The algebra homomorphism
c *: C Sym Ž 3 . ª C M Ž 10, 3 .
S L Ž5 .
is bijecti¨ e. That is, c gi¨ es a GLŽ3.-equi¨ ariant isomorphism: ;
Ž C 3 m L2 C 5 . rrSL Ž 5 . ª S 2 Ž C 3 . . Proof. We will apply Theorem 2.2 for G s SLŽ5., H s GLŽ3., V s L C 5 m C 3 , W s S 2 ŽC 3 . m det, and c : V ª W defined above. Condition Ža2. can be easily seen by w15x. In fact, by a straightforward calculation, the generic isotropy G¨ turns out to be a finite group. This yields Ža29.. Finally, we have to check condition Ža1.. The generic isotropy Hc Ž ¨ . is isomorphic to th < h g SO Ž3., t g GLŽ1., t 5 s 14 ( SO Ž3. = Z 5 . It is enough to construct an element of Ž G = H . ¨ whose image under p H has a nontrivial factor in Z 5 . Take a fifth root of unity z and consider Ž z I5 , zy2 I3 .. It is easy to see that it acts on V trivially and p H ŽŽ z I5 , zy2 I3 .. generates Z 5 . 2
The irreducible relative invariant F in w4, Theorem 1x is given by F Ž x, y, z . s y 12 detŽ c Ž x, y, z ... 4. THE SPACE Ž21.: Ž SpinŽ10. = GLŽ3., M Ž16, 3.. In this section, we will construct the map c for the prehomogeneous vector space Ž SpinŽ10. = GLŽ3., half-spin m L 1 , M Ž16, 3... The argument is parallel to the previous section. The calculation below gives the quotient for the space Ž20.: Ž SpinŽ10. = GLŽ2., half-spin m L 1 , M Ž16, 2... 4.1. First we define a SpinŽ10.-equivariant symmetric bilinear function
b : C 16 = C 16 ª C 10 . Here C 16 is a half-spin representation of SpinŽ10. and C 10 is a vector representation of SO Ž10.. The explicit form of b is also given in w3x and w4, p. 442x.
69
PREHOMOGENEOUS VECTOR SPACES
Next we define an SO Ž10.-invariant symmetric bilinear form ² ? , ? : : C 10 = C 10 ª C by Ž j , h . ¬ ² j , h : [ Ý5is1Ž j ihiq5 q j iq5hi .. 4.2. Now we define a map
c : M Ž 16, 3 . ª Sym Ž 3 . . We will identify an element of M Ž16, 3. with a triplet Ž x 1 , x 2 , x 3 . of three vectors in C 16 . For an x s Ž x 1 , x 2 , x 3 . g M Ž16, 3., we define
½
ai s ai Ž x . [ b Ž x i , x i . , bi s bi Ž x . [ b Ž x j , x k .
for i , j, k 4 s 1, 2, 3 4 .
These satisfy the following relations:
¡² a , a : s 0, i
i
² a i , bj : s 0
~²
for i / j,
a i , bi : s y2² bj , bk :
for i , j, k 4 s 1, 2, 3 4 ,
¢² a , a : s y2² b , b : i
j
k
for i , j, k 4 s 1, 2, 3 4 .
k
Using the above notation, we define c Ž x . s Ž c i j Ž x .. by c i j Ž x . [ Žy1. d i j ² bi Ž x ., bj Ž x .:. Here the Kronecker delta is d ii s 1 and d i j s 0 for i / j. LEMMA 4.1. ha¨ e
For g g SpinŽ10., h g GLŽ3., Ž x 1 , x 2 , x 3 . g M Ž16, 3. we
c Ž gx 1 , gx 2 , gx 3 . s c Ž x 1 , x 2 , x 3 . ,
Ž 4.1.
c Ž Ž x 1 , x 2 , x 3 . t h . s det h2 t hy1c Ž x 1 , x 2 , x 3 . hy1 .
Proof. As in the proof of Lemma 3.2, it is enough to prove Ž4.1. for hs
1 « 0 0 1 0 0 0 1
ž / . That is, c Ž x1 q « x 2 , x 2 , x 3 . s
ž
1 y« 0
0 1 0
0 0 1
/
c Ž x1 , x 2 , x 3 .
ž
1 0 0
y« 1 0
0 0 1
/
.
Ž 4.2.
70
HIROYUKI OCHIAI
The Ž2, 3.-entry of Ž4.2. is y² b 2 q « b1 < b 3 q « a2 : s y² b1 < b 3 : y « Ž ² b1 < b 3 : q ² b 2 < a2 : . y « 2 ² b1 < a2 : s y² b1 < b 3 : y « Ž y² b1 < b 3 : . . The Ž2, 2.-entry of Ž4.2. is ² b 2 q « b1 < b 2 q « b1 : s ² b 2 < b 2 : y 2 « Ž y² b1 < b 2 : . q « 2 ² b1 < b1 : . The other cases are similar. LEMMA 4.2. The map c is surjecti¨ e. Proof. We employ the notation in w15, 4x for spinors and let ¨ i g C 10 Ž i s 1, . . . , 10. be the standard unit vector. We set x 1 s 1 q e1 e2 e3 e4 , x 2 s e1 e5 q e2 e3 e 4 e5 , x 3 s e1 e 2 q e1 e 3 e4 e5 , xX3 s e1 e 2 q e 3 e5 q e1 e 2 e 4 e5 . Then b Ž x 1 , x 2 . s ¨ 1 q ¨ 6 , b Ž x 1 , x 3 . s y¨ 7 , b Ž x 1 , xX3 . s ¨ 3 q ¨ 8 , 0 y1 0 b Ž x 2 , x 3 . s b Ž x 2 , x X3 . s y ¨ 2 , and c Ž x 1 , x 2 , x 3 . s y 1 0 0 ,
ž
c Ž x1, x 2 ,
xX3 .
s
0 0 0 0 2 0 0 0 2
0
0 2
/
0 0 0 0 0 0 0 0 2
ž / , c Ž x , x , 0. s ž / . Finally, c Ž0, 0, 0. s 0. The 1
2
rest of the argument is the same as for Lemma 3.3. THEOREM 4.3. Ži.
The algebra homomorphism
c *: C Sym Ž 3 . ª C M Ž 16, 3 .
S p i n Ž10 .
is bijecti¨ e. That is, c gi¨ es a GLŽ3.-equi¨ ariant isomorphism: ;
Ž C 16 m C 3 . rrSpin Ž 10 . ª S 2 Ž Ž C 3 . * . . Žii.
The restriction to x 3 s 0 gi¨ es a GLŽ2.-equi¨ ariant isomorphism ;
c 33 : Ž C 16 m C 2 . rrSpin Ž 10 . ª Ž C, det 2 . . Proof. We will apply Theorem 2.2 for G s SpinŽ10., H s GLŽ3., V s half-spin m C 3 , W s S 2 ŽŽC 3 .*. m det 2 , and the equivariant surjective map c : V ª W defined above. Since condition Ža2. follows from w15x, it is enough to show Ža1.. By the action Ž4.1., the generic isotropy Hc Ž ¨ . ( SO Ž3. = Z 4 . Since Ž G = H . ¨ contains Ž ZG = ZH . l ker r , it is sufficient to give an element of p H ŽŽ ZG = ZH . l r . > Z 4 . Here ZG , ZH are the centers of G, H, respectively. Since the half-spin representation G ª GLŽ16. is irreducible, an element of the center ZG of G acts by some scalar l: ZG ª GLŽ1. s C=. It is known that the half-spin representation is faithful, hence l: ZG ª GLŽ1. is an injective homomorphism. Since the center ZG is isomor-
PREHOMOGENEOUS VECTOR SPACES
71
phic to Z 4 , the image of ZG consists of multiplications by the fourth root of unity. Then, for any multiplication in Z 4 ; Hc Ž ¨ . , its inverse is realized by an element in ZG . This must come from Ž ZG = ZH . l ker r . In w4, Theorems 2 and 3x, Gyoja gives the basic relative invariants F2 , F3 of the prehomogeneous vector spaces Ž20.: Ž SpinŽ10. = GLŽ2., M Ž16, 2.. and Ž21.: Ž SpinŽ10. = GLŽ3., M Ž16, 3.., respectively. These are written in terms of the map c as F2 Ž x 1 , x 2 . s c 33 Ž x 1 , x 2 , 0 . ,
F3 Ž x 1 , x 2 , x 3 . s yc Ž x 1 , x 2 , x 3 . .
5. APPLICATION We can apply the idea using the quotient variety for constructing relative invariants of some related prehomogeneous vector spaces. PROPOSITION 5.1. Let us consider the cases Ž10. or Ž21.: Ž G = GLŽ3., V1 m C 3 . with Ž G, r 1 , V1 . s Ž SLŽ5., L2 , C 10 . or Ž SpinŽ10., half-spin, C 16 .. Denote the set of lower triangular matrices in GLŽ3. by B Ž3., a Borel subgroup of GLŽ3.. Then Ži. Ž G = B Ž3., V1 m C 3 . is also a prehomogeneous ¨ ector space. Namely, the prehomogeneous spaces Ž G = GLŽ3., V1 m C 3 . are spherical with respect to the second factor GLŽ3.. Žii. Denote the ith principal minor Ž i s 1, 2, 3. of SymŽ3. by d i . Then a set of basic relati¨ e in¨ ariants of the prehomogeneous ¨ ector space Ž G = B Ž3., V1 m C 3 . is gi¨ en by d i ( c < i s 1, 2, 34 . Proof. It is obvious from Theorem 3.4 or Theorem 4.3 that any relative invariant of Ž G = B Ž3., V1 m C 3 . has to factor through SymŽ3.. The question of the construction of relative invariants is posed by Sato w13x in his work on zeta functions for some weakly spherical homogeneous spaces. We can also regard the construction of relative invariants of prehomogeneous vector spaces of commutative parabolic type w11x as an example of Theorem 2.2. By a standard the argument, Proposition 5.1 is stated in a slightly different form. Let P1, 1, 1, my3 be the standard parabolic subgroup of GLŽ m. whose Levi factor is GLŽ1. 3 = GLŽ m y 3.. Then SLŽ5. = P1, 1, 1, 7 has an open orbit on GLŽ10., and SpinŽ10. = P1, 1, 1, 13 has an open orbit on GLŽ16.. The defining equations of this open orbit are given by
g g GL Ž m .
d i ( c Ž Ž I3
0 . g . s 0 for i s 1, 2, 3 4 .
72
HIROYUKI OCHIAI
6. EXISTENCE OF THE QUOTIENT PROPOSITION 6.1. Let V be a Ž finite dimensional. representation of a linear algebraic group G = H. Suppose that the action of G on V is coregular and that H is reducti¨ e. Then there exists a representation W of H such that Cw V xG is isomorphic to Cw W x s S ŽW *. as an H-module. That is, VrrG ( W as an H-space. Proof. Denote the set of homogeneous invariants of degree d by Cw V xŽGd. . Denote by Jd the subalgebra generated by Cw V xŽGd9. Ž d9 - d .. Since Cw V xŽGd. is a completely reducible H-module, there is an H-submodule WŽ d. such that C w V x Ž d . s Ž C w V x Ž d . l Jd . [ WŽ d. G
G
` as H-modules. We set W [ [ds0 WŽ d. ; then W generates Cw V xG . By the coregularness, any basis of W is algebraically independent.
Remark 6.2. The coregular representations for all simple algebraic groups G are classified by Schwartz w16x. By his table, we had already known that Ž L2 ŽC 5 . m C 3 .rrSLŽ5. and ŽC 16 m C 3 .rrSpinŽ10. are affine.
ACKNOWLEDGMENTS The author would like to express his gratitude to Professors F. Sato, D. Vogan, and A. Yukie for their helpful discussions. He also thanks the referee.
REFERENCES 1. N. Bourbaki, ‘‘Groupes et algebres de Lie,’’ Chaps. 4]6, Hermann, Paris, 1968. ` 2. M. Brion, D. Luna, and T. Vust, Espaces homogenes ` spheriques, In¨ ent. Math. 84 Ž1986., 617]632. 3. C. Chevalley, ‘‘The Algebraic Theory of Spinors,’’ Columbia Univ. Press, 1954. 4. A. Gyoja, Construction of invariants, Tsukuba J. Math. 14 Ž1990., 437]457. 5. R. Howe, Remarks on classical in¨ ariant theory, Trans. Amer. Math. Soc. 313 Ž1989., 539]570. 6. R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 Ž1991., 565]619. 7. J. Igusa, On certain class of prehomogeneous vector spaces, J. Pure Appl. Algebra 47 Ž1987., 265]282. 8. M. Ishikawa and M. Wakayama, Minor summation formula of Pfaffians, preprint. 9. T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 Ž1982., 1]80. 10. D. Luna, Slices ˆ etales, Bull. Soc. Math. France, Memoirs 33 Ž1973., 81]105. ´
PREHOMOGENEOUS VECTOR SPACES
73
11. I. Muller, H. Rubenthaler, and G. Schiffmann, Structure des espace prehomogenes ´ ` associes ´ a de Lie graduees, ` certaines algebres ` ´ Math. Ann. 274 Ž1986., 95]123. 12. D. Mumford, ‘‘Geometric Invariant Theory,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1965. 13. F. Sato, Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, Comment. Math. Uni¨ . St. Paul. 44 Ž1995., 129]150. 14. F. Sato, Eisenstein series on Spin10 R GL16 , preprint. 15. M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 Ž1977., 1]155. 16. G. W. Schwartz, Representations of simple Lie groups with regular rings of invariants, In¨ ent. Math. 49 Ž1978., 167]191. 17. G. W. Schwartz, Representations of simple Lie groups with a free module of covariants, In¨ ent. Math. 50 Ž1978., 1]12. 18. H. Weyl, ‘‘Classical Groups,’’ Princeton Univ. Press, 1964 19. D. J. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions, In¨ ent. Math. 110 Ž1992., 283]314.
JOURNAL OF ALGEBRA ARTICLE NO.
Quotients of Some Prehomogeneous Vector Spaces Hiroyuki Ochiai* Department of Mathematics, Rikkyo Uni¨ ersity, Nishi-Ikebukuro, Tokyo 171, Japan Communicated by Robert Steinberg Received December 7, 1995
We give quotients VrrG s SpecŽCw V xG . for some prehomogeneous vector spaces Ž G = H, V .. Q 1997 Academic Press
1. INTRODUCTION 1.1. Let C be an algebraically closed field of characteristic zero. For a prehomogeneous vector space Ž G, r , V ., the quotient VrrG s SpecŽCw V xG . in the sense of geometric invariant theory w12x reduces to one point. Nothing interesting happens Žat least over an algebraically closed field.. Let N be a normal subgroup of G, and consider the quotient VrrN s SpecŽCw V x N .. It can be a nontrivial variety with an action of the quotient group GrN. Now we consider a special situation and change the notation slightly. Let us consider a prehomogeneous vector space of the form Ž G = H, r , V .. The quotient VrrG s SpecŽCw V xG . is an H-variety. A representation V of G is called coregular if the ring Cw V xG of invariants is isomorphic to a polynomial ring. If Ž G, r , V . is coregular, then it is easy to see ŽProposition 6.1. that the action of H on the affine space VrrG is linear. That is, we know the existence of the quotient prehomogeneous ¨ ector space Ž H, r , VrrG. in such a case. Several problems for Ž G = H, r , V . are expected to be reduced to that for the quotient Ž H, r , VrrG .. For example, we can construct a relative invariant of a prehomogeneous vector space Ž G = H, r , V . using a relative invariant of the quotient Ž H, r , VrrG .. * E-mail: [email protected]. 61 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
62
HIROYUKI OCHIAI
In the 1970s, Sato and Kimura w15x completed the classifications of irreducible reduced prehomogeneous vector spaces. At the same time, the relative invariants are constructed for almost all of these spaces. However, for some complicated prehomogeneous vector spaces, numbers Ž10., Ž21., and Ž24., the construction of relative invariants had not been settled. In 1990, Gyoja w4x constructed relative invariants explicitly for these spaces Ž10.: Ž SLŽ5. = GLŽ3., L 2 m L 1 ., Ž21.: Ž SpinŽ10. = GLŽ3., half-spin m L 1 ., and Ž24.: Ž GSpinŽ14., half-spin. by some complicated calculations. In this paper, we construct relative invariants for the spaces Ž10. and Ž21. using the quotient VrrG. Although the ingredients of the argument have already appeared in w3x or w4x, it seems new to stress the role of the quotient. Moreover, we can also construct several basic relative invariants for some related prehomogeneous vector spaces ŽProposition 5.1.. This construction is expected to be useful in obtaining an explicit formula of the functional equations of the associated zeta functions w13x. It is also helpful for orbit decompositions over a number field w14x. Observe that this method is not effective for the space Ž24. because GSpinŽ14. has only one simple factor. 1.2. Let Ž G = H, r , V . be a prehomogeneous vector space such that the quotient W [ VrrG is also a vector space. Then Ž H, r , W . is a prehomogeneous vector space such that the projection to the second factor of the generic isotropy of G = H at V coincides with the generic isotropy of H at W. The quotient map c : V ª W is surjective and G = H-equivariant. Conversely ŽTheorem 2.2., if two prehomogeneous vector spaces Ž G = H, r , V . and Ž H, r , W . are connected by a surjective equivariant morphism c : V ª W and they satisfy the condition above on the generic isotropy subgroups, then Ž H, r , W . is the quotient of Ž G = H, r , V .. 1.3. We explain the organization of the paper. In Section 2, we give a sufficient condition ŽTheorem 2.2. to obtain the quotient. We construct the equivariant surjective map c for the spaces Ž10., Ž20., and Ž21. in Sections 3 and 4. In Section 5, we give an application of the description of the quotient and Section 6 is devoted to proving the existence of an equivariant quotient. Notation. We denote the mth symmetric Žresp. skew symmetric. tensor representation of C n by S m ŽC n . Žresp. Lm ŽC n ... We often identify S 2 ŽC n . with the set SymŽ n. s X g M Ž n. < t X s X 4 of symmetric matrices, and L2 ŽC n . with the set Alt Ž n. of skew symmetric matrices.
PREHOMOGENEOUS VECTOR SPACES
63
A vector space or a linear algebraic group is defined over C. Note that, for example, the symbol SO Ž m. means a complex special orthogonal group SO Ž m, C., instead of a Žusual. real special orthogonal group SO Ž m, R..
2. MAIN THEOREM First we recall the notion of equivariant quotient of an affine variety. DEFINITION 2.1. Let G, H be linear algebraic groups and V, W be affine algebraic varieties. Suppose V is a G = H-variety and W is an H-variety. If c : V ª W induces an isomorphism ;
c *: C w W x ª C w V x
G
as C-algebras and H-modules, then the affine variety W Žor, more pre; cisely, the induced morphism c : VrrG ª W . is called the H-equi¨ ariant quotient of V by G. We introduce further standard terminology on prehomogeneous vector spaces. For a prehomogeneous vector space Ž G, r , V ., the open G-orbit in V is denoted by V s V V , and its complement by S s SV [ V y V V . A point ¨ in the open orbit V is called a generic point and the Žisomorphism class of. G¨ is called a generic isotropy. The following theorem is the main result of the abstract part of this paper. Let Ž G = H, r , V . and Ž H, r , W . be prehomogeneous
THEOREM 2.2.
¨ ector spaces and c : V ª W be a surjecti¨ e polynomial map such that
c Ž r Ž g , h. ¨ . s r Ž h. Ž c Ž ¨ . .
for all g g G, h g H , ¨ g V .
Let p H : G = H ª H be the second projection, p H ŽŽ g, h.. s h, and suppose the condition
Ž a.
p H Ž Ž G = H . ¨ . s Hc Ž ¨ .
for a ¨ g V V .
;
Then c : VrrG ª W is an H-equi¨ ariant quotient. Proof. It is enough to show the surjectivity of c *. First of all, we have V W s H ? c Ž ¨ . ( HrHc Ž ¨ . , V VrrG s Ž G = H . ? ¨ rrG ( Ž G = H . r Ž G = H . ¨ rrG s Hrp H Ž Ž G = H . ¨ . . By condition Ža., V W ( V VrrG.
64
HIROYUKI OCHIAI
Take some f g Cw V xG . Since Cw V xG ; Cw V V xG ( Cw V W x, there exists an f˜g Cw V W x such that f s f˜( c . We want to prove f˜g Cw W x. Let SW s S1 j ??? j Sl j S9 be the decomposition of SW such that the Si ’s are the irreducible components of codimension one and S9 is the union of components of codimension greater than one. Note that Si s w g W < pi Ž w . s 04 with an appropriate basic relative invariant pi . l Let p s Ł is1 pi . Then there is an integer N G 0 such that p N ? f˜ is regular outside of S9. Since S9 has codimension at least two, p N ? f˜g Cw W x. Suppose N G 1. We set g [ p N ? f˜g Cw W x, then g ( c s Ž p( c . N ? f. Since c is surjective, for a w g W, there is a ¨ g V such that w s c Ž ¨ .. Then g Ž w . s g Ž c Ž ¨ .. s pŽ c Ž ¨ .. N f Ž ¨ . s pŽ w . N f Ž ¨ .. Then g vanishes on S1 j ??? j Sl s w g W < pŽ w . s 04 . Again grp is regular outside of S9, g can be divided by p in Cw W x. That is, p Ny 1 ? f˜g Cw W x. By an induction on N, we have f˜g Cw W x. Condition Ža. in Theorem 2.2 may be stated in the following form: LEMMA 2.3. conditions:
Let ¨ g V V be a generic point and consider the following
Ža9. p H ŽŽ G = H . ¨ . > Hc Ž ¨ . . Ža1. Each connected component of Hc Ž ¨ . meets p H ŽŽ G = H . ¨ .. Ža2. dim p H ŽŽ g [ h . ¨ . s dim h c Ž ¨ . . Ža29. dim G y dim G¨ s dim V y dim W. Ži. Žii. Žiii.
Condition Ža. is equi¨ alent to Ža9.. Conditions Ža1. and Ža2. are equi¨ alent to Ža.. Condition Ža2. is equi¨ alent to Ža29..
Proof. Ži. It is easy to see
p H Ž Ž G = H . ¨ . ; Hc Ž ¨ .
for ¨ g V V .
Žii. Clear. Žiii. The left-hand side of Ža2. is dim Ž g [ h . ¨ y dim Ž g ¨ . s dim Ž G = H . y dim V y dim G¨ . The right-hand side of Ža2. is dim H y dim W. We give an example of Theorem 2.2. PROPOSITION 2.4.
For m ) n,
M Ž m, n . rrSO Ž m . 2 X ¬t XX g S 2 Ž C n . is a GLŽ n.-equi¨ ariant isomorphism.
PREHOMOGENEOUS VECTOR SPACES
65
The result is, of course, well known. This is easily derived from the classical invariant theory w5, 18x. However, we will give a proof using Theorem 2.2 in order to illustrate the examples in Sections 3 and 4 below. Proof. It is enough to check condition Ža. in Theorem 2.2 for G s SO Ž m., H s GLŽ n., V s M Ž m, n., W s SymŽ n., and c : V ª W defined by c Ž X . st XX. It is easily verified that c is surjective and that condition Ža2. holds. We prove condition Ža1.. Take a generic point ¨ s Ž I0n . g V. Then c Ž ¨ . s In g W and Hc Ž ¨ . s O Ž n., which has two connected components. It is enough to prove that diagŽy1, 1, . . . , 1. g O Ž n. can be obtained by p H ŽŽ G = H . ¨ .. In fact, we have
Ž diag Ž y1, 1, . . . , 1, y1. , diag Ž y1, 1, . . . , 1. . g Ž SO Ž m . = GL Ž n . . ¨ . 3. THE SPACE Ž10.: Ž SLŽ5. = GLŽ3., M Ž10, 3.. In this section, we will construct the map c for the prehomogeneous vector space Ž SLŽ5. = GLŽ3., L 2 m L 1 , M Ž10, 3... 3.1. In this section, x, y will be elements of Alt Ž5. ( L2 ŽC 5 . ( C 10 . First define an SLŽ5.-equivariant symmetric bilinear function
b : Alt Ž 5 . = Alt Ž 5 . ª Ž C 5 . * by
bi Ž x, y . [ x iq1 iq2 yiq3 iq4 y x iq1 iq3 yiq2 iq4 q x iq1 iq4 yiq2 iq3 q yiq1 iq2 x iq3 iq4 y yiq1iq3 x iq2 iq4 q yiq1 iq4 x iq2 iq3 s
1 4
Ý Ž sgn s . x iq s Ž1., iq s Ž2. yiq s Ž3., iq s Ž4.
sgS 4
for i s 1, . . . , 5. Here indices are numbered modulo 5. In w4x, bi is written as g i . As is known, ŽC 5 .* ( L4 ŽC 5 . appears in L2 ŽC 2 . m L2 ŽC 2 . with multiplicity one. The projection to this irreducible component is nothing but b . We can also give b in terms of the 4 by 4 principal minor Pfaffian w8x. For an x g Alt Ž5. and for i s 1, . . . , 5, define x Ž i. g Alt Ž4. by the matrix obtained by deleting the ith row and ith column from x. Then we have Žy1. iy1bi Ž x, x . s 2 PfŽ x Ž i. . and Žy1. iy1bi Ž x, y . s PfŽŽ x q y .Ž i. . y PfŽ x Ž i. .
66
HIROYUKI OCHIAI
y PfŽ y Ž i. .. That is, bi is a symmetric bilinear form corresponding to the quadratic form 2 PfŽ?Ž i. .. 3.2. Next we define an SLŽ5.-equivariant trilinear mapping ² ? < ? < ? : : Ž C 5 . * = Alt Ž 5 . = Ž C 5 . * ª C by Ž j , u, h . ¬ ² j < u
¡Ž a.
Ž b. Ž c. ~ Ž d. Ž e. Ž f. Ž g.
¢
²h < u < j : s y² j < u
Here x, y, z, u g Alt Ž5., j , h g ŽC 5 .*. Proof. Although these are proved in w4, Sect. 2x, we repeat the proof briefly. Ža. and Žb. follow from u g Alt Ž5.. Žd. is the differential of Žc. in the direction x y . To be more precise, substitute y q « x in y of Žc., and look at the coefficient of « 1. If we set h s b Ž x, y . in Žd., we have Že. with the help of Žb.. Žf. is ‘‘z x Že.’’ and Žg. is ‘‘z y Žd..’’ Now we prove Žc.. ² b Ž y, y . < x < j s
5
Ý bi Ž y, y . x i j s 2 Pf Ž A . , is1
where A s Ž a p q . g Alt Ž6. is defined by a p q s y p q , a p6 s x p j , a6 q s x jq if p, q F 5, and a66 s 0. In particular, Ž² b Ž y, y .< y < j . 2 s PfŽ A. 2 s detŽ A. s 0. 3.3. Now we define a map c : M Ž10, 3. ª SymŽ3.. We often identify an element of M Ž10, 3. with a triplet Ž x, y, z . of three alternative matrices in Alt Ž5.. c Ž x, y, z . [
ž
2² b Ž x, x . < z < b Ž x, y . : ² b Ž x, x . < z < b Ž y, y . : ² b Ž z, z . < y < b Ž x, x . : ² b Ž x, x . < z < b Ž y, y . : 2² b Ž y, y . < x < b Ž y, z . : ² b Ž y, y . < x < b Ž z, z . : . ² b Ž z, z . < y < b Ž x, x . : ² b Ž y, y . < x < b Ž z, z . : 2² b Ž z, z . < y < b Ž z, x . :
/
PREHOMOGENEOUS VECTOR SPACES
67
For e¨ ery g g SLŽ5., h g GLŽ3., Ž x, y, z . g M Ž10, 3., we
LEMMA 3.2. ha¨ e
c Ž gx, gy, gz . s c Ž x, y, z . , t
t
c Ž Ž x, y, z . h . s det Ž h . h c Ž x, y, z . h.
Ž 3.1.
Proof. We may prove the equivariance Ž3.1. for h’s of one of the 1 « 0
a 0 0
following three forms: h s 0 1 0 , 0 1 0 , or permutation matrices. 0 0 1 0 0 1 Checking Ž3.1. for diagonal or permutation matrices is easy. For h s 1 « 0 0 1 0 , we use relations Ža. ] Žg. listed in Section 3.2.
ž / ž /
ž /Ž 0 0 1
The 1, 1.-entry of c ŽŽ x, y, z . t h. s c Ž x q « , y, z . is calculated as 2² b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . < z < b Ž x, y . : q 2 « ² b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . < z < b Ž y, y . : Ž b.
s 2² b Ž x, x . < z < b Ž x, y . : q 2 « ² b Ž x, x . < z < b Ž y, y . : q Ž 2 « 2 y 4« 2 . ² b Ž y, y . < z < b Ž x, y . : s 2² b Ž x, x . < z < b Ž x, y . : q 2 « ² b Ž x, x . < z < b Ž y, y . : q 2 « 2 ² b Ž y, y . < x < b Ž y, z . : by using Ža. and Žf.. The Ž1, 3.-entry is ² b Ž z, z . < y < b Ž x, x . q 2 «b Ž x, y . q « 2b Ž y, y . : Žc.
s ² b Ž z, z . < y < b Ž x, x . : q 2 « ² b Ž z, z . < y < b Ž x, y . : s ² b Ž z, z . < y < b Ž x, x . : q « ² b Ž y, y . < x < b Ž z, z . : by Žd. and Ža.. Similarly, the Ž1, 2.-entry is checked by Žb., the Ž2, 2.-entry by Žc., the Ž2, 3.-entry by Žc., and the Ž3, 3.-entry by Že.. LEMMA 3.3. The map c is surjecti¨ e. Proof. Let ¨ i g C 5 Ž i s 1, . . . , 5. be the standard unit vector, let Ei j g Ž M 5. be the standard matrix unit, and define Yi j s Ei j y Eji g Alt Ž5. for a moment. We set x s Y12 q Y34 , y s Y23 q Y45 , z s Y13 q Y25 , z9 s Y15 , and z0 s Y13 . Then we have b Ž x, x . s 2 ¨ 5 , b Ž y, y . s 2 ¨ 1 , b Ž z, z . s 2 ¨ 4 , b Ž x, y . s ¨ 3 , b Ž x, z . s ¨ 1 , b Ž y, z . s ¨ 2 , b Ž x, z9. s ¨ 2 , b Ž y, z9. s ¨ 4 , b Ž y, z0 . s ¨ 2 , b Ž x, z0 . s 0, b Ž z9, z9. s 0, and b Ž z0, z0 . s 0. By a direct calculation, ² b Ž y, y .< x < s 2 ¨ 2 , ² b Ž x, x .< z < s y2 ¨ 2 , ² b Ž z, z .< y < s 2 ¨ 5 , ² b Ž x, x .< z 9 < s y2 ¨ 1 , ² b Ž x, x .< z 0 < s 0, ² b Ž z 9, z 9.< y < s 0, and ² b Ž z0, z0 .< y < s 0.
68
HIROYUKI OCHIAI
Finally, we
c Ž x, y, z0 . s
get
c Ž x, y, z . s
ž
0 0 4 0 4 0 4 0 0
/,
c Ž x, y, z 9. s
ž
0 y4 0 y4 0 0 0 0 0
/,
0 0 0 0 4 0 0 0 0
c Ž x, y, 0. s 0. Since the map c is GLŽ3.equivariant, the image is a GLŽ3.-stable subset of SymŽ3. containing the four elements above. It must be SymŽ3. itself.
ž / , and
THEOREM 3.4. The algebra homomorphism
c *: C Sym Ž 3 . ª C M Ž 10, 3 .
S L Ž5 .
is bijecti¨ e. That is, c gi¨ es a GLŽ3.-equi¨ ariant isomorphism: ;
Ž C 3 m L2 C 5 . rrSL Ž 5 . ª S 2 Ž C 3 . . Proof. We will apply Theorem 2.2 for G s SLŽ5., H s GLŽ3., V s L C 5 m C 3 , W s S 2 ŽC 3 . m det, and c : V ª W defined above. Condition Ža2. can be easily seen by w15x. In fact, by a straightforward calculation, the generic isotropy G¨ turns out to be a finite group. This yields Ža29.. Finally, we have to check condition Ža1.. The generic isotropy Hc Ž ¨ . is isomorphic to th < h g SO Ž3., t g GLŽ1., t 5 s 14 ( SO Ž3. = Z 5 . It is enough to construct an element of Ž G = H . ¨ whose image under p H has a nontrivial factor in Z 5 . Take a fifth root of unity z and consider Ž z I5 , zy2 I3 .. It is easy to see that it acts on V trivially and p H ŽŽ z I5 , zy2 I3 .. generates Z 5 . 2
The irreducible relative invariant F in w4, Theorem 1x is given by F Ž x, y, z . s y 12 detŽ c Ž x, y, z ... 4. THE SPACE Ž21.: Ž SpinŽ10. = GLŽ3., M Ž16, 3.. In this section, we will construct the map c for the prehomogeneous vector space Ž SpinŽ10. = GLŽ3., half-spin m L 1 , M Ž16, 3... The argument is parallel to the previous section. The calculation below gives the quotient for the space Ž20.: Ž SpinŽ10. = GLŽ2., half-spin m L 1 , M Ž16, 2... 4.1. First we define a SpinŽ10.-equivariant symmetric bilinear function
b : C 16 = C 16 ª C 10 . Here C 16 is a half-spin representation of SpinŽ10. and C 10 is a vector representation of SO Ž10.. The explicit form of b is also given in w3x and w4, p. 442x.
69
PREHOMOGENEOUS VECTOR SPACES
Next we define an SO Ž10.-invariant symmetric bilinear form ² ? , ? : : C 10 = C 10 ª C by Ž j , h . ¬ ² j , h : [ Ý5is1Ž j ihiq5 q j iq5hi .. 4.2. Now we define a map
c : M Ž 16, 3 . ª Sym Ž 3 . . We will identify an element of M Ž16, 3. with a triplet Ž x 1 , x 2 , x 3 . of three vectors in C 16 . For an x s Ž x 1 , x 2 , x 3 . g M Ž16, 3., we define
½
ai s ai Ž x . [ b Ž x i , x i . , bi s bi Ž x . [ b Ž x j , x k .
for i , j, k 4 s 1, 2, 3 4 .
These satisfy the following relations:
¡² a , a : s 0, i
i
² a i , bj : s 0
~²
for i / j,
a i , bi : s y2² bj , bk :
for i , j, k 4 s 1, 2, 3 4 ,
¢² a , a : s y2² b , b : i
j
k
for i , j, k 4 s 1, 2, 3 4 .
k
Using the above notation, we define c Ž x . s Ž c i j Ž x .. by c i j Ž x . [ Žy1. d i j ² bi Ž x ., bj Ž x .:. Here the Kronecker delta is d ii s 1 and d i j s 0 for i / j. LEMMA 4.1. ha¨ e
For g g SpinŽ10., h g GLŽ3., Ž x 1 , x 2 , x 3 . g M Ž16, 3. we
c Ž gx 1 , gx 2 , gx 3 . s c Ž x 1 , x 2 , x 3 . ,
Ž 4.1.
c Ž Ž x 1 , x 2 , x 3 . t h . s det h2 t hy1c Ž x 1 , x 2 , x 3 . hy1 .
Proof. As in the proof of Lemma 3.2, it is enough to prove Ž4.1. for hs
1 « 0 0 1 0 0 0 1
ž / . That is, c Ž x1 q « x 2 , x 2 , x 3 . s
ž
1 y« 0
0 1 0
0 0 1
/
c Ž x1 , x 2 , x 3 .
ž
1 0 0
y« 1 0
0 0 1
/
.
Ž 4.2.
70
HIROYUKI OCHIAI
The Ž2, 3.-entry of Ž4.2. is y² b 2 q « b1 < b 3 q « a2 : s y² b1 < b 3 : y « Ž ² b1 < b 3 : q ² b 2 < a2 : . y « 2 ² b1 < a2 : s y² b1 < b 3 : y « Ž y² b1 < b 3 : . . The Ž2, 2.-entry of Ž4.2. is ² b 2 q « b1 < b 2 q « b1 : s ² b 2 < b 2 : y 2 « Ž y² b1 < b 2 : . q « 2 ² b1 < b1 : . The other cases are similar. LEMMA 4.2. The map c is surjecti¨ e. Proof. We employ the notation in w15, 4x for spinors and let ¨ i g C 10 Ž i s 1, . . . , 10. be the standard unit vector. We set x 1 s 1 q e1 e2 e3 e4 , x 2 s e1 e5 q e2 e3 e 4 e5 , x 3 s e1 e 2 q e1 e 3 e4 e5 , xX3 s e1 e 2 q e 3 e5 q e1 e 2 e 4 e5 . Then b Ž x 1 , x 2 . s ¨ 1 q ¨ 6 , b Ž x 1 , x 3 . s y¨ 7 , b Ž x 1 , xX3 . s ¨ 3 q ¨ 8 , 0 y1 0 b Ž x 2 , x 3 . s b Ž x 2 , x X3 . s y ¨ 2 , and c Ž x 1 , x 2 , x 3 . s y 1 0 0 ,
ž
c Ž x1, x 2 ,
xX3 .
s
0 0 0 0 2 0 0 0 2
0
0 2
/
0 0 0 0 0 0 0 0 2
ž / , c Ž x , x , 0. s ž / . Finally, c Ž0, 0, 0. s 0. The 1
2
rest of the argument is the same as for Lemma 3.3. THEOREM 4.3. Ži.
The algebra homomorphism
c *: C Sym Ž 3 . ª C M Ž 16, 3 .
S p i n Ž10 .
is bijecti¨ e. That is, c gi¨ es a GLŽ3.-equi¨ ariant isomorphism: ;
Ž C 16 m C 3 . rrSpin Ž 10 . ª S 2 Ž Ž C 3 . * . . Žii.
The restriction to x 3 s 0 gi¨ es a GLŽ2.-equi¨ ariant isomorphism ;
c 33 : Ž C 16 m C 2 . rrSpin Ž 10 . ª Ž C, det 2 . . Proof. We will apply Theorem 2.2 for G s SpinŽ10., H s GLŽ3., V s half-spin m C 3 , W s S 2 ŽŽC 3 .*. m det 2 , and the equivariant surjective map c : V ª W defined above. Since condition Ža2. follows from w15x, it is enough to show Ža1.. By the action Ž4.1., the generic isotropy Hc Ž ¨ . ( SO Ž3. = Z 4 . Since Ž G = H . ¨ contains Ž ZG = ZH . l ker r , it is sufficient to give an element of p H ŽŽ ZG = ZH . l r . > Z 4 . Here ZG , ZH are the centers of G, H, respectively. Since the half-spin representation G ª GLŽ16. is irreducible, an element of the center ZG of G acts by some scalar l: ZG ª GLŽ1. s C=. It is known that the half-spin representation is faithful, hence l: ZG ª GLŽ1. is an injective homomorphism. Since the center ZG is isomor-
PREHOMOGENEOUS VECTOR SPACES
71
phic to Z 4 , the image of ZG consists of multiplications by the fourth root of unity. Then, for any multiplication in Z 4 ; Hc Ž ¨ . , its inverse is realized by an element in ZG . This must come from Ž ZG = ZH . l ker r . In w4, Theorems 2 and 3x, Gyoja gives the basic relative invariants F2 , F3 of the prehomogeneous vector spaces Ž20.: Ž SpinŽ10. = GLŽ2., M Ž16, 2.. and Ž21.: Ž SpinŽ10. = GLŽ3., M Ž16, 3.., respectively. These are written in terms of the map c as F2 Ž x 1 , x 2 . s c 33 Ž x 1 , x 2 , 0 . ,
F3 Ž x 1 , x 2 , x 3 . s yc Ž x 1 , x 2 , x 3 . .
5. APPLICATION We can apply the idea using the quotient variety for constructing relative invariants of some related prehomogeneous vector spaces. PROPOSITION 5.1. Let us consider the cases Ž10. or Ž21.: Ž G = GLŽ3., V1 m C 3 . with Ž G, r 1 , V1 . s Ž SLŽ5., L2 , C 10 . or Ž SpinŽ10., half-spin, C 16 .. Denote the set of lower triangular matrices in GLŽ3. by B Ž3., a Borel subgroup of GLŽ3.. Then Ži. Ž G = B Ž3., V1 m C 3 . is also a prehomogeneous ¨ ector space. Namely, the prehomogeneous spaces Ž G = GLŽ3., V1 m C 3 . are spherical with respect to the second factor GLŽ3.. Žii. Denote the ith principal minor Ž i s 1, 2, 3. of SymŽ3. by d i . Then a set of basic relati¨ e in¨ ariants of the prehomogeneous ¨ ector space Ž G = B Ž3., V1 m C 3 . is gi¨ en by d i ( c < i s 1, 2, 34 . Proof. It is obvious from Theorem 3.4 or Theorem 4.3 that any relative invariant of Ž G = B Ž3., V1 m C 3 . has to factor through SymŽ3.. The question of the construction of relative invariants is posed by Sato w13x in his work on zeta functions for some weakly spherical homogeneous spaces. We can also regard the construction of relative invariants of prehomogeneous vector spaces of commutative parabolic type w11x as an example of Theorem 2.2. By a standard the argument, Proposition 5.1 is stated in a slightly different form. Let P1, 1, 1, my3 be the standard parabolic subgroup of GLŽ m. whose Levi factor is GLŽ1. 3 = GLŽ m y 3.. Then SLŽ5. = P1, 1, 1, 7 has an open orbit on GLŽ10., and SpinŽ10. = P1, 1, 1, 13 has an open orbit on GLŽ16.. The defining equations of this open orbit are given by
g g GL Ž m .
d i ( c Ž Ž I3
0 . g . s 0 for i s 1, 2, 3 4 .
72
HIROYUKI OCHIAI
6. EXISTENCE OF THE QUOTIENT PROPOSITION 6.1. Let V be a Ž finite dimensional. representation of a linear algebraic group G = H. Suppose that the action of G on V is coregular and that H is reducti¨ e. Then there exists a representation W of H such that Cw V xG is isomorphic to Cw W x s S ŽW *. as an H-module. That is, VrrG ( W as an H-space. Proof. Denote the set of homogeneous invariants of degree d by Cw V xŽGd. . Denote by Jd the subalgebra generated by Cw V xŽGd9. Ž d9 - d .. Since Cw V xŽGd. is a completely reducible H-module, there is an H-submodule WŽ d. such that C w V x Ž d . s Ž C w V x Ž d . l Jd . [ WŽ d. G
G
` as H-modules. We set W [ [ds0 WŽ d. ; then W generates Cw V xG . By the coregularness, any basis of W is algebraically independent.
Remark 6.2. The coregular representations for all simple algebraic groups G are classified by Schwartz w16x. By his table, we had already known that Ž L2 ŽC 5 . m C 3 .rrSLŽ5. and ŽC 16 m C 3 .rrSpinŽ10. are affine.
ACKNOWLEDGMENTS The author would like to express his gratitude to Professors F. Sato, D. Vogan, and A. Yukie for their helpful discussions. He also thanks the referee.
REFERENCES 1. N. Bourbaki, ‘‘Groupes et algebres de Lie,’’ Chaps. 4]6, Hermann, Paris, 1968. ` 2. M. Brion, D. Luna, and T. Vust, Espaces homogenes ` spheriques, In¨ ent. Math. 84 Ž1986., 617]632. 3. C. Chevalley, ‘‘The Algebraic Theory of Spinors,’’ Columbia Univ. Press, 1954. 4. A. Gyoja, Construction of invariants, Tsukuba J. Math. 14 Ž1990., 437]457. 5. R. Howe, Remarks on classical in¨ ariant theory, Trans. Amer. Math. Soc. 313 Ž1989., 539]570. 6. R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 Ž1991., 565]619. 7. J. Igusa, On certain class of prehomogeneous vector spaces, J. Pure Appl. Algebra 47 Ž1987., 265]282. 8. M. Ishikawa and M. Wakayama, Minor summation formula of Pfaffians, preprint. 9. T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 Ž1982., 1]80. 10. D. Luna, Slices ˆ etales, Bull. Soc. Math. France, Memoirs 33 Ž1973., 81]105. ´
PREHOMOGENEOUS VECTOR SPACES
73
11. I. Muller, H. Rubenthaler, and G. Schiffmann, Structure des espace prehomogenes ´ ` associes ´ a de Lie graduees, ` certaines algebres ` ´ Math. Ann. 274 Ž1986., 95]123. 12. D. Mumford, ‘‘Geometric Invariant Theory,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1965. 13. F. Sato, Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, Comment. Math. Uni¨ . St. Paul. 44 Ž1995., 129]150. 14. F. Sato, Eisenstein series on Spin10 R GL16 , preprint. 15. M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 Ž1977., 1]155. 16. G. W. Schwartz, Representations of simple Lie groups with regular rings of invariants, In¨ ent. Math. 49 Ž1978., 167]191. 17. G. W. Schwartz, Representations of simple Lie groups with a free module of covariants, In¨ ent. Math. 50 Ž1978., 1]12. 18. H. Weyl, ‘‘Classical Groups,’’ Princeton Univ. Press, 1964 19. D. J. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions, In¨ ent. Math. 110 Ž1992., 283]314.