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A Classification of Simple Weakly Spherical Homogeneous Spaces, I
JOURNAL OF ALGEBRA ARTICLE NO.
182, 235]255 Ž1996.
0169
A Classification of Simple Weakly Spherical Homogeneous Spaces, I Shin-Ichi Kasai Department of Mathematics, Faculty of Education, Yamaguchi Uni¨ ersity, Yamaguchi, 753, Japan
and Tatsuo Kimura and Shin-Ichi Otani The Institute of Mathematics, Uni¨ ersity of Tsukuba, Ibaraki, 305, Japan Communicated by Corrado de Concini Received February 15, 1995
INTRODUCTION Let G be a connected linear algebraic group and H its closed subgroup. When a Borel subgroup of G has a Zariski-dense orbit in H _ G, we say that H _ G is spherical wBx. By wVx, symmetric spaces are spherical in this sense. More generally, when there exists a parabolic subgroup P of G such that H _ G has a Zariski-dense P-orbit, we say that H _ G is weakly spherical Žor P-spherical.. We shall consider everything over the complex number field C. To develop the theory of Eisenstein series on weakly spherical homogeneous spaces, we need many examples. When G s GLn , it is closely related with a prehomogeneous vector space wSx. A homogeneous space r Ž H . _ GLn is called simple when Ž H, r . is of the form Ž GLl1 = Gs r 1 q ??? q r l ., where each r i is an irreducible representation of a simple algebraic group Gs and GLl1 acts on each irreducible component as a scalar multiplication. In this paper ŽI., as the first trial toward the classification of weakly spherical homogeneous spaces, we shall classify all simple weakly spherical 235 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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homogeneous spaces r Ž H . _ GLn with
Ž H, r . /
ž
t GLkq 1
k !# "
= SL r , L 1 q ??? q L 1
ž
t !# "
Ž).
/
q 1 q ??? q 1 .
/
This paper consists of the following four sections. 1. Preliminaries 2. The case when r : H ª GLn is irreducible 3. Non-irreducible case r Ž H . _ GLn with k !# "
t = SL r , L 1 q ??? q L 1 Ž H , r . / GLkq 1
ž
ž
Ž).
/
t !# "
q 1 q ??? q 1
/
4. Table of simple weakly spherical homogeneous spaces r Ž H . _ GLŽ n. with
Ž H, r . /
ž
t GLkq Ž1.
k !# "
= SLŽ r . , L 1 q ??? q L 1
ž
Ž).
/
t !# "
q 1 q ??? q 1
/
1. PRELIMINARIES Let G be a connected linear algebraic group and H its closed subgroup. PROPOSITION 1.1. The following conditions are equi¨ alent. Ž1. Ž2. Ž3. Ž4.
H _ G is P-spherical. gHgy1 _ G is P-spherical for any g g G. H _ G is gPgy1 -spherical for any g g G. t y1 t y1 H _ G is tPy1-spherical.
Proof. Clearly H _ G is P-spherical if and only if G has a Zariski-dense orbit HyP. However, HyP is dense in G if and only if gHyP s Ž gHgy1 .Ž gy . P Žresp. HyPgy1 s H Ž ygy1 .Ž gPgy1 ., t Ž HyP .y1 s tHy1 ? t yy1 ? t y1 P . is dense in G. From now on, we shall assume that G s GLn Žs GLŽ n.. and let Pe1 , . . . , e r s P Ž e1 , . . . , e r .Ž e1 q ??? qe r s n. be a standard parabolic subgroup given as follows.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
P Ž e1 , . . . , e r .
¡P
P12
..
0
P22
..
0
0
..
0
0
0
11
s~
¢
. . .
237
¦
P1 r .. . . . g GLn ; Pi j g M Ž e i , e j . Ž 1 F i , j F n . . . Pr r
0
¥
§
Ž 1.1 . COROLLARY 1.2. The following conditions are equi¨ alent. Ž1. Ž2.
H _ GLn is P Ž e1 , e2 , . . . , e r .-spherical. Hy1 _ GLn is P Ž e r , e ry1 , . . . , e1 .-spherical.
t
In particular, if H is reducti¨ e, then H _ GLn is P Ž e r , e ry1 , . . . , e1 .spherical. Proof. Put 0 I˜n s
1 ?
?
?
1 1 1
0
0
.
˜ ˜t Ž .y1 ? Then we have I˜y1 n s In and one can easily check that In P e1 , . . . , e r y1 I˜n s P Ž e r , . . . , e1 .. Hence we have our result by Proposition 1.1. Put f 1 s e1 , f 2 s e1 q e2 , . . . , f i s e1 q ??? qe i , . . . , f r s e1 q ??? qe r s n. Let GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 . act on V s M Ž n, f ry1 . [ ??? [ y1 . M Ž f 2 , f 1 . by r Ž g . ¨ s Ž g r¨ ry1 gy1 for g s Ž g r , . . . , g 1 . g ry1 , . . . , g 2 ¨ 1 g 1 GLŽ f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ ry1 , . . . , ¨ 1 . g V s M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. Then it is a prehomogeneous vector space ŽPV.. The GLŽ f r .-part of a generic isotropy subgroup of a generic point ¨0 s
ž
I f ry 1 0
,...,
If1 0
/
is a parabolic subgroup P Ž e1 , . . . , e r .. Hence Ž H = P Ž e1 , . . . , e r ., M Ž n.. is a PV if and only if Ž H = GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n . [ M Ž n, f ry 1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where g ? ¨ s 1 y1 y1. Ž h¨ r gy for g s Ž h, g r , . . . , g 1 . g H = r , g r ¨ ry 1 g ry 1 , . . . , g 2 ¨ 1 g 1 GLŽ f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ r , . . . , ¨ 1 . g M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. It is clear that H _ GLn is P Ž e1 , . . . , e r .-spherical if and only if Ž H = P Ž e1 , . . . , e r ., M Ž n.. is a PV. Since a generic isotropy subgroup of
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Ž H = GLŽ f r ., M Ž n.. at a generic point In is given by Ž h, h. g H = GLŽ f r .; h g H 4 ( H, it is also PV-equivalent to Ž H = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 ... Thus we obtain the following proposition, which we learned from Professor Fumihiro Sato wSx. PROPOSITION 1.3 ŽF. Sato.. Put f 1 s e1 , f 2 s e1 q e2 , . . . , f r s e1 q ??? qe r s n. Then the following conditions are equi¨ alent. Ž1. H _ GLn is P Ž e1 , . . . , e r .-spherical. Ž2. Ž H = GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where the action is gi¨ en by g ? ¨ s y1 y1 . Ž h¨ r gy1 for g s Ž h, g r , . . . , g 1 . g H = GL r , g r ¨ ry1 g ry1 , . . . , g 2 ¨ 1 g 1 Ž f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ r , . . . , ¨ 1 . g M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. Ž3. Ž H = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where the action is gi¨ en by g ? ¨ s Ž h¨ ry1 gy1 g ry1¨ ry2 ry1 , y1 . Ž . Ž . gy1 , . . . , g ¨ g for g s h, g , . . . , g g H = GL f ry2 2 1 1 ry1 1 ry1 = ??? = GLŽ f 1 . and ¨ s Ž ¨ ry1 , . . . , ¨ 1 . g M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. As a corollary, we obtain the simple proof of Teranishi’s generalized castling transform. COROLLARY 1.4 ŽTeranishi wTx.. Let n s e1 q ??? qe r , and r : H ª GLŽ n. a rational representation. Then the following conditions are equi¨ alent. Ž1. Ž H = P Ž e1 , . . . , e ry1 ., r m LU1 , M Ž n, n y e r .. is a PV. Ž2. Ž H = P Ž e r , . . . , e2 ., r * m LU1 , M Ž n, n y e1 .. is a PV. Proof. Note that Ž1. is PV-equivalent to Ž3. in Proposition 1.3. Hence, by Corollary 1.2 and again by Proposition 1.3, it is PV-equivalent to Ž2.. Note that when r s 2, Corollary 1.4 gives a usual castling transformation.
2. THE CASE WHEN r : H ª GLn IS IRREDUCIBLE In this section, we shall consider an irreducible representation r : H ª GLn of a simple algebraic group H. PROPOSITION 2.1 ŽwSKx; p. 372 in wtype Ix.. We identify r Ž H . and H: r Ž H . s H. By Proposition 1.3, H _ GLn is P Ž m, n y m.-spherical if and only if Ž H = GLm , r m L 1 , M Ž n, m.. is a PV Ž n ) m G 1.. All such PV ’s are gi¨ en as follows.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
239
ŽI. Ž H = GLm , r m L 1 , M Ž n, m.. with only m s 1 or m s n y 1. Such Ž H, r . are gi¨ en as follows. Ž1. Ž SL k , 2L 1 . Ž k G 4. Ž2. Ž SL 2 k , L 2 . Ž k G 4. Ž3. Ž SL k , L 3 . Ž k s 6, 7, 8. Ž4. Ž SL 2 , 3L 1 . Ž5. Ž Sp 3 , L 3 . Ž6. Ž Spin k , the spin rep.. Ž k s 9, 11. Ž7. Ž Spin k , a half-spin rep.. Ž k s 12, 14. Ž8. Ž E7 , L 6 . with deg. 6L 6 s 56. ŽII. Ž H = GLm , r m L 1 , M Ž n, m.. with only m s 1, 2, n y 2, n y 1. Such Ž H, r . are gi¨ en as follows. Ž1. Ž SL 2 kq1 , L 2 . Ž k G 3. Ž2. Ž SL6 , L 2 . Ž3. Ž SL 3 , 2L 1 . Ž4. Ž G 2 , L 2 . with deg L 2 s 7 Ž5. Ž E6 , L 1 . with deg L 1 s 27. ŽIII. Ž1. Ž Spin 7 = GL m , the spin rep.m L 1 , V Ž8. m V Ž m.., m s 1, 2, 3, 5, 6, 7 Ž2. Ž Spin10 = GLm , a half-spin rep.m L 1 , V Ž16. m V Ž m.., m s 1, 2, 3, 13, 14, 15 Ž3. Ž SL5 = GLm , L 2 m L 1 , V Ž10. m V Ž m.., 1 F m F 9, m / 5 ŽIV. Ž1. Ž SOn = GLm , L 1 m L 1 , V Ž n. m V Ž m.., 1 F m F n y 1 Ž2. Ž Spk = GLm , L 1 m L 1 , V Ž n. m V Ž m.., 1 F m - n s 2 k. PROPOSITION 2.2. Let Ž H, r . with deg r s n be one of ŽI. in Proposition 2.1. Then H _ GLn is P-spherical if and only if P is Ž conjugate to . P Ž1, n y 1. or P Ž n y 1, 1.. Note that we identify H and r Ž H . Ž; GLn .. Proof. By Proposition 1.3, H _ GLn is P Ž e1 , e2 .-spherical if and only if Ž e1 , e2 . s Ž1, n y 1. or Ž n y 1, 1.. If H _ GLn is P Ž e1 , e2 , e3 .-spherical, then it must be P Ž e1 q e2 , e3 . and P Ž e1 , e2 q e3 .-spherical since P Ž e1 q e2 , e3 . > P Ž e1 , e2 , e3 ., etc. Hence we have Ž e1 , e2 , e3 . s Ž1, n y 2, 1.. By Proposition 1.3, H _ GLn is P Ž1, n y 2, 1.-spherical if and only if Ž H = GLny 1 = GL1 , r m LU1 m 1 q 1 m L 1 m LU1 . ( Ž GL21 = H = SL ny1 , r m LU1 q 1 m L 1 . is a PV. However, it is a non-PV by Theorem 2.1 and Lemma 2.2, page 375 in wtype Ix. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 2.3. Let H s L 2 Ž SL 2 kq1 . with k G 3 so that n s deg L 2 s k Ž2 k q 1.. Then H _ GLn is P-spherical if and only if P s P Ž e1 , e2 . with Ž e1 , e2 . s Ž1, n y 1., Ž2, n y 2., Ž n y 2, 2., Ž n y 1, 1. and P Ž1, 1, n y 2., P Ž n y 2, 1, 1.. Proof. We obtain our results for P Ž e1 , e2 . by Propositions 1.3 and 2.1. If H _ GLn is P Ž e1 , e2 , e3 .-spherical, then it must be P Ž e1 q e2 , e3 . and
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P Ž e 1 , e 2 q e 3 .-spherical. Hence Ž e 1 , e 2 , e 3 . s Ž1, 1, n y 2 ., Ž n y 2, 1, 1., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 3, 1., and Ž2, n y 4, 2.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1. and Ž2, n y 3, 1.. By Proposition 1.3, H _ GLn is P Ž e1 , e2 , e3 .-spherical if and only if Ž SL 2 kq1 = GLŽ e1 q e2 . = GLŽ e1 ., L 2 m LU1 m 1 q 1 m L 1 m LU1 . is a PV. When e1 s 1, it is equivalent to Ž GL21 = SL 2 kq1 = SLŽ e2 q 1., L 2 m LU1 q 1 m L 1 . and it is a PV when e2 s 1 ŽTheorem 2.11 for k s 3, Theorem 2.7 for k s 4, and Theorem 2.5 for k G 5 in wtype Ix.. However, it is a non-PV when e2 s n y 3 ŽLemma 2.2 for k s 3, 4 and Theorem 2.1 for k G 5 in wtype Ix. or when e2 s n y 2 ŽTheorem 2.12 for k s 3 and Theorem 2.7 for k G 4 in wtype Ix.. When e1 s 2 and e2 s n y 4, it is a non-PV by dimension reason. Now if H _ GLn is P Ž e1 , e2 , e3 , e4 .-spherical, then it must be P Ž e1 , e2 , e3 q e4 . and P Ž e1 q e2 , e3 , e4 .-spherical and hence e2 s e3 s 1, and e1 , e4 is 1 or n y 3. If it is P Ž1, 1, 1, n y 3. Žresp. P Ž n y 3, 1, 1, 1..-spherical, it must be P Ž1, 2, n y 3. Žresp. P Ž n y 3, 2, 1..spherical, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. PROPOSITION 2.4. Let H s L 2 Ž SL6 . with n s 15 Ž resp. 2L 1Ž SL 3 . with n s 6, L 1Ž E6 . with n s 27.. Then H _ GLn is P-spherical if and only if P s P Ž1, n y 1., P Ž2, n y 2., P Ž n y 2, 2., and P Ž n y 1, 1.. Proof. By Propositions 1.3 and 2.1, we have our assertion for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then Ž e1 q e2 , e3 . and Ž e1 , e2 q e3 . must be Ž1, n y 1., Ž2, n y 2., Ž n y 2, 2., or Ž n y 1, 1.. Hence we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, n y 4, 2. and Ž2, n y 3, 1., Ž n y 2, 1, 1.. By Corollary 1.2, it is not necessary to consider Ž2, n y 3, 1., Ž n y 2, 1, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV, where r : H ª GLn is an injection. When e1 s 1, it is equivalent to Ž GL21 = H = SLŽ e2 q 1., r m L 1 q 1 m LU1 ., which is a non-PV for e2 s 1, 12, 13 by Lemma 2.2 in wtype Ix Žresp. for e2 s 1, 3 by Lemma 2.2 in wtype Ix and for e2 s 4 by Theorem 2.1 in wtype Ix; resp. for e2 s 1, 24, 25 by Theorem 2.1 in wtype Ix.. By dimension reason, it is a non-PV for e1 s 2 and e2 s 11 Žresp. e2 s 2, e2 s 23.. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 2.5. Let H s L 2 Ž G 2 . with n s 7. Then H _ GL7 is Pspherical if and only if P s P Ž e1 , e2 . with Ž e1 , e2 . s Ž1, 6., Ž2, 5., Ž5, 2., Ž6, 1. and P s P Ž1, 1, 5., P Ž1, 5, 1., P Ž5, 1, 1.. Proof. For P s P Ž e1 , e2 ., we have our results by Propositions 1.3 and 2.1. If it is P Ž e1 , e2 , e3 .-spherical, we have Ž e1 , e2 , e3 . s Ž1, 1, 5., Ž1, 4, 2., Ž1, 5, 1., Ž2, 3, 2. Žand Ž2, 4, 1., Ž5, 1, 1... By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž G 2 = GLŽ e1 q e2 . = GLŽ e1 ., L 2 m
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
241
LU1 m 1 q 1 m L 1 m LU1 . is a PV. If e1 s 1, it is equivalent to Ž GL21 = G 2 = SLŽ e2 q 1., L 2 m L 1 q 1 m LU1 ., which is a PV for e1 s 1 and 5 by Theorem 2.5 in wtype Ix while it is a non-PV for e2 s 4 by Theorem 2.1 in wtype Ix. When e1 s 2 and e2 s 3, it is a non-PV by a dimension reason. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then it is P Ž e1 q e2 , e3 , e4 ., P Ž e1 , e2 q e3 , e4 ., P Ž e1 , e2 , e3 q e4 .-spherical and hence e1 , . . . , e4 s 1 or 5 with e1 q ??? qe4 s 7, which is impossible. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. PROPOSITION 2.6. Let H s LŽ Spin 7 . with n s 8 Ž resp. L9Ž Spin10 . with n s 16, L 2 Ž SL5 . with n s 10., where L Ž resp. L9. is the spin Ž resp. a half-spin. representation. Then H _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, 3, 5, 6, 7 Ž resp. 1, 2, 3, 13, 14, 15; resp. 1 F m F 9, m / 5.; P Ž1, 1, n y 2., P Ž1, 2, n y 3., P Ž1, n y 3, 2., P Ž1, n y 2, 1., P Ž2, 1, n y 3. Ž and P Ž2, n y 3, 1., P Ž n y 3, 1, 2., P Ž n y 3, 2, 1. P Ž n y 2, 1, 1..; P Ž1, 1, 1, n y 3., P Ž1, 1, n y 3, 1. Ž and P Ž1, n y 3, 1, 1., P Ž n y 3, 1, 1, 1... Proof. By Propositions 1.3 and 2.1, we have our assertion for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then similarly to the above, we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 4, 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 5, 3., Ž2, n y 4, 2., Ž3, n y 6, 3. Žand Ž2, n y 3, 1., Ž3, n y 5, 2., Ž3, n y 4, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1.; cf. Corollary 1.2.. Moreover, in the case H s L 2 Ž SL5 ., we have also Ž1, 3, 6 ., Ž1, 5, 4 ., Ž2, 2, 6 ., Ž2, 4, 4 ., Ž3, 1, 6 ., Ž3, 3, 4 ., Ž4, 2, 4 . Žand Ž4, 3, 3., Ž4, 4, 2., Ž4, 5, 1., Ž6, 1, 3., Ž6, 2, 2., Ž6, 3, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV, where r : H ª GLn is the inclusion. When e1 s 1, it is equivalent to Ž GL21 = H = SLŽ e2 q 1., r m L 1 q 1 m LU1 ., which is a PV for e2 s 1, 2, n y 3 by Theorem 2.5 Žfor L 2 Ž SL5 . by Theorems 2.8, 2.5, 2.7, respectively. and e2 s n y 2 by Theorem 2.7 Žfor L 2 Ž SL5 . by Theorem 2.9. in wtype Ix, while it is a non-PV for e2 s n y 4 Žfor L 2 Ž SL5 ., also e2 s 3, 5. by Theorem 2.1 in wtype Ix. If Ž e1 , e2 . s Ž2, 1., it is a PV by Theorem 2.5 in wtype Ix. If Ž e1 , e2 . s Ž2, n y 5., Ž3, n y 6., it is a non-PV by Lemma 2.17 in wKasaix. If Ž e1 , e2 . s Ž2, n y 4., it is a non-PV by Theorem 3.15 in wKasaix. When H s L 2 Ž SL5 . with Ž e1 , e2 . s Ž2, 2., Ž2, 4., Ž3, 1., Ž3, 3., and Ž4.2., it is a nonPV by Lemma 2.16 in wKasaix. Hence if it is P Ž e1 , e2 , e3 , e4 .-spherical, then Ž e1 q e2 , e3 , e4 ., Ž e1 , e2 q e3 , e4 ., Ž e1 , e2 , e3 q e4 . must be one of Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 3, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1., and we have Ž e1 , e2 , e3 , e4 . s Ž1, 1, 1, n y 3., Ž1, 1, n y 3, 1. Žand Ž1, n y 3, 1, 1., Ž n y 3, 1, 1, 1.; cf. Corollary 1.2.. It is P Ž1, 1, 1, n y 3.-spherical if and only if Ž H = GL3 = GL2 = GL1 , r m LU1 m 1 m 1 q 1 m L 1 m LU1 m 1 q 1 m 1 m L 1 m LU1 . is a PV. Since the
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GL3-part of a generic isotropy subgroup of Ž H = GL3 , r m LU1 . is SO 3 wSKx, it is a PV ŽTheorem 2.5 in wtype Ix.. Similarly it is P Ž1, 1, n y 3, 1.spherical if and only if Ž H = GLny 1 = GL2 = GL1 , r m LU1 m 1 m 1 q 1 m L 1 m LU1 m 1 q 1 m 1 m L 1 m LU1 . is a PV. When H s LŽ Spin 7 ., the GL7-part of a generic isotropy subgroup of Ž H = GL7 , r m LU1 . is G 2 wSKx, and hence it is a PV ŽTheorem 2.5 in wtype Ix.. When H s L9Ž Spin10 ., the Lie algebra LieŽ H . Ž; M Ž16.. is given by page 120 in wSKx. Then the isotropy subalgebra at the following point, x 0 , is given by LieŽ SL 2 . and it is a PV. x0 s
½
I15 y1, 0, . . . , 0
, Ž a 5 q a 6 , a 11 q a 12 . ,
1 1
ž /5
Ž 2.1.
g M Ž 16, 15 . [ M Ž 15, 2 . [ M Ž 2, 1 . , where a i is a column vector in C 15 with ith entry 1 and remaining parts 0. When H s L 2 Ž SL5 ., the isotropy subalgebra at Ž a 1 n a 3 q a 2 n a 4 , a 1 n a 2 , a 1 n a 4 , a 3 n a 4 , a 2 n a 3 , a 1 n a5 , a 2 n a5 , a 3 n a5 , a 4 n a 5 .,Ž b 2 q b4 , b 1 q b6 q b 7 ., g 1 .4 , where a i Žresp. bi , g i . is a column vector in C 5 Žresp. C 9 , C 2 . with ith entry 1 and remaining parts 0, is 04 and hence it is a PV. Now if it is P Ž e1 , . . . , e5 .-spherical, then e1 , . . . , e5 s 1 or n y 3 with e1 q ??? qe5 s n, which is impossible. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 5. Finally, the following proposition is well known. PROPOSITION 2.7 ŽwKrx; Th. 2.27 and Prop. 4.23 in wKKYx.. SOn _ GLn and Spm _ GL2 m are P-spherical for any parabolic subgroup P; i.e., they are spherical. 3. THE NON-IRREDUCIBLE CASE r Ž H . _ GLn WITH k !# "
t !# "
t Ž H, r . / Ž GLkq = SL r , Ž L 1 q ??? q L 1 .Ž*. q 1 q ??? q 1. 1
Let t !# "
r s r 1 q ??? qr k q 1 q ??? q 1
Ž k G 1, k q t G 2 .
t be a representation of H s GLkq = Gs with a simple algebraic group Gs , 1 kq t where GL1 acts on each irreducible component as a scalar multiplication. Let n s deg r . Then by Proposition 1.3, r Ž H . _ GLn is P Ž m, n y m.-spherical if and only if Ž H = GLm r m L 1 . is a PV.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
PROPOSITION 3.1.
Ž H, r . /
ž
243
All such PV ’s Ž H = GLm , r m L 1 . with
t GLkq 1
k !# "
Ž).
= SL r , L 1 q ??? q L 1
ž
t !# "
q 1 q ??? q 1
/
/
are gi¨ en as follows. ŽI. Ž H = GLm r m L 1 ., with only m s 1 or m s n y 1. Such Ž H, r . t with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
ky 1 !# "
t !# "
Gs s SL r ; r s L 1 q ??? q L 1 q LU1 q 1 q ??? q 1
Ž r q 1 G k G 2, t G 0 . . Ž 2.
t !# "
Gs s SL r ; r s 2L 1 q 1 q ??? q 1 Ž t G 1 . , t !# "
r s 2L 1 q LŽ1) . q 1 q ??? q 1 Ž t G 0 . .
Ž 3.
t !# "
Gs s SL 2 r 9 Ž r 9 G 3 . ; r s L 2 q 1 q ??? q 1 Ž t G 1 . ,
r s L2 q
s !# "
LŽ1) . q ??? q LŽ1) .
t !# "
q 1 q ??? q 1
Ž 1 F s F 3; t G 0 . . Ž 4.
t !# "
Gs s SL 2 r 9q1 Ž r 9 G 2 . ; r s L 2 q 1 q ??? q 1 Ž t G 4 . , t !# "
r s L 2 q L 2 q 1 q ??? q 1 Ž t G 0 . , s !# "
t !# "
r s L 2 q LŽ1) . q ??? q LŽ1) . q 1 q ??? q 1
Ž 1 F s F 3; t G 0 . except L 2 q L 1 q L 1 q LU1 q 1 q ??? q1. t !# "
Ž 5.
Gs s SL 2 ; r s 3L 1 q 1 q ??? q t Ž t G 1 . .
Ž 6.
Gs s SL 4 ; L 2 q L 1 q 1 q ??? q 1 Ž t G 1 . .
Ž 7.
t !# "
Gs s SL5 ; r s L 2 q L 2 q
LU1
t !# "
q 1 q ??? q 1 Ž t G 0 . .
244
Ž 8.
KASAI, KIMURA, AND OTANI t !# "
Gs s SL6 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . ,
r s L3 q
LŽ1) .
t !# "
q 1 q ??? q 1 Ž t G 0 . , t !# "
r s L 3 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . .
Ž 9.
t !# "
Gs s SL 7 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . , t !# "
r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0 . .
Ž 10 . Ž 11 .
t !# "
Gs s SL8 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . . t !# "
Gs s Spr ; r s L 1 q 1 q ??? q 1 Ž t G 4 . , t !# "
r s L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . , t !# "
r s L 1 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 12 .
Gs s Sp 2 ; r s L 2 q L 1 q 1 q ??? q 1 Ž t G 0 . .
Ž 13 .
Gs s Sp 3 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . ,
t !# "
r s L 3 q L 1 q 1 q ??? q1 Ž t G 0 . . t !# "
Ž 14 .
Gs s SOr ; r s L 1 q 1 q ??? q 1 Ž t G 2 . .
Ž 15 .
Gs s Spin 7 ; r s the spin rep. q1 q ??? q 1 Ž t G 2 . ,
t !# "
r s the spin rep. q the ¨ ector rep. t !# "
q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 16 . Gs s Spin n Ž n s 9, 11 . ; r s the spin rep. q1 q ??? q 1 Ž t G 1 . . Ž 17 .
t !# "
Gs s Spin8 ; r s a half-spin rep. q the ¨ ector rep. q1 q ??? q 1 Ž t G 0 . .
Ž 18 .
t !# "
Gs s Spin10 ; r s L e q 1 q ??? q 1 Ž t G 4 . ,
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
245
t !# "
r s L e q the ¨ ector rep. q1 q ??? q 1 Ž t G 0 . , t !# "
r s L e q L e q 1 q ??? q 1 Ž t G 0 . , where L e s the e¨ en half-spin representation.
Ž 19 .
t !# "
Gs s Spin12 ; r s a half-spin rep. q1 q ??? q 1 Ž t G 1 . ,
r s a half-spin rep. q the ¨ ector rep. t !# "
q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 20 .
Gs s Spin14 ; r s a half-spin rep. q1 q ??? q 1 Ž t G 1 . .
Ž 21 .
Gs s G 2 ; r s L 2 q 1 q ??? q 1 Ž t G 2 . .
Ž 22 .
Gs s E6 ; r s L 1 q 1 q ??? q 1 Ž t G 1 . .
Ž 23 .
Gs s E7 ; r s L 6 q 1 q ??? q 1 Ž t G 1 . .
t !# "
t !# "
t !# "
ŽII. Ž H = GLm , r m L 1 . with only m s 1, 2, n y 2 and n y 1. Such t Ž H, r . with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SL 2 r 9q1 ; r s L 2 q 1 Ž r 9 G 3 . ,
r s L 2 q 1 q 1 Ž r 9 G 2. , r s L 2 q 1 q 1 q 1 Ž r 9 G 2. .
Ž 2.
Gs s SL 4 ; r s L 2 q L 1 .
Ž 3.
Gs s Spin10 ; r s a half-spin rep. q1 q 1,
r s a half-spin rep. q1 q 1 q 1.
Ž 4.
Gs s G 2 ; r s L 2 q 1.
ŽIII. Ž H = GLm , r m L 1 . with only m s 1, 2, 3, n y 3, n y 2, and t n y 1. Such Ž H, r . with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SL5 ; r s L 2 q 1.
Ž 2.
Gs s Spin 7 ; r s the spin rep. q1.
Ž 3.
Gs s Spin10 ; r s a half-spin rep. q1.
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KASAI, KIMURA, AND OTANI
ŽIV. Ž H = GLm , r m L 1 . with 1 F m F n y 1. Such Ž H, r . with t H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SOr ; r s L 1 q 1.
Ž 2.
Gs s Spr ; r s L 1 q 1 q ??? q 1 Ž 1 F t F 3 . .
t !# "
Proof. We obtain our results by wSKx, wSimplex, wtype Ix, and Section 3 in wtype IIx. LEMMA 3.2.
ž
t !# "
GLkqtq1 = Gs = SL ny1 , Ž r 1 q???qr k . m L 1 q 1 m L 1 q ??? q L 1 q LU1 1
ž
/
with k G 1, k q t G 2 and n s deg r 1 q ??? qdeg r k q t, is a PV if and only if it is Ž GL31 = Spr = SL 2 r , L 1 m L 1 q 1 m Ž L 1 q LU1 ... Proof. If t G 1, then we have n y 1 G deg r 1 q ??? qdeg r k so that it must be of type II. If t s 0, then we have k G 2 y t s 2 so that n y 1 ) deg r i Ž1 F i F k . and hence it is also of type II. For t s 0 Žresp. t s 1, t G 2., we have our results by Proposition 3.8 Žresp. Theorems 3.10, 3.12. in wtype IIx. PROPOSITION 3.3. Let Ž H, r . with deg r s n be one of ŽI. in Proposition 3.1. Then r Ž H . _ GLn is P-spherical if and only if P is P Ž1, n y 1. or P Ž n y 1, 1.. Proof. By Propositions 1.3 and 3.1, r Ž H . _ GLn is P Ž e1 , e2 .-spherical if and only if Ž e1 , e2 . s Ž1, n y 1. or Ž n y 1, 1.. If r Ž H . _ GLn is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, n y 2, 1.. By Proposition 1.3, r Ž H . _ GLn is P Ž1, n y 2, 1.-spherical if and only if
Ž H = GLny 1 = GL1 , r m LU1 m 1 q 1 m L 1 m LU1 . ( GLtqkq1 = Gs = SL ny1 , Ž r 1 q ??? qr k . m L 1 1
ž
t !# "
q1 m L 1 q ??? q L 1 q LU1
ž
//
is a PV. By Lemma 3.2, it is a non-PV in our case ŽI.. Hence r Ž H . _ GLn is not P Ž e1 , . . . , e r .-spherical for any r G 3. t LEMMA 3.4. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ2. = GLŽ1., r m LU1 m 1 q 1 m L 1 m LU1 . tq1 ( Ž GLkq = Gs = SL 2 , r m L 1 q 1 m LU1 . is a PV if and only if Ž1. 1
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
247
Gs s SL 2 r 9q1 Ž r 9 G 2.; r s L 2 q 1, r s L 2 q 1 q 1, or Ž2. Gs s Spin10 ; r s L9 q 1, r s L9 q 1 q 1, where L9 s a half-spin representation. Proof. We have our results by wtype Ix. t LEMMA 3.5. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ n y 2. = GLŽ1., r m LU1 m 1 q 1 m L 1 m tq1 LU1 . ( Ž GLkq = Gs = SL ny2 , r m L 1 q 1 m LU1 . is a non-PV. 1
Proof. Assume that t s 1 and k s 1. Then it is of type I and we have our results by wtype Ix. If t s 0 and k G 2 Žresp. t s 1 and k G 2, resp. t G 2., then it is of type II, and by Proposition 3.8 Žresp. Theorem 3.10, resp. Theorem 3.12., we have our results. t LEMMA 3.6. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ n y 2. = GLŽ2., r m LU1 m 1 q 1 m L 1 m L*1 . is a non-PV.
Proof. Since dim G y dim V s dim H q 12 y 4 n - 0 for our case, we obtain our assertion. t PROPOSITION 3.7. Let Ž H, r . with H s GLkq = Gs Ž n s deg r . be one 1 Ž . Ž . of 1 Gs s SL 2 r 9q1 r 9 G 2 ; r s L 2 q 1 q 1 q 1, Ž2. Gs s SL 4 ; r s L 2 q L 1 , Ž3. Gs s Spin10 ; r s a half-spin rep. q1 q 1 q 1, Ž4. Gs s G 2 ; r s L 2 q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, n y 2, n y 1.
Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 4, 2. and Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. By Lemma 3.4 Žresp. Lemmas 3.2, 3.5, 3.6., it is a non-PV for Ž e1 , e2 . s Ž1, 1. Žresp. Ž1, n y 2., Ž1, n y 3., Ž2, n y 4... Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. t PROPOSITION 3.8. Let Ž H, r . with H s GLkq = Gs Ž n s deg r . be one 1 of Ž1. Gs s SL 2 r 9q1; r s L 2 q 1 Ž r 9 G 3., r s L 2 q 1 q 1 Ž r 9 G 2., Ž2. Gs s Spin10 ; r s a half-spin rep. q1 q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, n y 2, n y 1 and P Ž1, 1, n y 2., P Ž n y 2, 1, 1..
Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 4, 2. and Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . =
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KASAI, KIMURA, AND OTANI
GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. By Lemma 3.4, it is a PV for Ž e1 , e2 . s Ž1, 1.. It is a non-PV for Ž e1 , e2 . s Ž1, n y 2. Žresp. Ž1, n y 3., Ž2, n y 4.. by Lemma 3.2 Žresp. Lemmas 3.5, 3.6.. If it is P Ž e1 , . . . , e4 .spherical, we have e2 q e3 s 1, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. t LEMMA 3.9. Let Ž H, r . with H s GLkq = Gs be one of ŽIII. in Propo1 sition 3.1. Then Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a non-PV for Ž e1 , e2 . s Ž1, 2., Ž1, n y 4., Ž2, 1., Ž2, n y 5., and Ž3, n y 6..
Proof. For Ž e1 , e2 . s Ž1, 2., Ž2, 1. Žresp. Ž2, n y 5., Ž3, n y 6.., we have our results by Theorem 2.15 in wtype Ix Žresp. Theorem 3.15 in wKasaix.. Assume that Ž e1 , e2 . s Ž1, n y 4.. Then, for Ž1. Žresp. Ž2., Ž3.. of ŽIII. in Proposition 3.1, we have our results by Theorem 2.7 Žresp. Theorems 2.1, 2.5. in wtype Ix. PROPOSITION 3.10. Let Ž H, r . s Ž GL21 = Spin 7 , the spin rep. q1.. Then r Ž H . _ GL9 is P-spherical if and only if P s P Ž m, 9 y m. with m s 1, 2, 3, 6, 7, 8. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e 1 , e 2 , e 3 . -spherical, then we have Ž e 1 , e 2 , e 3 . s Ž1, 1, 7., Ž1, 2, 6., Ž1, 5, 3., Ž1, 6, 2., Ž1, 7, 1., Ž2, 1, 6., Ž2, 4, 3., Ž2, 5, 2., Ž3, 3, 3. Žand Ž2, 6, 1., Ž3, 4, 2., Ž3, 5, 1., Ž6, 1, 2., Ž6, 2, 1., Ž7, 1, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. However, it is a non-PV by Lemmas 3.2, 3.4]3.6, and 3.9. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 3.11. Let Ž H, r . s Ž GL21 = SL5 , L 2 q 1. with n s 11 or Ž GL21 = Spin10 , a half-spin rep. q1. with n s 17. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m., P Ž n y m, m. with m s 1, 2, 3 and P Ž1, 1, n y 2., P Ž n y 2, 1, 1.. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 4, 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 5, 3., Ž2, n y 4, 2., Ž3, n y 6, 3. Žand Ž2, n y 3, 1., Ž3, n y 5, 2., Ž3, n y 4, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. For Ž e1 , e2 . s Ž1, 1., it is a PV by Lemma 3.4. For the remaining eight cases, it is a non-PV by Lemmas 3.2, 3.5, 3.6, and 3.9. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then we have e2 q e3 s 1, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
249
PROPOSITION 3.12. Let Ž H, r . s Ž GL21 = SOr , L 1 q 1. with n s r q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with 1 F m F r s n y 1. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL21 = SOr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. Since the GLŽ e1 q e2 .-part of generic isotropy subgroup of Ž SOr = GLŽ e1 q e2 ., L 1 m LU1 . is SO Ž e1 q e 2 ., it is PV-equivalent to Ž GL21 = SO Ž e1 q e2 . = SLŽ e1 ., L 1 m 1 q L 1 m L 1 ., which is a non-PV by sublemma 2.4.2 in wtype Ix. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 3.13 ŽwKrx; Th. 2.24 in wKKYx.. Let Ž H, r . s Ž GL21 = Spr , L 1 q 1. with n s 2 r q 1. Then r Ž H . _ GLn is P-spherical for any parabolic subgroup P, i.e., it is spherical. PROPOSITION 3.14. Let Ž H, r . s Ž GL31 = Spr , L 1 q 1 q 1. with n s 2 r q 2. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. Ž1 F m F n y 1. and P Ž e1 , e2 , e3 . with e2 s odd. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL31 = Spr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. Then, by Proposition 13, page 40 in wSKx, when 2 r G e1 q e2 and Proposition 1.33 in wtype IIx, when e1 q e2 s 2 r q 1, it is PV-equivalent to Ž GL41 = SLŽ e1 q e2 . = SLŽ e1 ., Ž L 2 q L 1 q L 1 . m 1 q LU1 m L 1 .. When e1 q e2 s odd, it is a PV if and only if e1 s even, i.e., e2 s odd by Ž2. of Lemma 2.22 in wtype IIx. When e1 q e2 s even, it is PV-equivalent to Ž GL31 = SpŽ e . = SLŽ e1 ., L 1 m L 1 q Ž L 1 q L 1 . m 1. since the generic isotropy subgroup of Ž GLŽ e1 q e2 ., L 2 . is SpŽ e . with 2 e s e1 q e2 . By Lemma 2.20 and Theorem 2.24 in wtype Ix, it is a PV if and only if e1 s odd, i.e., e2 s odd. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then e2 , e3 and e2 q e3 must be odd, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .spherical for any r G 4. PROPOSITION 3.15. Let Ž H, r . s Ž GL41 = Spr , L 1 q 1 q 1 q 1. with n s 2 r q 3. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. Ž1 F m F n y 1.. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL41 = Spr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1 q 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. If 2 r q 1 G e1 q e2 , then similarly as in Proposition 3.14, we see that it is PV-equivalent to Ž GL51 = SLŽ e1 q e2 . = SLŽ e1 ., Ž L 2 q L 1 q L 1 q L 1 . m 1 q LU1 m L 1 .. When e1 q e2 s odd, it is a non-PV by Propo-
250
KASAI, KIMURA, AND OTANI
sition 2.26 in wtype IIx. When e1 q e2 s even, similarly as in Proposition 3 .1 4 , it is P V -e q u iv a le n t to Ž G L 41 = S p Ž e . = S L Ž e 1 . , L 1 m L 1 q Ž L 1 q L 1 q L 1 . m 1. with 2 e s e1 q e2 . By Lemmas 2.20 and 2.22 in wtype Ix, it is a non-PV. If e1 q e2 s 2 r q 2, then we have e3 s 1. Since P Ž e1 , e2 , e3 .-spherical if and only if P Ž e1 , e2 , e3 .-spherical by Corollary 1.2, we may assume that e1 s e3 s 1, e2 s 2 r q 1 since otherwise it reduces to the previous case. However, Ž GL51 = Spr = SLŽ2 r q 2., Ž L 1 q 1 q 1 q 1. m L 1 q 1 m LU1 . is a non-PV by Theorem 3.12 in wtype IIx. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. Remark 3.16. Finally, we mention the case
Ž H, r . s
ž
t GLkq 1
k !# "
t !# "
= SL r , L 1 q ??? q L 1 q 1 q ??? q 1 .
/
Assume that k G 2 and r ) f. Then r Ž H . _ GLn is P Ž f, kr q t y f .spherical if and only if
ž
t GLkq 1
k !# "
t !# "
= SL r = GL f , L q ??? q L 1 q 1 q ??? q 1 m L 1
ž
/
/
is a PV. By wtype IIx, it is so if and only if there exist m, s, and j with km F s, r s a jq1 s y a j m, f s a j s y a jy1 m, ta jq1 F m, where a j is defined by ay1 s y1, a0 s 0, and a j s ka jy1 y a jy2 Ž j G 2.. It is an open problem to determine when r Ž H . _ GLn is P Ž e1 , . . . , e t .-spherical for t G 3.
4. TABLE OF SIMPLE WEAKLY SPHERICAL HOMOGENEOUS SPACES r Ž H . _ GLn WITH k !# "
t !# "
t Ž H, r . / Ž GLkq = SL r , Ž L 1 q ??? q L 1 .Ž ) . q 1 q ??? q 1. 1
Note that if r Ž H . _ GLŽ n. is P Ž e1 , . . . , e r .-spherical, then it is also P-spherical for any parabolic subgroup P satisfying P > P Ž e1 , . . . , e r . or P > P Ž e r , . . . , e1 .. Therefore, in this table, we show only minimal parabolic subgroups P Ž e1 , . . . , e r . with e1 F e r Žor e2 F e ry1 , when e1 s e r . for which r Ž H . _ GLŽ n. is P Ž e1 , . . . , e r .-spherical ŽI. The irreducible case Ž1. Ž SLŽ k ., 2L 1 . _ GLŽ k Ž k q 1.r2. Ž k G 4., P Ž1, k Ž k q 1.r . 2y1 . Ž2. Ž SLŽ2 k ., L 2 . _ GLŽ2 k 2 y k . Ž k G 4., P Ž1, 2 k 2 y k y 1..
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
k y 2..
251
Ž3. Ž SLŽ2 k q 1., L 2 . _ GLŽ2 k 2 q k . Ž k G 3., P Ž1, 1, 2 k 2 q
Ž4. Ž5. Ž6. Ž7. Ž8. Ž9. Ž10. Ž11. Ž12. Ž13. Ž14. Ž15. Ž16. P Ž1, 1, 13, 1.. Ž17. Ž18. Ž19. Ž20. Ž21. Ž22. ŽII. The
Ž SLŽ2., 3L 1 . _ GLŽ4., P Ž1, 3.. Ž SLŽ3., 2L 1 . _ GLŽ6., P Ž1, 5., P Ž2, 4.. Ž SLŽ5., L 2 . _ GLŽ10., P Ž4, 6., P Ž1, 1, 1, 7., P Ž1, 1, 7, 1.. Ž SLŽ6., L 2 .rGLŽ15., P Ž1, 14., P Ž2, 13.. Ž SLŽ6., L 3 . _ GLŽ20., P Ž1, 19.. Ž SLŽ7., L 3 . _ GLŽ35., P Ž1, 34.. Ž SLŽ8., L 3 . _ GLŽ56., P Ž1, 55.. Ž SpŽ m., L 1 . _ GLŽ2 m., P Ž1, 1, . . . , 1.. Ž SpŽ3., L 3 . _ GLŽ14., P Ž1, 13.. Ž SO Ž n., L 1 . _ GLŽ n., P Ž1, 1, . . . , 1.. Ž SpinŽ7., the spin rep.. _ GLŽ8., P Ž1, 1, 1, 5., P Ž1, 1, 5, 1.. Ž SpinŽ9., the spin rep.. _ GLŽ16., P Ž1, 15.. Ž SpinŽ10., a half-spin rep.. _ GLŽ16., P Ž1, 1, 1, 13., Ž SpinŽ11., the spin rep.. _ GLŽ32., P Ž1, 31.. Ž SpinŽ12., a half-spin rep.. _ GLŽ32., P Ž1, 31.. Ž SpinŽ14., a half-spin rep.. _ GLŽ64., P Ž1, 63.. Ž G 2 , L 2 . _ GLŽ7., P Ž1, 1, 5., P Ž1, 5, 1.. Ž E6 , L 1 . _ GLŽ27., P Ž1, 26., P Ž2, 25.. Ž E7 , L 6 . _ GLŽ56., P Ž1, 55.. non-irreducible case r Ž H . _ GLŽ n. with
Ž H , r . / GL Ž 1 .
ž
kq t
k !# "
= SL Ž r . , L 1 q ??? q L 1
ž
U
/
t !# "
q 1 q ??? q 1
/
We write H s GLŽ1. kq t = Gs with a simple algebraic group Gs and t !# "
r s r 1 q ??? qr k q 1 q ??? q 1
Ž k G 1, k q t G 2 .
with non-trivial irreducible representations r 1 , . . . , r k . ky 1 !# "
t !# "
Ž1. Gs s SLŽ r ., r s L 1 q ??? q L 1 q LU1 q 1 q ??? q 1 Ž r q 1 G k G 2, t G 0., n s rk q t, P Ž1, n y 1.. t r Ž r q 1. !# " Ž2. Gs s SLŽ r ., r s 2L 1 q 1 q ??? q 1 Ž t G 1., n s q t, 2 P Ž1, n y 1..
252
KASAI, KIMURA, AND OTANI t !# "
Ž3. Gs s SL Ž r ., r s 2 L 1 q LŽ1) . q 1 q ??? q 1 Ž t G 0., r Ž r q 1. s q r q t, P Ž1, n y 1.. 2
n
t !# "
Ž4. Gs s SLŽ2 r 9. Ž r 9 G 3., r s L 2 q 1 q ??? q 1 Ž t G 1., n s r 9Ž2 r 9 . y 1 q t, P Ž1, n y 1.. s !# "
t !# "
Ž5. Gs s SLŽ2 r 9., r s L 2 q LŽ1) . q ??? q LŽ1) . q 1 q ??? q 1 Ž1 F s F 3; t G 0., n s r 9Ž2 r 9 y 1. q 2 r 9s q t, P Ž1, n y 1.. Ž6. Gs s SLŽ2 r 9 q 1. Ž r 9 G 3., r s L 2 q 1, n s r 9Ž2 r 9 q 1. q 1, P Ž1, 1, n y 2.. Ž7. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q 1, n s r 9Ž2 r 9 q 1. q 2, P Ž1, 1, n y 2.. Ž8. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q 1 q 1, n s r 9Ž2 r 9 q 1. q 3, P Ž1, n y 1., P Ž2, n y 2.. t !# "
Ž9. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q ??? q 1 Ž t G 4., n s Ž r 9 2 r 9 q 1. q t, P Ž1, n y 1.. t !# "
Ž10. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q L 2 q 1 q ??? q 1 Ž t G 0., n s 2 r 9Ž2 r 9 q 1. q t, P Ž1, n y 1.. Ž11.
t !# "
s !# "
Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q LŽ1) . q ??? q LŽ1) . q
1 q ??? q 1 Ž1 F s F 3; t G 0. except L 2 q L 1 q L 1 q LU1 q 1 q ??? q1, n s Ž r 9 q s .Ž2 r 9 q 1. q t, P Ž1, n y 1.. t !# "
Ž12.
Gs s SLŽ2., r s 3L 1 q 1 q ??? q 1 Ž t G 1., n s 4 q t, P Ž1, n y
Ž13.
Gs s SLŽ4., r s L 2 q L 1 , n s 10, P Ž1, 9., P Ž2, 8..
1.. t !# "
Ž14. Gs s SLŽ4., r s L 2 q L 1 q 1 q ??? q 1 Ž t G 1., n s 10 q t, Ž P 1, n y 1.. Ž15. Gs s SLŽ5., r s L 2 q 1, n s 11, P Ž3, 8., P Ž1, 1, 9.. t !# "
Ž16. Gs s SLŽ5., r s L 2 q L 2 q LU1 q 1 q ??? q 1 Ž t G 0., n s 25 q t, P Ž1, n y 1. Ž17. Gs s SLŽ6., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 20 q t, t !# "
Ž18. Gs s SLŽ6., r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0., n s 26 q t, P Ž1, n y 1..
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
253
t !# "
Ž19. Gs s SLŽ6., r s L 3 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 32 q t, P Ž1, n y 1.. Ž20. Gs s SLŽ7., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 35 q t, t !# "
Ž21. Gs s SLŽ7., r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0., n s 42 q t, P Ž1, n y 1.. Ž22. Gs s SLŽ8., Ž P 1, n y 1.. Ž23.
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 56 q t,
Gs s SpŽ r ., r s L 1 q 1, n s 2 r q 1, P Ž1, 1, . . . , 1..
Ž24. Gs s SpŽ r ., r s L 1 q 1 q 1, n s 2 r q 2, P Ž e1 , e2 , e3 . with e2 s odd. Ž25. Gs s SpŽ r ., r s L 1 q 1 q 1 q 1, n s 2 r q 3, P Ž m, n y m. Ž1 F m F n y m.. Ž26. Gs s SpŽ r ., P Ž1, n y 1..
t !# "
r s L 1 q 1 q ??? q 1 Ž t G 4., n s 2 r q t, t !# "
Ž27. Gs s SpŽ r ., r s L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 4 r q t, P Ž1, n y 1.. t !# "
Ž28. Gs s SpŽ r ., r s L 1 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 6 r q t, P Ž1, n y 1.. t !# "
Ž29. Gs s SpŽ2., r s L 2 q L 1 q 1 q ??? q 1 Ž t G 0., n s 9 q t, P Ž1, n y 1.. Ž30. Gs s SpŽ3., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 14 q t, t !# "
Ž31. Gs s SpŽ3., r s L 3 q L 1 q 1 q ??? q 1 Ž t G 0., n s 20 q t, P Ž1, n y 1.. Ž32. n y m..
Gs s SO Ž r ., r s L 1 q 1, n s r q 1, P Ž m, n y m. Ž1 F m F
Ž33. Gs s SO Ž r ., P Ž1, n y 1.. Ž34. P Ž3, 6..
t !# "
r s L 1 q 1 q ??? q 1 Ž t G 2., n s r q t,
Gs s SpinŽ7., r s the spin rep. q1, n s 9, P Ž1, 8., P Ž2, 7.,
254
KASAI, KIMURA, AND OTANI t !# "
Ž35. Gs s SpinŽ7., r s the spin rep. q1 q ??? q 1 Ž t G 2., n s 8 q t, P Ž1, n y 1.. t !# "
Ž36. Gs s SpinŽ7., r s the spin rep. q the vector rep. q1 q ??? q 1 Ž t G 0., n s 15 q t, P Ž1, n y 1.. Ž37. Gs s SpinŽ8., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 16 q t, P Ž1, n y 1..
t !# "
Ž38. Gs s SpinŽ9., r s the spin rep. q1 q ??? q 1 Ž t G 1., n s 16 q t, P Ž1, n y 1.. Ž39. Gs s SpinŽ10., r s a half-spin rep. q1, n s 17, PŽ3, 14., PŽ1, 1, 15.. Ž40. Gs s SpinŽ10., r s a half-spin rep. q1 q 1, n s 18, P Ž1, 1, 16.. Ž41. Gs s SpinŽ10., r s a half-spin rep. q1 q 1 q 1, n s 19, Ž P 1, 18., P Ž2, 17.. t !# "
Ž42. Gs s SpinŽ10., r s a half-spin rep. q1 q ??? q 1 Ž t G 4., n s 16 q t, P Ž1, n y 1.. Ž43. Gs s SpinŽ10., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 26 q t, P Ž1, n y 1..
t !# "
Ž44. Gs s SpinŽ10., r s L q L q 1 q ??? q 1 Ž t G 0., where L s the even half-spin representation, n s 32 q t, P Ž1, n y 1.. t !# "
Ž45. Gs s SpinŽ11., r s the spin rep. q1 q ??? q 1 Ž t G 1., n s 32 q t, P Ž1, n y 1.. t !# "
Ž46. Gs s SpinŽ12., r s a half-spin rep. q1 q ??? q 1 Ž t G 1., n s 32 q t, P Ž1, n y 1.. Ž47. Gs s SpinŽ12., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 44 q t, P Ž1, n y 1..
t !# "
Ž48. Gs s SpinŽ14., r s a half-spin rep. q1 q ??? q 1 Ž t G 1., n s 64 q t, P Ž1, n y 1.. Ž49. Gs s G 2 , r s L 2 q 1, n s 8, P Ž1, 7., P Ž2, 6.. t !# "
Ž50.
Gs s G 2 , r s L 2 q 1 q ??? q 1 Ž t G 2., n s 7 q t, P Ž1, n y 1..
Ž51.
Gs s E6 , r s L 1 q 1 q ??? q 1 Ž t G 1., n s 27 q t, P Ž1, n y 1..
Ž52.
Gs s E7 , r s L 6 q 1 q ??? q 1 Ž t G 1., n s 56 q t, P Ž1, n y 1..
t !# "
t !# "
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
255
REFERENCES wBx wKasaix wsimplex wtype Ix wtype IIx wKUYx wKKYx wKrx wSx wSKx wTx wVx
M. Brion, Classification des espaces homogenes spheriques, Compositio Math. 63 ` ´ Ž1987., 189]208. S. Kasai, A classification of reductive prehomogeneous vector spaces with two irreducible components, I, Japan J. Math. 14, No. 2 Ž1988., 385]418. T. Kimura, A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications, J. Algebra 83 Ž1983., 72]100. T. Kimura, S. Kasai, M. Inuzuka, and O. Yasukura, A classification of 2-simple prehomogeneous vector spaces of Type I, J. Algebra 114 Ž1988., 369]400. T. Kimura, S. Kasai, M. Taguchi, and M. Inuzuka, Some PV-equivalences and a classification of 2-simple prehomogeneous vector spaces of type II, Trans. Amer. Math. Soc. 308, No. 2 Ž1988., 433]494. T. Kimura, K. Ueda, and T. Yoshigaki, A classification of 3-simple prehomogeneous vector spaces of nontrivial type, Japanese J. Math., to appear. T. Kimura, S.-I. Kasai, and O. Yasukura, A classification of the representations of reductive algebraic groups which admit only a finite number of orbits, Amer. J. Math. 108 Ž1986., 643]691. M. Kramer, Spharische Untergruppen in kompakten zusammenhangenden ¨ ¨ ¨ Liegruppen, Compositio Math. 38 Ž1979., 129]153. F. Sato, Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, preprint. M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 Ž1977., 1]155. Y. Teranishi, Relative invariants and b-functions of prehomogeneous vector spaces Ž G = GLŽ d1 , . . . , d r ., r˜1 , M Ž n, C.., Nagoya Math. J. 98 Ž1985., 139]156. T. Vust, Operation de groupes reductifs dans un type de cone ´ ´ ˆ presque homogene, ` Bull. Soc. Math. France 102 Ž1974., 317]334.
182, 235]255 Ž1996.
0169
A Classification of Simple Weakly Spherical Homogeneous Spaces, I Shin-Ichi Kasai Department of Mathematics, Faculty of Education, Yamaguchi Uni¨ ersity, Yamaguchi, 753, Japan
and Tatsuo Kimura and Shin-Ichi Otani The Institute of Mathematics, Uni¨ ersity of Tsukuba, Ibaraki, 305, Japan Communicated by Corrado de Concini Received February 15, 1995
INTRODUCTION Let G be a connected linear algebraic group and H its closed subgroup. When a Borel subgroup of G has a Zariski-dense orbit in H _ G, we say that H _ G is spherical wBx. By wVx, symmetric spaces are spherical in this sense. More generally, when there exists a parabolic subgroup P of G such that H _ G has a Zariski-dense P-orbit, we say that H _ G is weakly spherical Žor P-spherical.. We shall consider everything over the complex number field C. To develop the theory of Eisenstein series on weakly spherical homogeneous spaces, we need many examples. When G s GLn , it is closely related with a prehomogeneous vector space wSx. A homogeneous space r Ž H . _ GLn is called simple when Ž H, r . is of the form Ž GLl1 = Gs r 1 q ??? q r l ., where each r i is an irreducible representation of a simple algebraic group Gs and GLl1 acts on each irreducible component as a scalar multiplication. In this paper ŽI., as the first trial toward the classification of weakly spherical homogeneous spaces, we shall classify all simple weakly spherical 235 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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KASAI, KIMURA, AND OTANI
homogeneous spaces r Ž H . _ GLn with
Ž H, r . /
ž
t GLkq 1
k !# "
= SL r , L 1 q ??? q L 1
ž
t !# "
Ž).
/
q 1 q ??? q 1 .
/
This paper consists of the following four sections. 1. Preliminaries 2. The case when r : H ª GLn is irreducible 3. Non-irreducible case r Ž H . _ GLn with k !# "
t = SL r , L 1 q ??? q L 1 Ž H , r . / GLkq 1
ž
ž
Ž).
/
t !# "
q 1 q ??? q 1
/
4. Table of simple weakly spherical homogeneous spaces r Ž H . _ GLŽ n. with
Ž H, r . /
ž
t GLkq Ž1.
k !# "
= SLŽ r . , L 1 q ??? q L 1
ž
Ž).
/
t !# "
q 1 q ??? q 1
/
1. PRELIMINARIES Let G be a connected linear algebraic group and H its closed subgroup. PROPOSITION 1.1. The following conditions are equi¨ alent. Ž1. Ž2. Ž3. Ž4.
H _ G is P-spherical. gHgy1 _ G is P-spherical for any g g G. H _ G is gPgy1 -spherical for any g g G. t y1 t y1 H _ G is tPy1-spherical.
Proof. Clearly H _ G is P-spherical if and only if G has a Zariski-dense orbit HyP. However, HyP is dense in G if and only if gHyP s Ž gHgy1 .Ž gy . P Žresp. HyPgy1 s H Ž ygy1 .Ž gPgy1 ., t Ž HyP .y1 s tHy1 ? t yy1 ? t y1 P . is dense in G. From now on, we shall assume that G s GLn Žs GLŽ n.. and let Pe1 , . . . , e r s P Ž e1 , . . . , e r .Ž e1 q ??? qe r s n. be a standard parabolic subgroup given as follows.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
P Ž e1 , . . . , e r .
¡P
P12
..
0
P22
..
0
0
..
0
0
0
11
s~
¢
. . .
237
¦
P1 r .. . . . g GLn ; Pi j g M Ž e i , e j . Ž 1 F i , j F n . . . Pr r
0
¥
§
Ž 1.1 . COROLLARY 1.2. The following conditions are equi¨ alent. Ž1. Ž2.
H _ GLn is P Ž e1 , e2 , . . . , e r .-spherical. Hy1 _ GLn is P Ž e r , e ry1 , . . . , e1 .-spherical.
t
In particular, if H is reducti¨ e, then H _ GLn is P Ž e r , e ry1 , . . . , e1 .spherical. Proof. Put 0 I˜n s
1 ?
?
?
1 1 1
0
0
.
˜ ˜t Ž .y1 ? Then we have I˜y1 n s In and one can easily check that In P e1 , . . . , e r y1 I˜n s P Ž e r , . . . , e1 .. Hence we have our result by Proposition 1.1. Put f 1 s e1 , f 2 s e1 q e2 , . . . , f i s e1 q ??? qe i , . . . , f r s e1 q ??? qe r s n. Let GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 . act on V s M Ž n, f ry1 . [ ??? [ y1 . M Ž f 2 , f 1 . by r Ž g . ¨ s Ž g r¨ ry1 gy1 for g s Ž g r , . . . , g 1 . g ry1 , . . . , g 2 ¨ 1 g 1 GLŽ f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ ry1 , . . . , ¨ 1 . g V s M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. Then it is a prehomogeneous vector space ŽPV.. The GLŽ f r .-part of a generic isotropy subgroup of a generic point ¨0 s
ž
I f ry 1 0
,...,
If1 0
/
is a parabolic subgroup P Ž e1 , . . . , e r .. Hence Ž H = P Ž e1 , . . . , e r ., M Ž n.. is a PV if and only if Ž H = GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n . [ M Ž n, f ry 1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where g ? ¨ s 1 y1 y1. Ž h¨ r gy for g s Ž h, g r , . . . , g 1 . g H = r , g r ¨ ry 1 g ry 1 , . . . , g 2 ¨ 1 g 1 GLŽ f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ r , . . . , ¨ 1 . g M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. It is clear that H _ GLn is P Ž e1 , . . . , e r .-spherical if and only if Ž H = P Ž e1 , . . . , e r ., M Ž n.. is a PV. Since a generic isotropy subgroup of
238
KASAI, KIMURA, AND OTANI
Ž H = GLŽ f r ., M Ž n.. at a generic point In is given by Ž h, h. g H = GLŽ f r .; h g H 4 ( H, it is also PV-equivalent to Ž H = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 ... Thus we obtain the following proposition, which we learned from Professor Fumihiro Sato wSx. PROPOSITION 1.3 ŽF. Sato.. Put f 1 s e1 , f 2 s e1 q e2 , . . . , f r s e1 q ??? qe r s n. Then the following conditions are equi¨ alent. Ž1. H _ GLn is P Ž e1 , . . . , e r .-spherical. Ž2. Ž H = GLŽ f r . = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where the action is gi¨ en by g ? ¨ s y1 y1 . Ž h¨ r gy1 for g s Ž h, g r , . . . , g 1 . g H = GL r , g r ¨ ry1 g ry1 , . . . , g 2 ¨ 1 g 1 Ž f r . = ??? = GLŽ f 1 . and ¨ s Ž ¨ r , . . . , ¨ 1 . g M Ž n. [ M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. Ž3. Ž H = GLŽ f ry1 . = ??? = GLŽ f 1 ., M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. is a PV, where the action is gi¨ en by g ? ¨ s Ž h¨ ry1 gy1 g ry1¨ ry2 ry1 , y1 . Ž . Ž . gy1 , . . . , g ¨ g for g s h, g , . . . , g g H = GL f ry2 2 1 1 ry1 1 ry1 = ??? = GLŽ f 1 . and ¨ s Ž ¨ ry1 , . . . , ¨ 1 . g M Ž n, f ry1 . [ ??? [ M Ž f 2 , f 1 .. As a corollary, we obtain the simple proof of Teranishi’s generalized castling transform. COROLLARY 1.4 ŽTeranishi wTx.. Let n s e1 q ??? qe r , and r : H ª GLŽ n. a rational representation. Then the following conditions are equi¨ alent. Ž1. Ž H = P Ž e1 , . . . , e ry1 ., r m LU1 , M Ž n, n y e r .. is a PV. Ž2. Ž H = P Ž e r , . . . , e2 ., r * m LU1 , M Ž n, n y e1 .. is a PV. Proof. Note that Ž1. is PV-equivalent to Ž3. in Proposition 1.3. Hence, by Corollary 1.2 and again by Proposition 1.3, it is PV-equivalent to Ž2.. Note that when r s 2, Corollary 1.4 gives a usual castling transformation.
2. THE CASE WHEN r : H ª GLn IS IRREDUCIBLE In this section, we shall consider an irreducible representation r : H ª GLn of a simple algebraic group H. PROPOSITION 2.1 ŽwSKx; p. 372 in wtype Ix.. We identify r Ž H . and H: r Ž H . s H. By Proposition 1.3, H _ GLn is P Ž m, n y m.-spherical if and only if Ž H = GLm , r m L 1 , M Ž n, m.. is a PV Ž n ) m G 1.. All such PV ’s are gi¨ en as follows.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
239
ŽI. Ž H = GLm , r m L 1 , M Ž n, m.. with only m s 1 or m s n y 1. Such Ž H, r . are gi¨ en as follows. Ž1. Ž SL k , 2L 1 . Ž k G 4. Ž2. Ž SL 2 k , L 2 . Ž k G 4. Ž3. Ž SL k , L 3 . Ž k s 6, 7, 8. Ž4. Ž SL 2 , 3L 1 . Ž5. Ž Sp 3 , L 3 . Ž6. Ž Spin k , the spin rep.. Ž k s 9, 11. Ž7. Ž Spin k , a half-spin rep.. Ž k s 12, 14. Ž8. Ž E7 , L 6 . with deg. 6L 6 s 56. ŽII. Ž H = GLm , r m L 1 , M Ž n, m.. with only m s 1, 2, n y 2, n y 1. Such Ž H, r . are gi¨ en as follows. Ž1. Ž SL 2 kq1 , L 2 . Ž k G 3. Ž2. Ž SL6 , L 2 . Ž3. Ž SL 3 , 2L 1 . Ž4. Ž G 2 , L 2 . with deg L 2 s 7 Ž5. Ž E6 , L 1 . with deg L 1 s 27. ŽIII. Ž1. Ž Spin 7 = GL m , the spin rep.m L 1 , V Ž8. m V Ž m.., m s 1, 2, 3, 5, 6, 7 Ž2. Ž Spin10 = GLm , a half-spin rep.m L 1 , V Ž16. m V Ž m.., m s 1, 2, 3, 13, 14, 15 Ž3. Ž SL5 = GLm , L 2 m L 1 , V Ž10. m V Ž m.., 1 F m F 9, m / 5 ŽIV. Ž1. Ž SOn = GLm , L 1 m L 1 , V Ž n. m V Ž m.., 1 F m F n y 1 Ž2. Ž Spk = GLm , L 1 m L 1 , V Ž n. m V Ž m.., 1 F m - n s 2 k. PROPOSITION 2.2. Let Ž H, r . with deg r s n be one of ŽI. in Proposition 2.1. Then H _ GLn is P-spherical if and only if P is Ž conjugate to . P Ž1, n y 1. or P Ž n y 1, 1.. Note that we identify H and r Ž H . Ž; GLn .. Proof. By Proposition 1.3, H _ GLn is P Ž e1 , e2 .-spherical if and only if Ž e1 , e2 . s Ž1, n y 1. or Ž n y 1, 1.. If H _ GLn is P Ž e1 , e2 , e3 .-spherical, then it must be P Ž e1 q e2 , e3 . and P Ž e1 , e2 q e3 .-spherical since P Ž e1 q e2 , e3 . > P Ž e1 , e2 , e3 ., etc. Hence we have Ž e1 , e2 , e3 . s Ž1, n y 2, 1.. By Proposition 1.3, H _ GLn is P Ž1, n y 2, 1.-spherical if and only if Ž H = GLny 1 = GL1 , r m LU1 m 1 q 1 m L 1 m LU1 . ( Ž GL21 = H = SL ny1 , r m LU1 q 1 m L 1 . is a PV. However, it is a non-PV by Theorem 2.1 and Lemma 2.2, page 375 in wtype Ix. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 2.3. Let H s L 2 Ž SL 2 kq1 . with k G 3 so that n s deg L 2 s k Ž2 k q 1.. Then H _ GLn is P-spherical if and only if P s P Ž e1 , e2 . with Ž e1 , e2 . s Ž1, n y 1., Ž2, n y 2., Ž n y 2, 2., Ž n y 1, 1. and P Ž1, 1, n y 2., P Ž n y 2, 1, 1.. Proof. We obtain our results for P Ž e1 , e2 . by Propositions 1.3 and 2.1. If H _ GLn is P Ž e1 , e2 , e3 .-spherical, then it must be P Ž e1 q e2 , e3 . and
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P Ž e 1 , e 2 q e 3 .-spherical. Hence Ž e 1 , e 2 , e 3 . s Ž1, 1, n y 2 ., Ž n y 2, 1, 1., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 3, 1., and Ž2, n y 4, 2.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1. and Ž2, n y 3, 1.. By Proposition 1.3, H _ GLn is P Ž e1 , e2 , e3 .-spherical if and only if Ž SL 2 kq1 = GLŽ e1 q e2 . = GLŽ e1 ., L 2 m LU1 m 1 q 1 m L 1 m LU1 . is a PV. When e1 s 1, it is equivalent to Ž GL21 = SL 2 kq1 = SLŽ e2 q 1., L 2 m LU1 q 1 m L 1 . and it is a PV when e2 s 1 ŽTheorem 2.11 for k s 3, Theorem 2.7 for k s 4, and Theorem 2.5 for k G 5 in wtype Ix.. However, it is a non-PV when e2 s n y 3 ŽLemma 2.2 for k s 3, 4 and Theorem 2.1 for k G 5 in wtype Ix. or when e2 s n y 2 ŽTheorem 2.12 for k s 3 and Theorem 2.7 for k G 4 in wtype Ix.. When e1 s 2 and e2 s n y 4, it is a non-PV by dimension reason. Now if H _ GLn is P Ž e1 , e2 , e3 , e4 .-spherical, then it must be P Ž e1 , e2 , e3 q e4 . and P Ž e1 q e2 , e3 , e4 .-spherical and hence e2 s e3 s 1, and e1 , e4 is 1 or n y 3. If it is P Ž1, 1, 1, n y 3. Žresp. P Ž n y 3, 1, 1, 1..-spherical, it must be P Ž1, 2, n y 3. Žresp. P Ž n y 3, 2, 1..spherical, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. PROPOSITION 2.4. Let H s L 2 Ž SL6 . with n s 15 Ž resp. 2L 1Ž SL 3 . with n s 6, L 1Ž E6 . with n s 27.. Then H _ GLn is P-spherical if and only if P s P Ž1, n y 1., P Ž2, n y 2., P Ž n y 2, 2., and P Ž n y 1, 1.. Proof. By Propositions 1.3 and 2.1, we have our assertion for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then Ž e1 q e2 , e3 . and Ž e1 , e2 q e3 . must be Ž1, n y 1., Ž2, n y 2., Ž n y 2, 2., or Ž n y 1, 1.. Hence we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, n y 4, 2. and Ž2, n y 3, 1., Ž n y 2, 1, 1.. By Corollary 1.2, it is not necessary to consider Ž2, n y 3, 1., Ž n y 2, 1, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV, where r : H ª GLn is an injection. When e1 s 1, it is equivalent to Ž GL21 = H = SLŽ e2 q 1., r m L 1 q 1 m LU1 ., which is a non-PV for e2 s 1, 12, 13 by Lemma 2.2 in wtype Ix Žresp. for e2 s 1, 3 by Lemma 2.2 in wtype Ix and for e2 s 4 by Theorem 2.1 in wtype Ix; resp. for e2 s 1, 24, 25 by Theorem 2.1 in wtype Ix.. By dimension reason, it is a non-PV for e1 s 2 and e2 s 11 Žresp. e2 s 2, e2 s 23.. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 2.5. Let H s L 2 Ž G 2 . with n s 7. Then H _ GL7 is Pspherical if and only if P s P Ž e1 , e2 . with Ž e1 , e2 . s Ž1, 6., Ž2, 5., Ž5, 2., Ž6, 1. and P s P Ž1, 1, 5., P Ž1, 5, 1., P Ž5, 1, 1.. Proof. For P s P Ž e1 , e2 ., we have our results by Propositions 1.3 and 2.1. If it is P Ž e1 , e2 , e3 .-spherical, we have Ž e1 , e2 , e3 . s Ž1, 1, 5., Ž1, 4, 2., Ž1, 5, 1., Ž2, 3, 2. Žand Ž2, 4, 1., Ž5, 1, 1... By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž G 2 = GLŽ e1 q e2 . = GLŽ e1 ., L 2 m
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
241
LU1 m 1 q 1 m L 1 m LU1 . is a PV. If e1 s 1, it is equivalent to Ž GL21 = G 2 = SLŽ e2 q 1., L 2 m L 1 q 1 m LU1 ., which is a PV for e1 s 1 and 5 by Theorem 2.5 in wtype Ix while it is a non-PV for e2 s 4 by Theorem 2.1 in wtype Ix. When e1 s 2 and e2 s 3, it is a non-PV by a dimension reason. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then it is P Ž e1 q e2 , e3 , e4 ., P Ž e1 , e2 q e3 , e4 ., P Ž e1 , e2 , e3 q e4 .-spherical and hence e1 , . . . , e4 s 1 or 5 with e1 q ??? qe4 s 7, which is impossible. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. PROPOSITION 2.6. Let H s LŽ Spin 7 . with n s 8 Ž resp. L9Ž Spin10 . with n s 16, L 2 Ž SL5 . with n s 10., where L Ž resp. L9. is the spin Ž resp. a half-spin. representation. Then H _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, 3, 5, 6, 7 Ž resp. 1, 2, 3, 13, 14, 15; resp. 1 F m F 9, m / 5.; P Ž1, 1, n y 2., P Ž1, 2, n y 3., P Ž1, n y 3, 2., P Ž1, n y 2, 1., P Ž2, 1, n y 3. Ž and P Ž2, n y 3, 1., P Ž n y 3, 1, 2., P Ž n y 3, 2, 1. P Ž n y 2, 1, 1..; P Ž1, 1, 1, n y 3., P Ž1, 1, n y 3, 1. Ž and P Ž1, n y 3, 1, 1., P Ž n y 3, 1, 1, 1... Proof. By Propositions 1.3 and 2.1, we have our assertion for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then similarly to the above, we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 4, 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 5, 3., Ž2, n y 4, 2., Ž3, n y 6, 3. Žand Ž2, n y 3, 1., Ž3, n y 5, 2., Ž3, n y 4, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1.; cf. Corollary 1.2.. Moreover, in the case H s L 2 Ž SL5 ., we have also Ž1, 3, 6 ., Ž1, 5, 4 ., Ž2, 2, 6 ., Ž2, 4, 4 ., Ž3, 1, 6 ., Ž3, 3, 4 ., Ž4, 2, 4 . Žand Ž4, 3, 3., Ž4, 4, 2., Ž4, 5, 1., Ž6, 1, 3., Ž6, 2, 2., Ž6, 3, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV, where r : H ª GLn is the inclusion. When e1 s 1, it is equivalent to Ž GL21 = H = SLŽ e2 q 1., r m L 1 q 1 m LU1 ., which is a PV for e2 s 1, 2, n y 3 by Theorem 2.5 Žfor L 2 Ž SL5 . by Theorems 2.8, 2.5, 2.7, respectively. and e2 s n y 2 by Theorem 2.7 Žfor L 2 Ž SL5 . by Theorem 2.9. in wtype Ix, while it is a non-PV for e2 s n y 4 Žfor L 2 Ž SL5 ., also e2 s 3, 5. by Theorem 2.1 in wtype Ix. If Ž e1 , e2 . s Ž2, 1., it is a PV by Theorem 2.5 in wtype Ix. If Ž e1 , e2 . s Ž2, n y 5., Ž3, n y 6., it is a non-PV by Lemma 2.17 in wKasaix. If Ž e1 , e2 . s Ž2, n y 4., it is a non-PV by Theorem 3.15 in wKasaix. When H s L 2 Ž SL5 . with Ž e1 , e2 . s Ž2, 2., Ž2, 4., Ž3, 1., Ž3, 3., and Ž4.2., it is a nonPV by Lemma 2.16 in wKasaix. Hence if it is P Ž e1 , e2 , e3 , e4 .-spherical, then Ž e1 q e2 , e3 , e4 ., Ž e1 , e2 q e3 , e4 ., Ž e1 , e2 , e3 q e4 . must be one of Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 3, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1., and we have Ž e1 , e2 , e3 , e4 . s Ž1, 1, 1, n y 3., Ž1, 1, n y 3, 1. Žand Ž1, n y 3, 1, 1., Ž n y 3, 1, 1, 1.; cf. Corollary 1.2.. It is P Ž1, 1, 1, n y 3.-spherical if and only if Ž H = GL3 = GL2 = GL1 , r m LU1 m 1 m 1 q 1 m L 1 m LU1 m 1 q 1 m 1 m L 1 m LU1 . is a PV. Since the
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GL3-part of a generic isotropy subgroup of Ž H = GL3 , r m LU1 . is SO 3 wSKx, it is a PV ŽTheorem 2.5 in wtype Ix.. Similarly it is P Ž1, 1, n y 3, 1.spherical if and only if Ž H = GLny 1 = GL2 = GL1 , r m LU1 m 1 m 1 q 1 m L 1 m LU1 m 1 q 1 m 1 m L 1 m LU1 . is a PV. When H s LŽ Spin 7 ., the GL7-part of a generic isotropy subgroup of Ž H = GL7 , r m LU1 . is G 2 wSKx, and hence it is a PV ŽTheorem 2.5 in wtype Ix.. When H s L9Ž Spin10 ., the Lie algebra LieŽ H . Ž; M Ž16.. is given by page 120 in wSKx. Then the isotropy subalgebra at the following point, x 0 , is given by LieŽ SL 2 . and it is a PV. x0 s
½
I15 y1, 0, . . . , 0
, Ž a 5 q a 6 , a 11 q a 12 . ,
1 1
ž /5
Ž 2.1.
g M Ž 16, 15 . [ M Ž 15, 2 . [ M Ž 2, 1 . , where a i is a column vector in C 15 with ith entry 1 and remaining parts 0. When H s L 2 Ž SL5 ., the isotropy subalgebra at Ž a 1 n a 3 q a 2 n a 4 , a 1 n a 2 , a 1 n a 4 , a 3 n a 4 , a 2 n a 3 , a 1 n a5 , a 2 n a5 , a 3 n a5 , a 4 n a 5 .,Ž b 2 q b4 , b 1 q b6 q b 7 ., g 1 .4 , where a i Žresp. bi , g i . is a column vector in C 5 Žresp. C 9 , C 2 . with ith entry 1 and remaining parts 0, is 04 and hence it is a PV. Now if it is P Ž e1 , . . . , e5 .-spherical, then e1 , . . . , e5 s 1 or n y 3 with e1 q ??? qe5 s n, which is impossible. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 5. Finally, the following proposition is well known. PROPOSITION 2.7 ŽwKrx; Th. 2.27 and Prop. 4.23 in wKKYx.. SOn _ GLn and Spm _ GL2 m are P-spherical for any parabolic subgroup P; i.e., they are spherical. 3. THE NON-IRREDUCIBLE CASE r Ž H . _ GLn WITH k !# "
t !# "
t Ž H, r . / Ž GLkq = SL r , Ž L 1 q ??? q L 1 .Ž*. q 1 q ??? q 1. 1
Let t !# "
r s r 1 q ??? qr k q 1 q ??? q 1
Ž k G 1, k q t G 2 .
t be a representation of H s GLkq = Gs with a simple algebraic group Gs , 1 kq t where GL1 acts on each irreducible component as a scalar multiplication. Let n s deg r . Then by Proposition 1.3, r Ž H . _ GLn is P Ž m, n y m.-spherical if and only if Ž H = GLm r m L 1 . is a PV.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
PROPOSITION 3.1.
Ž H, r . /
ž
243
All such PV ’s Ž H = GLm , r m L 1 . with
t GLkq 1
k !# "
Ž).
= SL r , L 1 q ??? q L 1
ž
t !# "
q 1 q ??? q 1
/
/
are gi¨ en as follows. ŽI. Ž H = GLm r m L 1 ., with only m s 1 or m s n y 1. Such Ž H, r . t with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
ky 1 !# "
t !# "
Gs s SL r ; r s L 1 q ??? q L 1 q LU1 q 1 q ??? q 1
Ž r q 1 G k G 2, t G 0 . . Ž 2.
t !# "
Gs s SL r ; r s 2L 1 q 1 q ??? q 1 Ž t G 1 . , t !# "
r s 2L 1 q LŽ1) . q 1 q ??? q 1 Ž t G 0 . .
Ž 3.
t !# "
Gs s SL 2 r 9 Ž r 9 G 3 . ; r s L 2 q 1 q ??? q 1 Ž t G 1 . ,
r s L2 q
s !# "
LŽ1) . q ??? q LŽ1) .
t !# "
q 1 q ??? q 1
Ž 1 F s F 3; t G 0 . . Ž 4.
t !# "
Gs s SL 2 r 9q1 Ž r 9 G 2 . ; r s L 2 q 1 q ??? q 1 Ž t G 4 . , t !# "
r s L 2 q L 2 q 1 q ??? q 1 Ž t G 0 . , s !# "
t !# "
r s L 2 q LŽ1) . q ??? q LŽ1) . q 1 q ??? q 1
Ž 1 F s F 3; t G 0 . except L 2 q L 1 q L 1 q LU1 q 1 q ??? q1. t !# "
Ž 5.
Gs s SL 2 ; r s 3L 1 q 1 q ??? q t Ž t G 1 . .
Ž 6.
Gs s SL 4 ; L 2 q L 1 q 1 q ??? q 1 Ž t G 1 . .
Ž 7.
t !# "
Gs s SL5 ; r s L 2 q L 2 q
LU1
t !# "
q 1 q ??? q 1 Ž t G 0 . .
244
Ž 8.
KASAI, KIMURA, AND OTANI t !# "
Gs s SL6 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . ,
r s L3 q
LŽ1) .
t !# "
q 1 q ??? q 1 Ž t G 0 . , t !# "
r s L 3 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . .
Ž 9.
t !# "
Gs s SL 7 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . , t !# "
r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0 . .
Ž 10 . Ž 11 .
t !# "
Gs s SL8 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . . t !# "
Gs s Spr ; r s L 1 q 1 q ??? q 1 Ž t G 4 . , t !# "
r s L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . , t !# "
r s L 1 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 12 .
Gs s Sp 2 ; r s L 2 q L 1 q 1 q ??? q 1 Ž t G 0 . .
Ž 13 .
Gs s Sp 3 ; r s L 3 q 1 q ??? q 1 Ž t G 1 . ,
t !# "
r s L 3 q L 1 q 1 q ??? q1 Ž t G 0 . . t !# "
Ž 14 .
Gs s SOr ; r s L 1 q 1 q ??? q 1 Ž t G 2 . .
Ž 15 .
Gs s Spin 7 ; r s the spin rep. q1 q ??? q 1 Ž t G 2 . ,
t !# "
r s the spin rep. q the ¨ ector rep. t !# "
q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 16 . Gs s Spin n Ž n s 9, 11 . ; r s the spin rep. q1 q ??? q 1 Ž t G 1 . . Ž 17 .
t !# "
Gs s Spin8 ; r s a half-spin rep. q the ¨ ector rep. q1 q ??? q 1 Ž t G 0 . .
Ž 18 .
t !# "
Gs s Spin10 ; r s L e q 1 q ??? q 1 Ž t G 4 . ,
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245
t !# "
r s L e q the ¨ ector rep. q1 q ??? q 1 Ž t G 0 . , t !# "
r s L e q L e q 1 q ??? q 1 Ž t G 0 . , where L e s the e¨ en half-spin representation.
Ž 19 .
t !# "
Gs s Spin12 ; r s a half-spin rep. q1 q ??? q 1 Ž t G 1 . ,
r s a half-spin rep. q the ¨ ector rep. t !# "
q 1 q ??? q 1 Ž t G 0 . . t !# "
Ž 20 .
Gs s Spin14 ; r s a half-spin rep. q1 q ??? q 1 Ž t G 1 . .
Ž 21 .
Gs s G 2 ; r s L 2 q 1 q ??? q 1 Ž t G 2 . .
Ž 22 .
Gs s E6 ; r s L 1 q 1 q ??? q 1 Ž t G 1 . .
Ž 23 .
Gs s E7 ; r s L 6 q 1 q ??? q 1 Ž t G 1 . .
t !# "
t !# "
t !# "
ŽII. Ž H = GLm , r m L 1 . with only m s 1, 2, n y 2 and n y 1. Such t Ž H, r . with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SL 2 r 9q1 ; r s L 2 q 1 Ž r 9 G 3 . ,
r s L 2 q 1 q 1 Ž r 9 G 2. , r s L 2 q 1 q 1 q 1 Ž r 9 G 2. .
Ž 2.
Gs s SL 4 ; r s L 2 q L 1 .
Ž 3.
Gs s Spin10 ; r s a half-spin rep. q1 q 1,
r s a half-spin rep. q1 q 1 q 1.
Ž 4.
Gs s G 2 ; r s L 2 q 1.
ŽIII. Ž H = GLm , r m L 1 . with only m s 1, 2, 3, n y 3, n y 2, and t n y 1. Such Ž H, r . with H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SL5 ; r s L 2 q 1.
Ž 2.
Gs s Spin 7 ; r s the spin rep. q1.
Ž 3.
Gs s Spin10 ; r s a half-spin rep. q1.
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ŽIV. Ž H = GLm , r m L 1 . with 1 F m F n y 1. Such Ž H, r . with t H s GLkq = Gs are gi¨ en as follows. 1
Ž 1.
Gs s SOr ; r s L 1 q 1.
Ž 2.
Gs s Spr ; r s L 1 q 1 q ??? q 1 Ž 1 F t F 3 . .
t !# "
Proof. We obtain our results by wSKx, wSimplex, wtype Ix, and Section 3 in wtype IIx. LEMMA 3.2.
ž
t !# "
GLkqtq1 = Gs = SL ny1 , Ž r 1 q???qr k . m L 1 q 1 m L 1 q ??? q L 1 q LU1 1
ž
/
with k G 1, k q t G 2 and n s deg r 1 q ??? qdeg r k q t, is a PV if and only if it is Ž GL31 = Spr = SL 2 r , L 1 m L 1 q 1 m Ž L 1 q LU1 ... Proof. If t G 1, then we have n y 1 G deg r 1 q ??? qdeg r k so that it must be of type II. If t s 0, then we have k G 2 y t s 2 so that n y 1 ) deg r i Ž1 F i F k . and hence it is also of type II. For t s 0 Žresp. t s 1, t G 2., we have our results by Proposition 3.8 Žresp. Theorems 3.10, 3.12. in wtype IIx. PROPOSITION 3.3. Let Ž H, r . with deg r s n be one of ŽI. in Proposition 3.1. Then r Ž H . _ GLn is P-spherical if and only if P is P Ž1, n y 1. or P Ž n y 1, 1.. Proof. By Propositions 1.3 and 3.1, r Ž H . _ GLn is P Ž e1 , e2 .-spherical if and only if Ž e1 , e2 . s Ž1, n y 1. or Ž n y 1, 1.. If r Ž H . _ GLn is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, n y 2, 1.. By Proposition 1.3, r Ž H . _ GLn is P Ž1, n y 2, 1.-spherical if and only if
Ž H = GLny 1 = GL1 , r m LU1 m 1 q 1 m L 1 m LU1 . ( GLtqkq1 = Gs = SL ny1 , Ž r 1 q ??? qr k . m L 1 1
ž
t !# "
q1 m L 1 q ??? q L 1 q LU1
ž
//
is a PV. By Lemma 3.2, it is a non-PV in our case ŽI.. Hence r Ž H . _ GLn is not P Ž e1 , . . . , e r .-spherical for any r G 3. t LEMMA 3.4. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ2. = GLŽ1., r m LU1 m 1 q 1 m L 1 m LU1 . tq1 ( Ž GLkq = Gs = SL 2 , r m L 1 q 1 m LU1 . is a PV if and only if Ž1. 1
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
247
Gs s SL 2 r 9q1 Ž r 9 G 2.; r s L 2 q 1, r s L 2 q 1 q 1, or Ž2. Gs s Spin10 ; r s L9 q 1, r s L9 q 1 q 1, where L9 s a half-spin representation. Proof. We have our results by wtype Ix. t LEMMA 3.5. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ n y 2. = GLŽ1., r m LU1 m 1 q 1 m L 1 m tq1 LU1 . ( Ž GLkq = Gs = SL ny2 , r m L 1 q 1 m LU1 . is a non-PV. 1
Proof. Assume that t s 1 and k s 1. Then it is of type I and we have our results by wtype Ix. If t s 0 and k G 2 Žresp. t s 1 and k G 2, resp. t G 2., then it is of type II, and by Proposition 3.8 Žresp. Theorem 3.10, resp. Theorem 3.12., we have our results. t LEMMA 3.6. Let Ž H, r . with H s GLkq = Gs be one of ŽII., ŽIII. in 1 Proposition 3.1. Then Ž H = GLŽ n y 2. = GLŽ2., r m LU1 m 1 q 1 m L 1 m L*1 . is a non-PV.
Proof. Since dim G y dim V s dim H q 12 y 4 n - 0 for our case, we obtain our assertion. t PROPOSITION 3.7. Let Ž H, r . with H s GLkq = Gs Ž n s deg r . be one 1 Ž . Ž . of 1 Gs s SL 2 r 9q1 r 9 G 2 ; r s L 2 q 1 q 1 q 1, Ž2. Gs s SL 4 ; r s L 2 q L 1 , Ž3. Gs s Spin10 ; r s a half-spin rep. q1 q 1 q 1, Ž4. Gs s G 2 ; r s L 2 q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, n y 2, n y 1.
Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 4, 2. and Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. By Lemma 3.4 Žresp. Lemmas 3.2, 3.5, 3.6., it is a non-PV for Ž e1 , e2 . s Ž1, 1. Žresp. Ž1, n y 2., Ž1, n y 3., Ž2, n y 4... Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. t PROPOSITION 3.8. Let Ž H, r . with H s GLkq = Gs Ž n s deg r . be one 1 of Ž1. Gs s SL 2 r 9q1; r s L 2 q 1 Ž r 9 G 3., r s L 2 q 1 q 1 Ž r 9 G 2., Ž2. Gs s Spin10 ; r s a half-spin rep. q1 q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with m s 1, 2, n y 2, n y 1 and P Ž1, 1, n y 2., P Ž n y 2, 1, 1..
Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, n y 2, 1., Ž1, n y 3, 2., Ž2, n y 4, 2. and Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Corollary 1.2, it is not necessary to consider Ž n y 2, 1, 1., Ž2, n y 3, 1.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . =
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GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. By Lemma 3.4, it is a PV for Ž e1 , e2 . s Ž1, 1.. It is a non-PV for Ž e1 , e2 . s Ž1, n y 2. Žresp. Ž1, n y 3., Ž2, n y 4.. by Lemma 3.2 Žresp. Lemmas 3.5, 3.6.. If it is P Ž e1 , . . . , e4 .spherical, we have e2 q e3 s 1, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4. t LEMMA 3.9. Let Ž H, r . with H s GLkq = Gs be one of ŽIII. in Propo1 sition 3.1. Then Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a non-PV for Ž e1 , e2 . s Ž1, 2., Ž1, n y 4., Ž2, 1., Ž2, n y 5., and Ž3, n y 6..
Proof. For Ž e1 , e2 . s Ž1, 2., Ž2, 1. Žresp. Ž2, n y 5., Ž3, n y 6.., we have our results by Theorem 2.15 in wtype Ix Žresp. Theorem 3.15 in wKasaix.. Assume that Ž e1 , e2 . s Ž1, n y 4.. Then, for Ž1. Žresp. Ž2., Ž3.. of ŽIII. in Proposition 3.1, we have our results by Theorem 2.7 Žresp. Theorems 2.1, 2.5. in wtype Ix. PROPOSITION 3.10. Let Ž H, r . s Ž GL21 = Spin 7 , the spin rep. q1.. Then r Ž H . _ GL9 is P-spherical if and only if P s P Ž m, 9 y m. with m s 1, 2, 3, 6, 7, 8. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e 1 , e 2 , e 3 . -spherical, then we have Ž e 1 , e 2 , e 3 . s Ž1, 1, 7., Ž1, 2, 6., Ž1, 5, 3., Ž1, 6, 2., Ž1, 7, 1., Ž2, 1, 6., Ž2, 4, 3., Ž2, 5, 2., Ž3, 3, 3. Žand Ž2, 6, 1., Ž3, 4, 2., Ž3, 5, 1., Ž6, 1, 2., Ž6, 2, 1., Ž7, 1, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. However, it is a non-PV by Lemmas 3.2, 3.4]3.6, and 3.9. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 3.11. Let Ž H, r . s Ž GL21 = SL5 , L 2 q 1. with n s 11 or Ž GL21 = Spin10 , a half-spin rep. q1. with n s 17. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m., P Ž n y m, m. with m s 1, 2, 3 and P Ž1, 1, n y 2., P Ž n y 2, 1, 1.. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. If it is P Ž e1 , e2 , e3 .-spherical, then we have Ž e1 , e2 , e3 . s Ž1, 1, n y 2., Ž1, 2, n y 3., Ž1, n y 4, 3., Ž1, n y 3, 2., Ž1, n y 2, 1., Ž2, 1, n y 3., Ž2, n y 5, 3., Ž2, n y 4, 2., Ž3, n y 6, 3. Žand Ž2, n y 3, 1., Ž3, n y 5, 2., Ž3, n y 4, 1., Ž n y 3, 1, 2., Ž n y 3, 2, 1., Ž n y 2, 1, 1.; cf. Corollary 1.2.. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if Ž H = GLŽ e1 q e2 . = GLŽ e1 ., r m LU1 m 1 q 1 m L 1 m LU1 . is a PV. For Ž e1 , e2 . s Ž1, 1., it is a PV by Lemma 3.4. For the remaining eight cases, it is a non-PV by Lemmas 3.2, 3.5, 3.6, and 3.9. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then we have e2 q e3 s 1, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 4.
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
249
PROPOSITION 3.12. Let Ž H, r . s Ž GL21 = SOr , L 1 q 1. with n s r q 1. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. with 1 F m F r s n y 1. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL21 = SOr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. Since the GLŽ e1 q e2 .-part of generic isotropy subgroup of Ž SOr = GLŽ e1 q e2 ., L 1 m LU1 . is SO Ž e1 q e 2 ., it is PV-equivalent to Ž GL21 = SO Ž e1 q e2 . = SLŽ e1 ., L 1 m 1 q L 1 m L 1 ., which is a non-PV by sublemma 2.4.2 in wtype Ix. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. PROPOSITION 3.13 ŽwKrx; Th. 2.24 in wKKYx.. Let Ž H, r . s Ž GL21 = Spr , L 1 q 1. with n s 2 r q 1. Then r Ž H . _ GLn is P-spherical for any parabolic subgroup P, i.e., it is spherical. PROPOSITION 3.14. Let Ž H, r . s Ž GL31 = Spr , L 1 q 1 q 1. with n s 2 r q 2. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. Ž1 F m F n y 1. and P Ž e1 , e2 , e3 . with e2 s odd. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL31 = Spr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. Then, by Proposition 13, page 40 in wSKx, when 2 r G e1 q e2 and Proposition 1.33 in wtype IIx, when e1 q e2 s 2 r q 1, it is PV-equivalent to Ž GL41 = SLŽ e1 q e2 . = SLŽ e1 ., Ž L 2 q L 1 q L 1 . m 1 q LU1 m L 1 .. When e1 q e2 s odd, it is a PV if and only if e1 s even, i.e., e2 s odd by Ž2. of Lemma 2.22 in wtype IIx. When e1 q e2 s even, it is PV-equivalent to Ž GL31 = SpŽ e . = SLŽ e1 ., L 1 m L 1 q Ž L 1 q L 1 . m 1. since the generic isotropy subgroup of Ž GLŽ e1 q e2 ., L 2 . is SpŽ e . with 2 e s e1 q e2 . By Lemma 2.20 and Theorem 2.24 in wtype Ix, it is a PV if and only if e1 s odd, i.e., e2 s odd. If it is P Ž e1 , e2 , e3 , e4 .-spherical, then e2 , e3 and e2 q e3 must be odd, i.e., a contradiction. Hence it is not P Ž e1 , . . . , e r .spherical for any r G 4. PROPOSITION 3.15. Let Ž H, r . s Ž GL41 = Spr , L 1 q 1 q 1 q 1. with n s 2 r q 3. Then r Ž H . _ GLn is P-spherical if and only if P s P Ž m, n y m. Ž1 F m F n y 1.. Proof. By Propositions 1.3 and 3.1, we have our results for P s P Ž e1 , e2 .. By Proposition 1.3, it is P Ž e1 , e2 , e3 .-spherical if and only if ŽŽ GL41 = Spr . = GLŽ e1 q e2 . = GLŽ e1 ., Ž L 1 q 1 q 1 q 1. m LU1 m 1 q 1 m L 1 m LU1 . is a PV. If 2 r q 1 G e1 q e2 , then similarly as in Proposition 3.14, we see that it is PV-equivalent to Ž GL51 = SLŽ e1 q e2 . = SLŽ e1 ., Ž L 2 q L 1 q L 1 q L 1 . m 1 q LU1 m L 1 .. When e1 q e2 s odd, it is a non-PV by Propo-
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sition 2.26 in wtype IIx. When e1 q e2 s even, similarly as in Proposition 3 .1 4 , it is P V -e q u iv a le n t to Ž G L 41 = S p Ž e . = S L Ž e 1 . , L 1 m L 1 q Ž L 1 q L 1 q L 1 . m 1. with 2 e s e1 q e2 . By Lemmas 2.20 and 2.22 in wtype Ix, it is a non-PV. If e1 q e2 s 2 r q 2, then we have e3 s 1. Since P Ž e1 , e2 , e3 .-spherical if and only if P Ž e1 , e2 , e3 .-spherical by Corollary 1.2, we may assume that e1 s e3 s 1, e2 s 2 r q 1 since otherwise it reduces to the previous case. However, Ž GL51 = Spr = SLŽ2 r q 2., Ž L 1 q 1 q 1 q 1. m L 1 q 1 m LU1 . is a non-PV by Theorem 3.12 in wtype IIx. Hence it is not P Ž e1 , . . . , e r .-spherical for any r G 3. Remark 3.16. Finally, we mention the case
Ž H, r . s
ž
t GLkq 1
k !# "
t !# "
= SL r , L 1 q ??? q L 1 q 1 q ??? q 1 .
/
Assume that k G 2 and r ) f. Then r Ž H . _ GLn is P Ž f, kr q t y f .spherical if and only if
ž
t GLkq 1
k !# "
t !# "
= SL r = GL f , L q ??? q L 1 q 1 q ??? q 1 m L 1
ž
/
/
is a PV. By wtype IIx, it is so if and only if there exist m, s, and j with km F s, r s a jq1 s y a j m, f s a j s y a jy1 m, ta jq1 F m, where a j is defined by ay1 s y1, a0 s 0, and a j s ka jy1 y a jy2 Ž j G 2.. It is an open problem to determine when r Ž H . _ GLn is P Ž e1 , . . . , e t .-spherical for t G 3.
4. TABLE OF SIMPLE WEAKLY SPHERICAL HOMOGENEOUS SPACES r Ž H . _ GLn WITH k !# "
t !# "
t Ž H, r . / Ž GLkq = SL r , Ž L 1 q ??? q L 1 .Ž ) . q 1 q ??? q 1. 1
Note that if r Ž H . _ GLŽ n. is P Ž e1 , . . . , e r .-spherical, then it is also P-spherical for any parabolic subgroup P satisfying P > P Ž e1 , . . . , e r . or P > P Ž e r , . . . , e1 .. Therefore, in this table, we show only minimal parabolic subgroups P Ž e1 , . . . , e r . with e1 F e r Žor e2 F e ry1 , when e1 s e r . for which r Ž H . _ GLŽ n. is P Ž e1 , . . . , e r .-spherical ŽI. The irreducible case Ž1. Ž SLŽ k ., 2L 1 . _ GLŽ k Ž k q 1.r2. Ž k G 4., P Ž1, k Ž k q 1.r . 2y1 . Ž2. Ž SLŽ2 k ., L 2 . _ GLŽ2 k 2 y k . Ž k G 4., P Ž1, 2 k 2 y k y 1..
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
k y 2..
251
Ž3. Ž SLŽ2 k q 1., L 2 . _ GLŽ2 k 2 q k . Ž k G 3., P Ž1, 1, 2 k 2 q
Ž4. Ž5. Ž6. Ž7. Ž8. Ž9. Ž10. Ž11. Ž12. Ž13. Ž14. Ž15. Ž16. P Ž1, 1, 13, 1.. Ž17. Ž18. Ž19. Ž20. Ž21. Ž22. ŽII. The
Ž SLŽ2., 3L 1 . _ GLŽ4., P Ž1, 3.. Ž SLŽ3., 2L 1 . _ GLŽ6., P Ž1, 5., P Ž2, 4.. Ž SLŽ5., L 2 . _ GLŽ10., P Ž4, 6., P Ž1, 1, 1, 7., P Ž1, 1, 7, 1.. Ž SLŽ6., L 2 .rGLŽ15., P Ž1, 14., P Ž2, 13.. Ž SLŽ6., L 3 . _ GLŽ20., P Ž1, 19.. Ž SLŽ7., L 3 . _ GLŽ35., P Ž1, 34.. Ž SLŽ8., L 3 . _ GLŽ56., P Ž1, 55.. Ž SpŽ m., L 1 . _ GLŽ2 m., P Ž1, 1, . . . , 1.. Ž SpŽ3., L 3 . _ GLŽ14., P Ž1, 13.. Ž SO Ž n., L 1 . _ GLŽ n., P Ž1, 1, . . . , 1.. Ž SpinŽ7., the spin rep.. _ GLŽ8., P Ž1, 1, 1, 5., P Ž1, 1, 5, 1.. Ž SpinŽ9., the spin rep.. _ GLŽ16., P Ž1, 15.. Ž SpinŽ10., a half-spin rep.. _ GLŽ16., P Ž1, 1, 1, 13., Ž SpinŽ11., the spin rep.. _ GLŽ32., P Ž1, 31.. Ž SpinŽ12., a half-spin rep.. _ GLŽ32., P Ž1, 31.. Ž SpinŽ14., a half-spin rep.. _ GLŽ64., P Ž1, 63.. Ž G 2 , L 2 . _ GLŽ7., P Ž1, 1, 5., P Ž1, 5, 1.. Ž E6 , L 1 . _ GLŽ27., P Ž1, 26., P Ž2, 25.. Ž E7 , L 6 . _ GLŽ56., P Ž1, 55.. non-irreducible case r Ž H . _ GLŽ n. with
Ž H , r . / GL Ž 1 .
ž
kq t
k !# "
= SL Ž r . , L 1 q ??? q L 1
ž
U
/
t !# "
q 1 q ??? q 1
/
We write H s GLŽ1. kq t = Gs with a simple algebraic group Gs and t !# "
r s r 1 q ??? qr k q 1 q ??? q 1
Ž k G 1, k q t G 2 .
with non-trivial irreducible representations r 1 , . . . , r k . ky 1 !# "
t !# "
Ž1. Gs s SLŽ r ., r s L 1 q ??? q L 1 q LU1 q 1 q ??? q 1 Ž r q 1 G k G 2, t G 0., n s rk q t, P Ž1, n y 1.. t r Ž r q 1. !# " Ž2. Gs s SLŽ r ., r s 2L 1 q 1 q ??? q 1 Ž t G 1., n s q t, 2 P Ž1, n y 1..
252
KASAI, KIMURA, AND OTANI t !# "
Ž3. Gs s SL Ž r ., r s 2 L 1 q LŽ1) . q 1 q ??? q 1 Ž t G 0., r Ž r q 1. s q r q t, P Ž1, n y 1.. 2
n
t !# "
Ž4. Gs s SLŽ2 r 9. Ž r 9 G 3., r s L 2 q 1 q ??? q 1 Ž t G 1., n s r 9Ž2 r 9 . y 1 q t, P Ž1, n y 1.. s !# "
t !# "
Ž5. Gs s SLŽ2 r 9., r s L 2 q LŽ1) . q ??? q LŽ1) . q 1 q ??? q 1 Ž1 F s F 3; t G 0., n s r 9Ž2 r 9 y 1. q 2 r 9s q t, P Ž1, n y 1.. Ž6. Gs s SLŽ2 r 9 q 1. Ž r 9 G 3., r s L 2 q 1, n s r 9Ž2 r 9 q 1. q 1, P Ž1, 1, n y 2.. Ž7. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q 1, n s r 9Ž2 r 9 q 1. q 2, P Ž1, 1, n y 2.. Ž8. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q 1 q 1, n s r 9Ž2 r 9 q 1. q 3, P Ž1, n y 1., P Ž2, n y 2.. t !# "
Ž9. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q 1 q ??? q 1 Ž t G 4., n s Ž r 9 2 r 9 q 1. q t, P Ž1, n y 1.. t !# "
Ž10. Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q L 2 q 1 q ??? q 1 Ž t G 0., n s 2 r 9Ž2 r 9 q 1. q t, P Ž1, n y 1.. Ž11.
t !# "
s !# "
Gs s SLŽ2 r 9 q 1. Ž r 9 G 2., r s L 2 q LŽ1) . q ??? q LŽ1) . q
1 q ??? q 1 Ž1 F s F 3; t G 0. except L 2 q L 1 q L 1 q LU1 q 1 q ??? q1, n s Ž r 9 q s .Ž2 r 9 q 1. q t, P Ž1, n y 1.. t !# "
Ž12.
Gs s SLŽ2., r s 3L 1 q 1 q ??? q 1 Ž t G 1., n s 4 q t, P Ž1, n y
Ž13.
Gs s SLŽ4., r s L 2 q L 1 , n s 10, P Ž1, 9., P Ž2, 8..
1.. t !# "
Ž14. Gs s SLŽ4., r s L 2 q L 1 q 1 q ??? q 1 Ž t G 1., n s 10 q t, Ž P 1, n y 1.. Ž15. Gs s SLŽ5., r s L 2 q 1, n s 11, P Ž3, 8., P Ž1, 1, 9.. t !# "
Ž16. Gs s SLŽ5., r s L 2 q L 2 q LU1 q 1 q ??? q 1 Ž t G 0., n s 25 q t, P Ž1, n y 1. Ž17. Gs s SLŽ6., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 20 q t, t !# "
Ž18. Gs s SLŽ6., r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0., n s 26 q t, P Ž1, n y 1..
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
253
t !# "
Ž19. Gs s SLŽ6., r s L 3 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 32 q t, P Ž1, n y 1.. Ž20. Gs s SLŽ7., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 35 q t, t !# "
Ž21. Gs s SLŽ7., r s L 3 q LŽ1) . q 1 q ??? q 1 Ž t G 0., n s 42 q t, P Ž1, n y 1.. Ž22. Gs s SLŽ8., Ž P 1, n y 1.. Ž23.
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 56 q t,
Gs s SpŽ r ., r s L 1 q 1, n s 2 r q 1, P Ž1, 1, . . . , 1..
Ž24. Gs s SpŽ r ., r s L 1 q 1 q 1, n s 2 r q 2, P Ž e1 , e2 , e3 . with e2 s odd. Ž25. Gs s SpŽ r ., r s L 1 q 1 q 1 q 1, n s 2 r q 3, P Ž m, n y m. Ž1 F m F n y m.. Ž26. Gs s SpŽ r ., P Ž1, n y 1..
t !# "
r s L 1 q 1 q ??? q 1 Ž t G 4., n s 2 r q t, t !# "
Ž27. Gs s SpŽ r ., r s L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 4 r q t, P Ž1, n y 1.. t !# "
Ž28. Gs s SpŽ r ., r s L 1 q L 1 q L 1 q 1 q ??? q 1 Ž t G 0., n s 6 r q t, P Ž1, n y 1.. t !# "
Ž29. Gs s SpŽ2., r s L 2 q L 1 q 1 q ??? q 1 Ž t G 0., n s 9 q t, P Ž1, n y 1.. Ž30. Gs s SpŽ3., P Ž1, n y 1..
t !# "
r s L 3 q 1 q ??? q 1 Ž t G 1., n s 14 q t, t !# "
Ž31. Gs s SpŽ3., r s L 3 q L 1 q 1 q ??? q 1 Ž t G 0., n s 20 q t, P Ž1, n y 1.. Ž32. n y m..
Gs s SO Ž r ., r s L 1 q 1, n s r q 1, P Ž m, n y m. Ž1 F m F
Ž33. Gs s SO Ž r ., P Ž1, n y 1.. Ž34. P Ž3, 6..
t !# "
r s L 1 q 1 q ??? q 1 Ž t G 2., n s r q t,
Gs s SpinŽ7., r s the spin rep. q1, n s 9, P Ž1, 8., P Ž2, 7.,
254
KASAI, KIMURA, AND OTANI t !# "
Ž35. Gs s SpinŽ7., r s the spin rep. q1 q ??? q 1 Ž t G 2., n s 8 q t, P Ž1, n y 1.. t !# "
Ž36. Gs s SpinŽ7., r s the spin rep. q the vector rep. q1 q ??? q 1 Ž t G 0., n s 15 q t, P Ž1, n y 1.. Ž37. Gs s SpinŽ8., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 16 q t, P Ž1, n y 1..
t !# "
Ž38. Gs s SpinŽ9., r s the spin rep. q1 q ??? q 1 Ž t G 1., n s 16 q t, P Ž1, n y 1.. Ž39. Gs s SpinŽ10., r s a half-spin rep. q1, n s 17, PŽ3, 14., PŽ1, 1, 15.. Ž40. Gs s SpinŽ10., r s a half-spin rep. q1 q 1, n s 18, P Ž1, 1, 16.. Ž41. Gs s SpinŽ10., r s a half-spin rep. q1 q 1 q 1, n s 19, Ž P 1, 18., P Ž2, 17.. t !# "
Ž42. Gs s SpinŽ10., r s a half-spin rep. q1 q ??? q 1 Ž t G 4., n s 16 q t, P Ž1, n y 1.. Ž43. Gs s SpinŽ10., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 26 q t, P Ž1, n y 1..
t !# "
Ž44. Gs s SpinŽ10., r s L q L q 1 q ??? q 1 Ž t G 0., where L s the even half-spin representation, n s 32 q t, P Ž1, n y 1.. t !# "
Ž45. Gs s SpinŽ11., r s the spin rep. q1 q ??? q 1 Ž t G 1., n s 32 q t, P Ž1, n y 1.. t !# "
Ž46. Gs s SpinŽ12., r s a half-spin rep. q1 q ??? q 1 Ž t G 1., n s 32 q t, P Ž1, n y 1.. Ž47. Gs s SpinŽ12., r s a half-spin rep. q the vector rep. t !# "
q1 q ??? q 1 Ž t G 0., n s 44 q t, P Ž1, n y 1..
t !# "
Ž48. Gs s SpinŽ14., r s a half-spin rep. q1 q ??? q 1 Ž t G 1., n s 64 q t, P Ž1, n y 1.. Ž49. Gs s G 2 , r s L 2 q 1, n s 8, P Ž1, 7., P Ž2, 6.. t !# "
Ž50.
Gs s G 2 , r s L 2 q 1 q ??? q 1 Ž t G 2., n s 7 q t, P Ž1, n y 1..
Ž51.
Gs s E6 , r s L 1 q 1 q ??? q 1 Ž t G 1., n s 27 q t, P Ž1, n y 1..
Ž52.
Gs s E7 , r s L 6 q 1 q ??? q 1 Ž t G 1., n s 56 q t, P Ž1, n y 1..
t !# "
t !# "
WEAKLY SPHERICAL HOMOGENEOUS SPACES, I
255
REFERENCES wBx wKasaix wsimplex wtype Ix wtype IIx wKUYx wKKYx wKrx wSx wSKx wTx wVx
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