  # On K-spherical flag varieties joint work with Xuhua He, Hiroyuki - PowerPoint PPT Presentation

## On K-spherical flag varieties joint work with Xuhua He, Hiroyuki Ochiai & Yoshiki Oshima Kyo Nishiyama Aoyama Gakuin University Representations of Reductive Groups University of Utah (July 8-12, 2013) . . . . . . Nishiyama (AGU)

1. Motivation & problems Multiple Flag Varieties . . 2 Affine symmetric space G / K # B \ G / K < ∞ ⇝ G / K : spherical C [ G / K ] ≃ ⊕ λ V λ : mult-free decomp ( V K λ ̸ = 0 only appears) ∴ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 5 / 27

2. Motivation & problems Multiple Flag Varieties . . 2 Affine symmetric space G / K # B \ G / K < ∞ ⇝ G / K : spherical C [ G / K ] ≃ ⊕ λ V λ : mult-free decomp ( V K λ ̸ = 0 only appears) ∴ Interesting analytic result: L 2 ( G R / K R ) is also mult-free (with continuous spectrum) Harish-Chandra, van den Ban, Schrichtkrull, Oshima, ... . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 5 / 27

3. � � Motivation & problems Multiple Flag Varieties . . 2 Affine symmetric space G / K # B \ G / K < ∞ ⇝ G / K : spherical C [ G / K ] ≃ ⊕ λ V λ : mult-free decomp ( V K λ ̸ = 0 only appears) ∴ Interesting analytic result: L 2 ( G R / K R ) is also mult-free (with continuous spectrum) Harish-Chandra, van den Ban, Schrichtkrull, Oshima, ... Extra feature (KGB-theory): ■ K -orbits on X B = G / B with local system ← → K -equiv D -module on X B localization Harish-Chandra ( g , K )-modules . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 5 / 27

4. Motivation & problems Multiple Flag Varieties 3 G ↷ X = X P 1 × X P 2 × X B : triple flag variety . . ⇝ G \ X ≃ B \ ( X P 1 × X P 2 ) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 6 / 27

5. Motivation & problems Multiple Flag Varieties 3 G ↷ X = X P 1 × X P 2 × X B : triple flag variety . . ⇝ G \ X ≃ B \ ( X P 1 × X P 2 ) # G \ X < ∞ ⇐ ⇒ X P 1 × X P 2 is G -spherical . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 6 / 27

6. Motivation & problems Multiple Flag Varieties 3 G ↷ X = X P 1 × X P 2 × X B : triple flag variety . . ⇝ G \ X ≃ B \ ( X P 1 × X P 2 ) # G \ X < ∞ ⇐ ⇒ X P 1 × X P 2 is G -spherical Recall highest weight variety X λ = G · v λ s . t . P ( X λ ) = X P 1 X µ = G · v µ s . t . P ( X µ ) = X P 2 ⇒ X λ × X µ : G × C × × C × -spherical = ℓµ ≃ ⊕ ⇝ V ∗ k λ ⊗ V ∗ η V η : mult-free decomp ( ∀ k , ℓ ≥ 0) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 6 / 27

7. Motivation & problems Multiple Flag Varieties 3 G ↷ X = X P 1 × X P 2 × X B : triple flag variety . . ⇝ G \ X ≃ B \ ( X P 1 × X P 2 ) # G \ X < ∞ ⇐ ⇒ X P 1 × X P 2 is G -spherical Recall highest weight variety X λ = G · v λ s . t . P ( X λ ) = X P 1 X µ = G · v µ s . t . P ( X µ ) = X P 2 ⇒ X λ × X µ : G × C × × C × -spherical = ℓµ ≃ ⊕ ⇝ V ∗ k λ ⊗ V ∗ η V η : mult-free decomp ( ∀ k , ℓ ≥ 0) Classification : Panyushev (1993), Littelman (1994) · · · P 1 , P 2 : max psg · · · ∀ P 1 , P 2 Stembridge (2003) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 6 / 27

8. Motivation & problems Multiple Flag Varieties 3 G ↷ X = X P 1 × X P 2 × X B : triple flag variety . . ⇝ G \ X ≃ B \ ( X P 1 × X P 2 ) # G \ X < ∞ ⇐ ⇒ X P 1 × X P 2 is G -spherical Recall highest weight variety X λ = G · v λ s . t . P ( X λ ) = X P 1 X µ = G · v µ s . t . P ( X µ ) = X P 2 ⇒ X λ × X µ : G × C × × C × -spherical = ℓµ ≃ ⊕ ⇝ V ∗ k λ ⊗ V ∗ η V η : mult-free decomp ( ∀ k , ℓ ≥ 0) Classification : Panyushev (1993), Littelman (1994) · · · P 1 , P 2 : max psg · · · ∀ P 1 , P 2 Stembridge (2003) Interesting generalization: Next to spherical (complexity 1) · · · Ponomareva (2012, arXiv) ∃ Open orbit on mult flag var · · · Popov (2007) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 6 / 27

9. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

10. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic G = GL n ⊃ B : Borel & P = P ( n − 1 , 1) : max parabolic (mirabolic) ⇝ G / P ≃ P ( C n ) P : psg with diag blocks ( n − 1 , 1) X B × X B × X P ≃ F ℓ n × F ℓ n × P ( C n ) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

11. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic G = GL n ⊃ B : Borel & P = P ( n − 1 , 1) : max parabolic (mirabolic) ⇝ G / P ≃ P ( C n ) P : psg with diag blocks ( n − 1 , 1) X B × X B × X P ≃ F ℓ n × F ℓ n × P ( C n ) Many good properties are known due to Travkin, Finkelberg-Ginzburg-Travkin, Achar-Henderson . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

12. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic G = GL n ⊃ B : Borel & P = P ( n − 1 , 1) : max parabolic (mirabolic) ⇝ G / P ≃ P ( C n ) P : psg with diag blocks ( n − 1 , 1) X B × X B × X P ≃ F ℓ n × F ℓ n × P ( C n ) Many good properties are known due to Travkin, Finkelberg-Ginzburg-Travkin, Achar-Henderson Analogue of Robinson-Schensted-Knuth algorithm for Springer fiber micro-local cells and action of Hecke algebra, etc. . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

13. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic G = GL n ⊃ B : Borel & P = P ( n − 1 , 1) : max parabolic (mirabolic) ⇝ G / P ≃ P ( C n ) P : psg with diag blocks ( n − 1 , 1) X B × X B × X P ≃ F ℓ n × F ℓ n × P ( C n ) Many good properties are known due to Travkin, Finkelberg-Ginzburg-Travkin, Achar-Henderson Analogue of Robinson-Schensted-Knuth algorithm for Springer fiber micro-local cells and action of Hecke algebra, etc. Enhanced nilpotent cone and orbits on N ( g ) × C n , local intersection theory on the closure of nilpotent orbits . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

14. Motivation & problems Mirabolic, enhanced, exotic nilcone Mirabolic (= miraculous parabolic?) case For type A, ∃ special wonderful case called mirabolic G = GL n ⊃ B : Borel & P = P ( n − 1 , 1) : max parabolic (mirabolic) ⇝ G / P ≃ P ( C n ) P : psg with diag blocks ( n − 1 , 1) X B × X B × X P ≃ F ℓ n × F ℓ n × P ( C n ) Many good properties are known due to Travkin, Finkelberg-Ginzburg-Travkin, Achar-Henderson Analogue of Robinson-Schensted-Knuth algorithm for Springer fiber micro-local cells and action of Hecke algebra, etc. Enhanced nilpotent cone and orbits on N ( g ) × C n , local intersection theory on the closure of nilpotent orbits · · · · · · want to extend it to a symmetric pair . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 7 / 27

15. . . . . . . Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution Ex. ( G , K ) =( GL p + q , GL p × GL q ) , ( SL n , O n ) , ( SL 2 n , Sp 2 n ) , ( Sp 2 n , GL n ) , . . . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

16. . . . . . . Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

17. . . . Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . Fact . . For ∀ Q ⊂ K : psg in K , ∃ P ′ ⊂ G : θ -stable psg s.t. Q = P ′ ∩ K . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

18. Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . Fact . . For ∀ Q ⊂ K : psg in K , ∃ P ′ ⊂ G : θ -stable psg s.t. Q = P ′ ∩ K . Definition (Double flag variety) . X P := G / P : partial flag var of G & Z Q := K / Q : partial flag of K . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

19. Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . Fact . . For ∀ Q ⊂ K : psg in K , ∃ P ′ ⊂ G : θ -stable psg s.t. Q = P ′ ∩ K . Definition (Double flag variety) . X P := G / P : partial flag var of G & Z Q := K / Q : partial flag of K K ↷ X P ×Z Q : double flag variety ( K acts diagonally) . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

20. Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . Fact . . For ∀ Q ⊂ K : psg in K , ∃ P ′ ⊂ G : θ -stable psg s.t. Q = P ′ ∩ K . Definition (Double flag variety) . X P := G / P : partial flag var of G & Z Q := K / Q : partial flag of K K ↷ X P ×Z Q : double flag variety ( K acts diagonally) K ↷ X P ×Z Q is of finite type ⇐ ⇒ # K \ X P ×Z Q < ∞ . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

21. Double flag variety for symmetric pair Double flag variety Double flag variety — definition ( G , K ) : symmetric pair / C K ↔ θ : involution P ′ : θ -stable parabolic of G P : parabolic & ⇝ Q := P ′ ∩ K : parabolic of K . Fact . . For ∀ Q ⊂ K : psg in K , ∃ P ′ ⊂ G : θ -stable psg s.t. Q = P ′ ∩ K . Definition (Double flag variety) . X P := G / P : partial flag var of G & Z Q := K / Q : partial flag of K K ↷ X P ×Z Q : double flag variety ( K acts diagonally) K ↷ X P ×Z Q is of finite type ⇐ ⇒ # K \ X P ×Z Q < ∞ . H ( G , B ) ↷ H ∗ ( X P ×Z Q ) ↶ H ( K , B K ) Hecke alg module structure: . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 8 / 27

22. . Double flag variety for symmetric pair Double flag variety X P = G / P : PFV of G , Z Q = K / Q : PFV of K . Examples of X P ×Z Q : double flag var (DFV) of finite type . Type AI : G / K = SL n / SO n ( n ≥ 3) Z Q P Q X P extra condition Grass m ( C n ) maximal any Z Q LGrass( C n ) ( λ 1 , λ 2 , λ 3 ) Siegel X P n is even . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 9 / 27

23. . Double flag variety for symmetric pair Double flag variety X P = G / P : PFV of G , Z Q = K / Q : PFV of K . Examples of X P ×Z Q : double flag var (DFV) of finite type . Type AI : G / K = SL n / SO n ( n ≥ 3) Z Q P Q X P extra condition Grass m ( C n ) maximal any Z Q LGrass( C n ) ( λ 1 , λ 2 , λ 3 ) Siegel X P n is even Type AII : G / K = SL 2 n / Sp 2 n ( n ≥ 2) Z Q P Q X P Grass m ( C n ) maximal any Z Q LGrass m ( C 2 n ) ( λ 1 , λ 2 , λ 3 ) Siegel X P . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 9 / 27

24. . Double flag variety for symmetric pair Double flag variety X P = G / P : PFV of G , Z Q = K / Q : PFV of K . Examples of X P ×Z Q : double flag var (DFV) of finite type . Type AI : G / K = SL n / SO n ( n ≥ 3) Z Q P Q X P extra condition Grass m ( C n ) maximal any Z Q LGrass( C n ) ( λ 1 , λ 2 , λ 3 ) Siegel X P n is even Type AII : G / K = SL 2 n / Sp 2 n ( n ≥ 2) Z Q P Q X P Grass m ( C n ) maximal any Z Q LGrass m ( C 2 n ) ( λ 1 , λ 2 , λ 3 ) Siegel X P Type AIII : G / K = GL n / GL p × GL q ( n = p + q ) P Q 1 Q 2 Z Q X P P ( C p ) any mirabolic GL q X P P ( C q ) any GL p mirabolic X P Grass m ( C n ) maximal any any Z Q Grass k ( C p ) × Grass ℓ ( C q ) ( λ 1 , λ 2 , λ 3 ) maximal maximal X P . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 9 / 27

25. . . . Double flag variety for symmetric pair Relation to MFV Relation to multiple flag varieties for G . . 1 Triple flag variety X P 1 × X P 2 × X P 3 with G -action · · · special case of double flag variety X P ×Z Q with K -action . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 10 / 27

26. . Double flag variety for symmetric pair Relation to MFV Relation to multiple flag varieties for G . . 1 Triple flag variety X P 1 × X P 2 × X P 3 with G -action · · · special case of double flag variety X P ×Z Q with K -action . ( ∵ ) Take G = G × G and K = ∆ G as usual P = P 1 × P 2 , Q = ∆ P 3 ⇝ G / P × K / Q = G / P 1 × G / P 2 × G / P 3 . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 10 / 27

27. Double flag variety for symmetric pair Relation to MFV Relation to multiple flag varieties for G . . 1 Triple flag variety X P 1 × X P 2 × X P 3 with G -action · · · special case of double flag variety X P ×Z Q with K -action . ( ∵ ) Take G = G × G and K = ∆ G as usual P = P 1 × P 2 , Q = ∆ P 3 ⇝ G / P × K / Q = G / P 1 × G / P 2 × G / P 3 . 2 Z Q ≃ K · P ′ / P ′ � � closed � X P ′ . i.e. Z Q is a closed K -orbit in K \ X P ′ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 10 / 27

28. Double flag variety for symmetric pair Relation to MFV Relation to multiple flag varieties for G . . 1 Triple flag variety X P 1 × X P 2 × X P 3 with G -action · · · special case of double flag variety X P ×Z Q with K -action . ( ∵ ) Take G = G × G and K = ∆ G as usual P = P 1 × P 2 , Q = ∆ P 3 ⇝ G / P × K / Q = G / P 1 × G / P 2 × G / P 3 . 2 Z Q ≃ K · P ′ / P ′ � � closed � X P ′ . i.e. Z Q is a closed K -orbit in K \ X P ′ Thus we get a closed embedding: closed � X P × X P ′ with diag K -action X P ×Z Q � � . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 10 / 27

29. Double flag variety for symmetric pair Relation to MFV Relation to multiple flag varieties for G . . 1 Triple flag variety X P 1 × X P 2 × X P 3 with G -action · · · special case of double flag variety X P ×Z Q with K -action . ( ∵ ) Take G = G × G and K = ∆ G as usual P = P 1 × P 2 , Q = ∆ P 3 ⇝ G / P × K / Q = G / P 1 × G / P 2 × G / P 3 . 2 Z Q ≃ K · P ′ / P ′ � � closed � X P ′ . i.e. Z Q is a closed K -orbit in K \ X P ′ Thus we get a closed embedding: closed � X P × X P ′ with diag K -action X P ×Z Q � � In general # K \ ( X P × X P ′ ) = ∞ however, # of closed K -orbits on X P × X P ′ < ∞ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 10 / 27

30. . . . . . . . . . Description of K -orbits Strategy Key idea to describe K orbits on X P ×Z Q . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 11 / 27

31. . . Description of K -orbits Strategy Key idea to describe K orbits on X P ×Z Q . . 1 Reduction to triple flag var: # G \ ( X P × X θ ( P ) × X P ′ ) < ∞ = ⇒ # K \ ( X P ×Z Q ) < ∞ Unfortunately “ ⇐ ⇒ ” does not hold . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 11 / 27

32. . Description of K -orbits Strategy Key idea to describe K orbits on X P ×Z Q . . 1 Reduction to triple flag var: # G \ ( X P × X θ ( P ) × X P ′ ) < ∞ = ⇒ # K \ ( X P ×Z Q ) < ∞ Unfortunately “ ⇐ ⇒ ” does not hold . . 2 Bruhat reduction: G \ X P × X P ′ ≃ P \ G / P ′ ≃ W P \ W / W P ′ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 11 / 27

33. Description of K -orbits Strategy Key idea to describe K orbits on X P ×Z Q . . 1 Reduction to triple flag var: # G \ ( X P × X θ ( P ) × X P ′ ) < ∞ = ⇒ # K \ ( X P ×Z Q ) < ∞ Unfortunately “ ⇐ ⇒ ” does not hold . . 2 Bruhat reduction: G \ X P × X P ′ ≃ P \ G / P ′ ≃ W P \ W / W P ′ . . 3 Reduction to smaller affine symm spaces (KGB reduction): # P \ G / K < ∞ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 11 / 27

34. . . Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

35. . . Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding ⇝ ∆ θ × ι : X P × Z Q ֒ → X P × X θ ( P ) × X P ′ image X = (∆ θ × ι )( X P × Z Q ) : closed subvariety of TFV . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

36. Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding ⇝ ∆ θ × ι : X P × Z Q ֒ → X P × X θ ( P ) × X P ′ image X = (∆ θ × ι )( X P × Z Q ) : closed subvariety of TFV . O ∈ ( X P × X θ ( P ) × X P ′ ) / G : orbit for TFV . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

37. Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding ⇝ ∆ θ × ι : X P × Z Q ֒ → X P × X θ ( P ) × X P ′ image X = (∆ θ × ι )( X P × Z Q ) : closed subvariety of TFV . O ∈ ( X P × X θ ( P ) × X P ′ ) / G : orbit for TFV (∆ θ ( X P ) × X P ′ ) / ∆ θ ( G ) ∼ ( X P × X P ′ ) / G ∼ W P \ W / W P ′ ∋ ∋ � w O θ w � . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

38. Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding ⇝ ∆ θ × ι : X P × Z Q ֒ → X P × X θ ( P ) × X P ′ image X = (∆ θ × ι )( X P × Z Q ) : closed subvariety of TFV . O ∈ ( X P × X θ ( P ) × X P ′ ) / G : orbit for TFV (∆ θ ( X P ) × X P ′ ) / ∆ θ ( G ) ∼ ( X P × X P ′ ) / G ∼ W P \ W / W P ′ ∋ ∋ � w O θ w � ⇒ conn comp’s of O ∩ O θ w ∩ X are precisely K -orbits!! = . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

39. Description of K -orbits θ -twisted embedding Strategy 1: θ -twisted embedding ` a la Miliˇ ci´ c { ∆ θ : X P ֒ → X P × X θ ( P ) : θ -twisted embedding ι : Z Q ֒ → X P ′ : closed embedding ⇝ ∆ θ × ι : X P × Z Q ֒ → X P × X θ ( P ) × X P ′ image X = (∆ θ × ι )( X P × Z Q ) : closed subvariety of TFV . O ∈ ( X P × X θ ( P ) × X P ′ ) / G : orbit for TFV (∆ θ ( X P ) × X P ′ ) / ∆ θ ( G ) ∼ ( X P × X P ′ ) / G ∼ W P \ W / W P ′ ∋ ∋ � w O θ w � ⇒ conn comp’s of O ∩ O θ w ∩ X are precisely K -orbits!! = . ⇝ parametrization of ( X P × Z Q ) / K roughly by ( ) ( W P \ W / W P ′ ) ( X P × X θ ( P ) × X P ′ ) / G × × (conn comp) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 12 / 27

40. . . . Description of K -orbits Bruhat Reduction Strategy 2: Bruhat Reduction G ⊃ P : psg of G ∃ P ′ ⊂ G : θ -stable psg of G s.t. Q = P ′ ∩ K K ⊃ Q : psg of K . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 13 / 27

41. . . . Description of K -orbits Bruhat Reduction Strategy 2: Bruhat Reduction G ⊃ P : psg of G ∃ P ′ ⊂ G : θ -stable psg of G s.t. Q = P ′ ∩ K K ⊃ Q : psg of K K \ X P ×Z Q = K \ ( G / P × K / Q ) ≃ P \ G / Q . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 13 / 27

42. . . . � � � Description of K -orbits Bruhat Reduction Strategy 2: Bruhat Reduction G ⊃ P : psg of G ∃ P ′ ⊂ G : θ -stable psg of G s.t. Q = P ′ ∩ K K ⊃ Q : psg of K K \ X P ×Z Q = K \ ( G / P × K / Q ) ≃ P \ G / Q Thus ∼ � P \ G / Q K \ X P ×Z Q proj = ⨿ w ∈ J W J ′ PwP ′ : Bruhat P \ G / P ′ Φ ≃ J W J ′ = W P \ W / W P ′ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 13 / 27

43. � � � Description of K -orbits Bruhat Reduction Strategy 2: Bruhat Reduction G ⊃ P : psg of G ∃ P ′ ⊂ G : θ -stable psg of G s.t. Q = P ′ ∩ K K ⊃ Q : psg of K K \ X P ×Z Q = K \ ( G / P × K / Q ) ≃ P \ G / Q Thus ∼ � P \ G / Q K \ X P ×Z Q proj = ⨿ w ∈ J W J ′ PwP ′ : Bruhat P \ G / P ′ Φ ≃ J W J ′ = W P \ W / W P ′ . Parametrization . Reduces to paramet’n of P \ PwP ′ / Q for w ∈ J W J ′ . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 13 / 27

44. Description of K -orbits KGB Reduction Strategy 2 (continued): KGB Reduction B ↔ ∆ + ⊃ Π : simple roots Assume B ⊃ T : θ -stable P ′ ↔ J ′ ⊂ Π P ↔ J ⊂ Π and P = LU , P ′ = L ′ U ′ : Levi decomposition . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 14 / 27

45. Description of K -orbits KGB Reduction Strategy 2 (continued): KGB Reduction B ↔ ∆ + ⊃ Π : simple roots Assume B ⊃ T : θ -stable P ′ ↔ J ′ ⊂ Π P ↔ J ⊂ Π and P = LU , P ′ = L ′ U ′ : Levi decomposition w ∈ J W J ′ : minimal representatives for W J \ W / W J ′ P \ PwP ′ / Q Want to analyze the fiber of Bruhat reduction . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 14 / 27

46. Description of K -orbits KGB Reduction Strategy 2 (continued): KGB Reduction B ↔ ∆ + ⊃ Π : simple roots Assume B ⊃ T : θ -stable P ′ ↔ J ′ ⊂ Π P ↔ J ⊂ Π and P = LU , P ′ = L ′ U ′ : Levi decomposition w ∈ J W J ′ : minimal representatives for W J \ W / W J ′ P \ PwP ′ / Q Want to analyze the fiber of Bruhat reduction P L ′ ( w ) := w − 1 Pw ∩ L ′ : psg of L ′ Put K := L ′ ∩ K : symmetric subgrp of L ′ L ′ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 14 / 27

47. Description of K -orbits KGB Reduction Strategy 2 (continued): KGB Reduction B ↔ ∆ + ⊃ Π : simple roots Assume B ⊃ T : θ -stable P ′ ↔ J ′ ⊂ Π P ↔ J ⊂ Π and P = LU , P ′ = L ′ U ′ : Levi decomposition w ∈ J W J ′ : minimal representatives for W J \ W / W J ′ P \ PwP ′ / Q Want to analyze the fiber of Bruhat reduction P L ′ ( w ) := w − 1 Pw ∩ L ′ : psg of L ′ Put K := L ′ ∩ K : symmetric subgrp of L ′ L ′ ⇒ P L ′ ( w ) \ L ′ / L ′ = K =: V ( w ) : finite set ( ∵ smaller P \ G / K ) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 14 / 27

48. Description of K -orbits KGB Reduction Strategy 2 (continued): KGB Reduction B ↔ ∆ + ⊃ Π : simple roots Assume B ⊃ T : θ -stable P ′ ↔ J ′ ⊂ Π P ↔ J ⊂ Π and P = LU , P ′ = L ′ U ′ : Levi decomposition w ∈ J W J ′ : minimal representatives for W J \ W / W J ′ P \ PwP ′ / Q Want to analyze the fiber of Bruhat reduction P L ′ ( w ) := w − 1 Pw ∩ L ′ : psg of L ′ Put K := L ′ ∩ K : symmetric subgrp of L ′ L ′ ⇒ P L ′ ( w ) \ L ′ / L ′ = K =: V ( w ) : finite set ( ∵ smaller P \ G / K ) surj � � P L ′ ( w ) \ L ′ / L ′ Reduction map : P \ PwP ′ / Q K = V ( w ) ∋ ∋ � P L ′ ( w ) ℓ a L ′ PwaQ ✤ K where a = ℓ a u a is Levi decomp along P ′ = L ′ U ′ . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 14 / 27

49. . . . Description of K -orbits Orbit parametrization For w ∈ J W J ′ , v ∈ V ( w ), put { U ( w , v ) := ( U ′ ∩ P ( wv )) \ U ′ / ( U ′ ∩ K ) : variety of unipotent elts K ( w , v ) := L ′ ∩ K ∩ P ( wv ) ⊂ L ′ L ′ K P ( g ) := g − 1 Pg ∈ X P Notation: . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 15 / 27

50. . . . Description of K -orbits Orbit parametrization For w ∈ J W J ′ , v ∈ V ( w ), put { U ( w , v ) := ( U ′ ∩ P ( wv )) \ U ′ / ( U ′ ∩ K ) : variety of unipotent elts K ( w , v ) := L ′ ∩ K ∩ P ( wv ) ⊂ L ′ L ′ K P ( g ) := g − 1 Pg ∈ X P Notation: L ′ K ( w , v ) acts on U ( w , v ) by conjugation ⇝ U ( w , v ) / Ad( L ′ K ( w , v )) : quotient sp . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 15 / 27

51. Description of K -orbits Orbit parametrization For w ∈ J W J ′ , v ∈ V ( w ), put { U ( w , v ) := ( U ′ ∩ P ( wv )) \ U ′ / ( U ′ ∩ K ) : variety of unipotent elts K ( w , v ) := L ′ ∩ K ∩ P ( wv ) ⊂ L ′ L ′ K P ( g ) := g − 1 Pg ∈ X P Notation: L ′ K ( w , v ) acts on U ( w , v ) by conjugation ⇝ U ( w , v ) / Ad( L ′ K ( w , v )) : quotient sp . Theorem (He-N-Ochiai-Y.Oshima) . Recall J W J ′ = W J \ W / W J ′ and V ( w ) = P L ′ ( w ) \ L ′ / L ′ K . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 15 / 27

52. Description of K -orbits Orbit parametrization For w ∈ J W J ′ , v ∈ V ( w ), put { U ( w , v ) := ( U ′ ∩ P ( wv )) \ U ′ / ( U ′ ∩ K ) : variety of unipotent elts K ( w , v ) := L ′ ∩ K ∩ P ( wv ) ⊂ L ′ L ′ K P ( g ) := g − 1 Pg ∈ X P Notation: L ′ K ( w , v ) acts on U ( w , v ) by conjugation ⇝ U ( w , v ) / Ad( L ′ K ( w , v )) : quotient sp . Theorem (He-N-Ochiai-Y.Oshima) . Recall J W J ′ = W J \ W / W J ′ and V ( w ) = P L ′ ( w ) \ L ′ / L ′ K We have bijection of orbits (parametrization): ⨿ ⨿ U ( w , v ) / Ad( L ′ K \ X P ×Z Q ≃ K ( w , v )) w ∈ J W J ′ v ∈ V ( w ) . ⇝ criterion of finiteness of orbits . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 15 / 27

53. . . . Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

54. . . . Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

55. . . . Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp X P ×Z B K : finite type ⇐ ⇒ X P = G / P : K -spherical . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

56. . . . Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp X P ×Z B K : finite type ⇐ ⇒ X P = G / P : K -spherical P ′ = B = TU 0 ⊂ G : θ -stable Borel subgrp s.t. B K = B ∩ K ( T : max torus, U 0 max unip subgrp) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

57. . . . Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp X P ×Z B K : finite type ⇐ ⇒ X P = G / P : K -spherical P ′ = B = TU 0 ⊂ G : θ -stable Borel subgrp s.t. B K = B ∩ K ( T : max torus, U 0 max unip subgrp) L ′ = T = ⇒ V ( w ) = P L ′ ( w ) \ L ′ / L ′ K reduces to { e } (1 pt) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

58. Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp X P ×Z B K : finite type ⇐ ⇒ X P = G / P : K -spherical P ′ = B = TU 0 ⊂ G : θ -stable Borel subgrp s.t. B K = B ∩ K ( T : max torus, U 0 max unip subgrp) L ′ = T = ⇒ V ( w ) = P L ′ ( w ) \ L ′ / L ′ K reduces to { e } (1 pt) . Corollary . ( ) / ⨿ K \ X P ×Z B K ≃ ( U 0 ∩ P ( w )) \ U 0 / ( U 0 ∩ K ) Ad T w ∈ J W . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

59. Spherical Flag Variety Q = B K is Borel Application 1: when Q = B K is Borel Assume Q = B K ⊂ K : Borel subgrp X P ×Z B K : finite type ⇐ ⇒ X P = G / P : K -spherical P ′ = B = TU 0 ⊂ G : θ -stable Borel subgrp s.t. B K = B ∩ K ( T : max torus, U 0 max unip subgrp) L ′ = T = ⇒ V ( w ) = P L ′ ( w ) \ L ′ / L ′ K reduces to { e } (1 pt) . Corollary . ( ) / ⨿ K \ X P ×Z B K ≃ ( U 0 ∩ P ( w )) \ U 0 / ( U 0 ∩ K ) Ad T w ∈ J W In particular, X P ×Z B K is of finite type ( ) / ⇐ ⇒ # ( U 0 ∩ P ( w )) \ U 0 / ( U 0 ∩ K ) Ad T < ∞ for ∀ w . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 16 / 27

60. . . . Spherical Flag Variety Q = B K is Borel Spherical K -action Want to know if X P = G / P is K -spherical . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 17 / 27

61. . . . Spherical Flag Variety Q = B K is Borel Spherical K -action Want to know if X P = G / P is K -spherical Idea: Concentrate on open orbit � O Ask ∃ ? open B K -orbit & � O ↔ w 0 ∈ J W : longest element . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 17 / 27

62. Spherical Flag Variety Q = B K is Borel Spherical K -action Want to know if X P = G / P is K -spherical Idea: Concentrate on open orbit � O Ask ∃ ? open B K -orbit & � O ↔ w 0 ∈ J W : longest element We can linearize the double coset space to get . Theorem . X P ×Z B K is of finite type ⇐ ⇒ P := L P ∩ K : reductive ↷ u − θ L θ is mult-free ( or spherical ) . P . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 17 / 27

63. Spherical Flag Variety Q = B K is Borel Spherical K -action Want to know if X P = G / P is K -spherical Idea: Concentrate on open orbit � O Ask ∃ ? open B K -orbit & � O ↔ w 0 ∈ J W : longest element We can linearize the double coset space to get . Theorem . X P ×Z B K is of finite type ⇐ ⇒ P := L P ∩ K : reductive ↷ u − θ L θ is mult-free ( or spherical ) . P Interesting connection to (co-)normal bundles: O X P ≃ K × R u − θ T ∗ : conormal bundle over O ( R := K ∩ P ) P . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 17 / 27

64. . . . . Spherical Flag Variety Q = B K is Borel Geometric & Representation Theoretic interpretation . . 1 We can deduce the former theorem from Panyushev’s thm . Theorem (Panyushev) . TFAE . . X P is K-spherical ( ⇐ ⇒ X P ×Z B K is of finite type) 1 . . conormal bundle T ∗ O X P is K-spherical ( O given above) 2 . . T ∗ O ′ X P is K-spherical for ∀O ′ . 3 . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 18 / 27

65. Spherical Flag Variety Q = B K is Borel Geometric & Representation Theoretic interpretation . . 1 We can deduce the former theorem from Panyushev’s thm . Theorem (Panyushev) . TFAE . . X P is K-spherical ( ⇐ ⇒ X P ×Z B K is of finite type) 1 . . conormal bundle T ∗ O X P is K-spherical ( O given above) 2 . . T ∗ O ′ X P is K-spherical for ∀O ′ . 3 . . 2 Multiplicity free derived functor modules: . Theorem (Y.Oshima) . Assume P is θ -stable X P ×Z B K is of finite type ⇐ ⇒ derived functor module A p ( λ ) has mult-free K-types for ∀ λ : dom regular integral . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 18 / 27

66. Spherical Flag Variety Q = B K is Borel Geometric & Representation Theoretic interpretation . . 1 We can deduce the former theorem from Panyushev’s thm . Theorem (Panyushev) . TFAE . . X P is K-spherical ( ⇐ ⇒ X P ×Z B K is of finite type) 1 . . conormal bundle T ∗ O X P is K-spherical ( O given above) 2 . . T ∗ O ′ X P is K-spherical for ∀O ′ . 3 . . 2 Multiplicity free derived functor modules: . Theorem (Y.Oshima) . Assume P is θ -stable X P ×Z B K is of finite type ⇐ ⇒ derived functor module A p ( λ ) has mult-free K-types for ∀ λ : dom regular integral . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 18 / 27

67. . . . . . . . . . . . . . . . . . . . Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

68. . . . . . . . . . . . . . . . . Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . Theorem (HNOO) . Complete cassification of X P ×Z B K of finite type ( including exceptional type ) . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

69. . . . . . . . . . . . . . . . . Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . Theorem (HNOO) . Complete cassification of X P ×Z B K of finite type ( including exceptional type ) . Howe, Wallach and Horvath classified them in 80’s (unpublished note) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

70. . . . . . . . . . . . . . . . . Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . Theorem (HNOO) . Complete cassification of X P ×Z B K of finite type ( including exceptional type ) . Howe, Wallach and Horvath classified them in 80’s (unpublished note) ... we catch up them 30 years later! . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

71. . . . Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . Theorem (HNOO) . Complete cassification of X P ×Z B K of finite type ( including exceptional type ) . Howe, Wallach and Horvath classified them in 80’s (unpublished note) ... we catch up them 30 years later! . Strategy . . 1 Dimension restriction: dim X P ×Z B K ≤ dim K . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

72. Spherical Flag Variety Q = B K is Borel Classification of K -spherical flag variety G : simply connected, connected simple group Possible to classify ( G , K , P ) for which X P ×Z B K is of finite type . Theorem (HNOO) . Complete cassification of X P ×Z B K of finite type ( including exceptional type ) . Howe, Wallach and Horvath classified them in 80’s (unpublished note) ... we catch up them 30 years later! . Strategy . . 1 Dimension restriction: dim X P ×Z B K ≤ dim K . . 2 Use criterion in Theorem (Existence of open orbit) . P := L P ∩ K : reductive ↷ u − θ . L θ is mult-free (or spherical) P ∃ classification of mult-free space by Benson-Ratcliff (2004) . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 19 / 27

73. � . Spherical Flag Variety Q = B K is Borel g k Π \ J ( P = P J ) α 1 α 2 α n sl n +1 ◦ ◦ ◦ sl n +1 so n +1 { α i } ( ∀ i ) { α i } ( ∀ i ), { α i , α i +1 } ( ∀ i ), sl 2 m sp m 2 m = n + 1 { α 1 , α i } ( ∀ i ), { α i , α n } ( ∀ i ), { α 1 , α 2 , α 3 } , { α n − 2 , α n − 1 , α n } , { α 1 , α 2 , α n } , { α 1 , α n − 1 , α n } sl p ⊕ sl q ⊕ C { α i } ( ∀ i ), { α i , α i +1 } ( ∀ i ), sl p + q p + q = n + 1 1 ≤ p ≤ q { α 1 , α i } ( ∀ i ), { α i , α n } ( ∀ i ), { α i , α j } ( ∀ i , j ) if p = 2, any subset of Π if p = 1 α n − 1 α 1 α 2 α n so 2 n +1 ◦ ◦ ◦ ◦ so p + q so p ⊕ so q { α 1 } , { α n } , p + q = 2 n + 1 1 ≤ p ≤ q { α i } ( ∀ i ) if p = 2, any subset of Π if p = 1 α n − 1 ◦ α n − 2 ◦ α 1 α 2 so 2 n ◦ ◦ ◦ α n so p ⊕ so q { α 1 } , { α n − 1 } , { α n } , so p + q p + q = 2 n 1 ≤ p ≤ q { α i } ( ∀ i ) if p = 2, n ≥ 4 { α i , α n − 1 } ( ∀ i ) if p = 2, { α i , α n } ( ∀ i ) if p = 2, any subset of Π if p = 1 so 2 n sl n ⊕ C { α 1 } , { α 2 } , { α 3 } , { α n − 1 } , { α n } , n ≥ 4 { α 1 , α 2 } , { α 1 , α n − 1 } , { α 1 , α n } , { α n − 1 , α n } , { α 2 , α 3 } if n = 4 . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 20 / 27

74. � � . Spherical Flag Variety Q = B K is Borel α n − 1 α n α 1 α 2 sp n ◦ ◦ ◦ ◦ sl n ⊕ C { α 1 } , { α n } sp n sp p ⊕ sp q { α 1 } , { α 2 } , { α 3 } , { α n } , { α 1 , α 2 } , sp p + q 1 ≤ p ≤ q { α i } ( ∀ i ) if p ≤ 2, p + q = n { α i , α j } ( ∀ i , j ) if p = 1 α 1 α 2 α 3 α 4 f 4 ◦ ◦ ◦ ◦ { α 1 } , { α 2 } , { α 3 } , { α 4 } , { α 1 , α 4 } f 4 so 9 α 2 ◦ α 1 α 3 α 5 α 6 e 6 ◦ ◦ ◦ ◦ ◦ α 4 e 6 sp 4 { α 1 } , { α 6 } e 6 sl 6 ⊕ sl 2 { α 1 } , { α 6 } so 10 ⊕ C { α 1 } , { α 2 } , { α 3 } , { α 5 } , { α 6 } , { α 1 , α 6 } e 6 { α 1 } , { α 2 } , { α 3 } , { α 5 } , { α 6 } , e 6 f 4 { α 1 , α 2 } , { α 2 , α 6 } , { α 1 , α 3 } , { α 5 , α 6 } α 2 ◦ α 1 α 3 α 5 α 6 α 7 e 7 ◦ ◦ ◦ ◦ ◦ ◦ α 4 e 7 sl 8 { α 7 } so 12 ⊕ sl 2 { α 7 } e 7 e 6 ⊕ C { α 1 } , { α 2 } , { α 7 } e 7 . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 21 / 27

75. Spherical fiber bundle over affine symmetric space P = B is Borel Application 2: when P = B is Borel . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 22 / 27

76. Spherical fiber bundle over affine symmetric space P = B is Borel Application 2: when P = B is Borel assume P = B = TU 0 ⊂ G : θ -stable Borel subgrp WLOG ( T : max torus, U 0 max unip subgrp) . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 22 / 27

77. � Spherical fiber bundle over affine symmetric space P = B is Borel Application 2: when P = B is Borel assume P = B = TU 0 ⊂ G : θ -stable Borel subgrp WLOG ( T : max torus, U 0 max unip subgrp) X B ×Z Q : finite type ⇐ ⇒ G / Q : G -spherical G / Q fiber = K / Q : flag var G / K . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 22 / 27

78. � Spherical fiber bundle over affine symmetric space P = B is Borel Application 2: when P = B is Borel assume P = B = TU 0 ⊂ G : θ -stable Borel subgrp WLOG ( T : max torus, U 0 max unip subgrp) X B ×Z Q : finite type ⇐ ⇒ G / Q : G -spherical G / Q fiber = K / Q : flag var G / K Recall θ -stable P ′ s.t. Q = P ′ ∩ K P ′ = L ′ U ′ : Levi decomp ⇝ Q = L ′ K U ′ K : Levi decomp . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 22 / 27

79. . . . . . . Spherical fiber bundle over affine symmetric space P = B is Borel . Theorem . TFAE . . 1 X B ×Z Q is of finite type . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 23 / 27

80. . . . . . Spherical fiber bundle over affine symmetric space P = B is Borel . Theorem . TFAE . . 1 X B ×Z Q is of finite type . . 2 G / Q is G-spherical . . . . . . . Nishiyama (AGU) K-spherical flag varieties 2013/07/08 23 / 27

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