ochiai/orthogonal

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 $n=1$.
• 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