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Discretization of new Weyl group orbit functions J. Hrivn ak, L. Motlochov a, J. Patera Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, B rehov a 7, 115 19 Praha 1 Centre de recherches

  1. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − and S l − functions for σ ∈ { 1 , σ e , σ s , σ l } , b ∈ P are the complex functions b : R n → C ϕ σ � ϕ σ σ ( w ) e 2 πi � wb, a � , a ∈ R n b ( a ) = w ∈ W σ = σ e . . . S − functions (known from the Weyl character formula) σ = 1 . . . C − functions σ = σ s . . . S s − functions σ = σ l . . . S l − functions (anti)symmetry with respect to w ∈ W ϕ s b ( wa ) = σ s ( w ) ϕ s b ( a ) ϕ s wb ( a ) = σ s ( w ) ϕ s b ( a ) invariance with respect to shifts from q ∨ ∈ Q ∨ ϕ s b ( a + q ∨ ) = ϕ s b ( a ) J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  2. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  3. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  4. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  5. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  6. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  7. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  8. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  9. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  10. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  11. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  12. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  13. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  14. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  15. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  16. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  17. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  18. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  19. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  20. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  21. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  22. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  23. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  24. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  25. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  26. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  27. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  28. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  29. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  30. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  31. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  32. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  33. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  34. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  35. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  36. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  37. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  38. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  39. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  40. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  41. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  42. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  43. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  44. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  45. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  46. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  47. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  48. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  49. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  50. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  51. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  52. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  53. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  54. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − function ϕ s (1 , 0) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  55. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (1 , 0) ( x, y ) and ϕ s (1 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  56. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (2 , 0) ( x, y ) and ϕ s (2 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  57. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (3 , 1) ( x, y ) and ϕ s (3 , 2) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  58. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (0 , 1) ( x, y ) and ϕ l (1 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  59. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 2) ( x, y ) and ϕ l (2 , 1) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  60. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (2 , 3) ( x, y ) and ϕ l (3 , 2) ( x, y ) of C 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  61. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (0 , 1) ( x, y ) and ϕ s (1 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  62. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (1 , 2) ( x, y ) and ϕ s (2 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  63. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S s − functions ϕ s (2 , 3) ( x, y ) and ϕ s (3 , 2) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  64. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 0) ( x, y ) and ϕ l (1 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  65. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (1 , 2) ( x, y ) and ϕ l (2 , 1) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  66. Lie groups/Lie algebras Short and long orbit functions Sign homomorphisms Discretization of orbit functions Ss − and Sl − functions S l − functions ϕ l (2 , 3) ( x, y ) and ϕ l (3 , 2) ( x, y ) of G 2 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  67. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  68. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M M P ∨ 1 W -invariant lattice M P ∨ /Q ∨ 1 M ∈ N , W -invariant finite group M P ∨ /Q ∨ is cM n 1 number of elements of The grid F M F M ≡ 1 M P ∨ /Q ∨ ∩ F The grids F s M ⊂ F M and F l M ⊂ F M M ≡ 1 M P ∨ /Q ∨ ∩ F s F s M ≡ 1 M P ∨ /Q ∨ ∩ F l F l the symbols u s i , u l i ∈ R , i = 0 , . . . , n : u s u l i ∈ Z ≥ 0 , r i ∈ R s i ∈ N , u s i ∈ Z ≥ 0 , u l r i ∈ R l i ∈ N , J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  69. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s 4 and F l 4 of C 2 η α 2 = α ∨ ω 2 = ω ∨ ξ = ω ∨ 2 2 1 H l H s F ω 1 = 1 2 ω ∨ 1 α ∨ α 1 1 r 0 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  70. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s 4 and F l 4 of C 2 η α 2 = α ∨ ω 2 = ω ∨ ξ = ω ∨ 2 2 1 H l H s F ω 1 = 1 2 ω ∨ 1 α ∨ α 1 1 r 0 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  71. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M the explicit form of F s and F l : � � u s � � 1 + · · · + u s � F s M ω ∨ 1 M ω ∨ n � u s 0 + u s 1 m 1 + · · · + u s M = n m n = M n � u l � � � 1 + · · · + u l � F l M ω ∨ 1 M ω ∨ n � u l 0 + u l 1 m 1 + · · · + u l M = n m n = M n Proposition Let m s and m l be the short and long Coxeter numbers, respectively. Then   M < m s M < m l  0  0   | F s | F l M | = M = m s , M | = M = m l 1 1     M > m s . M > m l . | F M − m s | | F M − m l | J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  72. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M Theorem The numbers of points of grids F s M and F l M of Lie algebras B n , C n are given by the following relations. C n , n ≥ 2, 1 � k + 1 � � k � | F s 2 k ( C n ) | = + n n � k + 1 � | F s 2 k +1 ( C n ) | = 2 n � n + k − 1 � � n + k − 2 � | F l 2 k ( C n ) | = + n n � n + k − 1 � | F l 2 k +1 ( C n ) | = 2 n B n , n ≥ 3, 2 | F s M ( B n ) | = | F l M ( C n ) | | F l M ( B n ) | = | F s M ( C n ) | J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  73. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s M and F l M Theorem The numbers of points of grids F s M and F l M of Lie algebra G 2 are given by the following relations. 6 k +1 ( G 2 ) | = 3 k 2 + k | F s 6 k ( G 2 ) | = 3 k 2 , | F s 6 k +2 ( G 2 ) | = 3 k 2 + 2 k, 6 k +3 ( G 2 ) | = 3 k 2 + 3 k + 1 | F s | F s 6 k +4 ( G 2 ) | = 3 k 2 + 4 k + 1 , 6 k +5 ( G 2 ) | = 3 k 2 + 5 k + 2 . | F s | F s | F l M ( G 2 ) | = | F s M ( G 2 ) | F 4 : the counting formula is also known J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  74. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  75. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions The dual Lie algebra ∆ → ∆ ∨ : 2 α i α ∨ i = � α i , α i � , i ∈ { 1 , . . . , n } the set ∆ ∨ = { α ∨ 1 , . . . , α ∨ n } is a system of simple roots of some simple Lie algebra span R ∆ ∨ = R n the highest dual root η ≡ − α ∨ 0 = m ∨ 1 α ∨ 1 + · · · + m ∨ n α ∨ n m ∨ j . . . the dual marks of G the dual Cartan matrix C ∨ ij = 2 � α ∨ i , α ∨ j � C ∨ j � = C ji , i, j ∈ { 1 , . . . , n } � α ∨ j , α ∨ J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  76. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Dual affine Weyl group the group generated by n reflections r α ∨ = r α , α ∨ ∈ ∆ ∨ coincides with W W aff is generated by reflections R ∨ = { r ∨ � 0 , r 1 , . . . , r n } , where 2 η r η a = a − 2 � a, η � r ∨ a ∈ R n 0 a = r η a + � η, η � , � η, η � η, Dual affine Weyl group � W aff W aff = Q ⋊ W � the fundamental domain F ∨ of � W aff � � 0 , ω 1 , . . . , ω n F ∨ = m ∨ m ∨ 1 n κ a decomposition of R ∨ = R s ∨ ∪ R l ∨ : R s ∨ = { r α | α ∈ ∆ s } ∪ { r ∨ 0 } R l ∨ = { r α | α ∈ ∆ l } J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  77. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Dual fundamental domains two subsets of boundaries of F ∨ : H s ∨ = { a ∈ F ∨ | ( ∃ r ∈ R s ∨ )( ra = a ) } � � H l ∨ = a ∈ F ∨ | ( ∃ r ∈ R l ∨ )( ra = a ) fundamental domains F s ∨ ⊂ F ∨ , F l ∨ ⊂ F ∨ F s ∨ = F ∨ \ H s ∨ F l ∨ = F ∨ \ H l ∨ the symbols z s i , z l i ∈ R , i = 0 , . . . , n z s z l r i ∈ R s ∨ i > 0 , i ≥ 0 , z s z l r i ∈ R l ∨ i ≥ 0 , i > 0 , � � � � F s ∨ = � z s 1 ω 1 + · · · + z s � z s 0 + z s 1 m ∨ 1 + · · · + z s n m ∨ n ω n n = 1 � � � � F l ∨ = � z l 1 ω 1 + · · · + z l � z l 0 + z l 1 m ∨ 1 + · · · + z l n m ∨ n ω n n = 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  78. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids Λ s M and Λ l M M ∈ N , W -invariant lattice P W -invariant finite group P/MQ number of elements of P/MQ is cM n The grid Λ M Λ M ≡ MF ∨ ∩ P/MQ The grids Λ s M ⊂ Λ M and Λ l M ⊂ Λ M M ≡ MF s ∨ ∩ P/MQ Λ s M ≡ MF l ∨ ∩ P/MQ Λ l Proposition For the numbers of elements of the sets Λ s M and Λ l M it holds that | Λ s M | = | F s M | , | Λ l M | = | F l M | . J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  79. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Grids F s ∨ and F l ∨ of C 2 4 4 4 α 2 r ∨ η 0 , 4 4 H s ∨ 4 F ∨ 4 H l ∨ α 2 = α ∨ ω 2 = ω ∨ 2 2 ξ = ω ∨ 1 1 r ∨ 2 ω 2 0 F ∨ ω 1 α ∨ α 1 4 α 1 1 r 2 r 1 J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  80. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions functions ϕ s , ϕ l on the grids F s M and F l M ϕ s b + MQ ( u ) = ϕ s b ( u ) , u ∈ F s M ϕ l b + MQ ( u ) = ϕ l b ( u ) , u ∈ F l M ϕ s λ , ϕ l λ with λ ∈ P/MQ , moreover λ ∈ Λ M zero values: λ ∈ MH s ∨ ∩ Λ M , u ∈ F s ϕ s λ ( u ) = 0 , M λ ∈ MH l ∨ ∩ Λ M , u ∈ F l ϕ l λ ( u ) = 0 , M Corollary ϕ s λ ( u ) , u ∈ F s M , λ ∈ Λ s M ϕ l λ ( u ) , u ∈ F l M , λ ∈ Λ l M J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  81. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Outline 1 Short and long orbit functions Lie groups/Lie algebras Sign homomorphisms S s − and S l − functions 2 Discretization of orbit functions Grids F s M and F l M Grids Λ s M and Λ l M Discrete orthogonality of S s − and S l − functions J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  82. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions x ∈ R n /Q ∨ , the isotropy group Stab( x ) = { w ∈ W | wx = x } the orbit Wx = { wx ∈ R n /Q ∨ | w ∈ W } h x ≡ | Stab( x ) | , ε ( x ) ≡ | Wx | ε ( x ) = | W | h x λ ∈ R n /MQ , the isotropy group Stab ∨ ( λ ) = { w ∈ W | wλ = λ } λ ≡ | Stab ∨ ( λ ) | h ∨ J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  83. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete orthogonality of S s − and S l − functions a scalar product for f, g : F s M → C � � f, g � F s M = ε ( x ) f ( x ) g ( x ) x ∈ F s M a scalar product for f, g : F l M → C � � f, g � F l M = ε ( x ) f ( x ) g ( x ) x ∈ F l M Theorem For λ, λ ′ ∈ Λ s M it holds that M = c | W | M n h ∨ � ϕ s λ , ϕ s λ ′ � F s λ δ λ,λ ′ and for λ, λ ′ ∈ Λ l M it holds that M = c | W | M n h ∨ � ϕ l λ , ϕ l λ ′ � F l λ δ λ,λ ′ . J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

  84. Grids F s M and F l Short and long orbit functions M Grids Λ s M and Λ l Discretization of orbit functions M Discrete orthogonality of Ss − and Sl − functions Discrete S s − and S l − transforms interpolating functions I s M , I l M I s � c s λ ϕ s I l � c l λ ϕ l x ∈ R n M ( x ) := λ ( x ) , M ( x ) := λ ( x ) , λ ∈ Λ s λ ∈ Λ l M M the interpolation of f : F M → C : find c s λ (or c l λ ) I s x ∈ F s M ( x ) = f ( x ) , M I l x ∈ F l M ( x ) = f ( x ) , M Proposition � f, ϕ s λ � F s c s = ( c | W | M n h ∨ λ ) − 1 � M ε ( x ) f ( x ) ϕ s λ = λ ( x ) � ϕ s λ , ϕ s λ � F s x ∈ F s M M � f, ϕ l λ � F l c l = ( c | W | M n h ∨ λ ) − 1 � λ = M ε ( x ) f ( x ) ϕ l λ ( x ) � ϕ l λ , ϕ l λ � F l x ∈ F l M M J. Hrivn´ ak, L. Motlochov´ a, J. Patera Orbit functions discretization

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