Toshihiko Matsuki; Orthogonal multiple flag varieties of finite type
Part I
- p01. 1. Introduction.
- p04. 2. Exclusion of multiple flag varieties of infinite type
- p04. 2.1. A technical lemma.
- p05. 2.2. The case of .
- p05. 2.3. Proof of Proposition 1.2.
- p06. 2.4. A lemma for O_5(\mathbb{F}).
- p07. 2.5. Proof of Proposition 1.3.
- p07. 2.6. First lemma for O_6(\mathbb{F}).
- p09. 2.7. Second lemma for O_6(\mathbb{F}).
- p09. 2.8. Some conditions on a,b and c.
- p10. 2.9. A lemma for O_7(\mathbb{F}).
- p11. 2.10. Exclusion by Lemma 2.16.
- p11. 2.11. Conclusion.
- p11. 2.12. Proof of Proposition 1.4. *
- p12. 3. Orbits on \mathcal{T}_{(\alpha),(\beta),(\gamma)}.
- p12. 3.1. Preliminaries.
- p15. 3.3. Structure of R=....
- p16. 3.4. Invariants of the R-orbit of V \in M = M_{(n)}.
- p17. 3.5. Representative of the R-orbit of V.
- p23. 3.6. Proof of Theorem 3.15. (i)-(ix).
- p31. 3.7. Construction of elements in R_V|_{U_+}.
- p40. 4. Finiteness of \mathcal{T}_{(\alpha_1,\alpha_2), (\beta),(n)}.
- p40. 4.1. First reduction.
- p41. 4.2. Second reduction.
- p41. 4.3. Third reduction.
- p43. 4.4. Finiteness of orbits for S_3-part.
- p43. 4.5. A standards form of S_4.
- p45. 4.6. Normalization of g_4 h_5(S_5\oplus S_6).
- p46. 4.7. Construction of a subgroup of R'_V.
- p47. 4.8. Finiteness of S_5\oplus S_6-part.
- p47. 5. Finiteness of \mathcal{T}_{(\alpha),(1),(1^n)}.
- p48. 5.1. Case of a_-=1.
- p48. 5.1.1. Case of b_4=1.
- p49. 5.1.2. Case of b_{11}=1.
- p49. 5.1.3. Case of b_7=1.
- p51. 5.2. Case of a_1=1.
- p51. 5.2.1. Case of b_5=1.
- p52. 5.2.2. Case of b_6=1.
- p52. 5.2.3. Case of b_{15}=1.
- p53. 5.2.4. Case of b_8=1.
- p54. 5.2.5. Case of b_{13}=1. *
- p55. 6. Appendix.
- p55. 6.1. An elementary lemma for GL_n(\mathbb{F}).
- p56. 6.2. Finiteness of some orbit decomposition on the Grassmann variety.
- p57. 6.3. An orbit decomposition of GL_n(\mathbb{F})/B.
- p60. References.
Part II
- p01. 1. Introduction
- p04. 2. Preliminaries
- p06. 3. Exclusion of multiple flag varieties of infinite type
- p06. 3.1. Proof of Proposition 1.2.
- p08. 3.2. Proof of Proposition 1.3.
- p10. 3.3. Proof of Proposition 1.4.
- p11. 3.4. Proof of Proposition 1.5.
- p14. 3.5. Case of a=(n), b=(\beta) with 4\le \beta \le n-2.
- p16. 3.5.1. Case of r=3 and \gamma_1+\gamma_2+\gamma_3=n.
- p17. 3.5.2. Case of r=3 and \gamma_1+\gamma_2+\gamma_3 \lt n.
- p17. 3.5.3. Case of r=4 and \gamma_1+\gamma_2+\gamma_3+\gamma_4=n.
- p18. 3.5.4. Case of r=4 and \gamma_1+\gamma_2+\gamma_3+\gamma_4\lt n.
- p18. 3.5.5. Case of r\ge 5.
- p19. 3.6. Proof of Proposition 1.6(ii). *
- p20. 3.7. Proof of Proposition 1.6(iii). *
- p21. 4. Triple flag varieties of finite type.
- p22. 5. Review of technical results in [M15].
- p22. 5.1. Normalization of U_+ and U_-.
- p22. 5.2. Invariants of the R-orbit of V\in M = M_{(n)}.
- p24. 5.3. Representative of the R-orbit of V.
- p25. 5.4. Construction of elements in R_V.
- p28. 6. Proof of Proposition 4.1.
- p28. 6.1. Reduction of flags in U_+.
- p31. 6.2. Proof of (i) and (ii).
- p33. 6.3. Proof of (iii). (A),(B),(C).
- p33. 6.4. Proof of (iv). (A),(B),(B.1),(B.2),(B.2.1), (B.2.2), (B.2.3), (C),(D),(E).
- p36. 6.5. Proof of (v). (A),(A.1), (A.2), (B),(C).
- p37. 7. Proof of Proposition 4.2 and 4.3.
- p37. 7.1. Proof of Proposition 4.2.
(A), (B), (C), (D), (D.1), (D.1.1), (D.1.2), (D.1.3), (D.1.4), (D.1.5), (D.1.6), (D.2), (D.3), (D.3.1), (D.3.2), (D.3.3), (D.3.4), (D.3.5), (D.3.6), (D.4), (D.5), (D.5.1), (D.5.2), (D.5.3), (D.6), (D.6.1), (D.6.2), (D.6.2.1), (D.6.2.2), (D.6.2.3), (D.6.2.4), (D.6.2.5), (D.6.3).
- p45. 7.2. Proof of Proposition 4.3.
(A),(B)*,(C),(C.0), (C.1),(C.2),(C.3).
- p47. 8. Proof of Proposition 4.4.
- p47. 8.1. Proof of the case of \mathbf{b}=(\beta)=(n-1).
(A), (B),(C).
- p49. 8.2. Proof of the case of \mathbf{b}=(1,n-1).
(A),(B),(C),(D),(E),(F).
- p54. 9. Proof of Proposition 4.5.
(A),(B),(C),(D),(E),(F)(i)*, (F)(ii), (G)(i)*, (G)(ii). p57
(A), (A.1),(A.2),(B),(B.1), (B.2), (B.3), (B.4), (B.5), (C),
- p61. 10. Proof of Proposition 4.6.
(A),(A.1), (A.2), (A.3), (B), (B.1),(B.2),(B.3),(C),(D)(i)*,(D)(ii),(E),(E.1),(E.2),(F),(G),(H),(I)(i)*,(I)(ii)
- p69. 11. Proof of Proposition 4.7.
(A),(B),(C),(D),(E), (E.1),(E.2), (E.3) *, (E.4)
- p76. 12. Appendix
- p76. 12.1. Two lemmas.
- p77. 12.2. Extensions of Proposition 6.3 in [M15].
- p82. 12.3. Orbit decompositions of the full flag variety.
(A)-(K).
- p91. 12.4. A lemma.
- p92. 12.5. Lemmas related with SL_2(\mathbb{F}).
- p96. References