image/svg+xml 1. The algebraic story 1. The algebraic story 2. Relation to W-algebras Answer: (McDowell) Example: e-actionf-actionh-action ... 2 0 -2 -4 2 1 -1 1 2 3 4 Case 1: completely degenerate Case 2: non-degenerate Verma module! The second natural question: 2. Relation to W-algebras 2. Relation to W-algebras 3. The geometric story 3. The geometric story 3. The geometric story 3. The geometric story Set-up: example: A remarkable bridge: Twisted Harish-Chandra sheaves: Example: C(w) X= open Bruhat cell= Choose 4. The algorithm 4. The algorithm 4. The algorithm 3 1 2 3 4 3 3 3 1 1 1 1 Set-up: Relating Deodar's theorem to D-modules: The punchline: These are holonomic D-moduleswith irregular singularities So the polynomials in Deodhar's algorithm compute composition multiplicities of Whittaker modules! Thank you! 1. The algebraic story Example: e-actionf-actionh-action ... 2 0 -2 -4 2 1 -1 1 2 3 4 Case 1: completely degenerate 1. The algebraic story Example: e-actionf-actionh-action ... 2 0 -2 -4 2 1 -1 1 2 3 4 Case 1: completely degenerate Verma module! 1. The algebraic story Example: e-actionf-actionh-action ... 2 0 -2 -4 2 1 -1 1 2 3 4 Case 1: completely degenerate Case 2: non-degenerate Verma module! 1. The algebraic story The second natural question: 1. The algebraic story The second natural question: 1. The algebraic story The second natural question: 1. The algebraic story The second natural question: 3. The geometric story Set-up: example: 3. The geometric story Set-up: example: orbits: C(1) C(w) 3. The geometric story Set-up: example: orbits: C(1) C(w) twisted sheafof differential operators on X 3. The geometric story Set-up: example: orbits: C(1) C(w) twisted sheafof differential operators on X trivial central character 3. The geometric story A remarkable bridge: 3. The geometric story A remarkable bridge: 3. The geometric story A remarkable bridge: 3. The geometric story Twisted Harish-Chandra sheaves: 3. The geometric story Twisted Harish-Chandra sheaves: 3. The geometric story Twisted Harish-Chandra sheaves: 3. The geometric story Twisted Harish-Chandra sheaves: 3. The geometric story Twisted Harish-Chandra sheaves: 3. The geometric story Twisted Harish-Chandra sheaves: These are holonomic D-moduleswith irregular singularities 3. The geometric story Twisted Harish-Chandra sheaves: These are holonomic D-moduleswith irregular singularities 3. The geometric story Example: 3. The geometric story Example: X= open Bruhat cell= 3. The geometric story Example: C(w) X= open Bruhat cell= Choose 3. The geometric story Example: C(w) X= open Bruhat cell= Choose 3. The geometric story Example: C(w) X= open Bruhat cell= Choose 3 1 2 3 4 3 3 3 1 1 1 1 3. The geometric story Twisted Harish-Chandra sheaves: These are holonomic D-moduleswith irregular singularities C(w) 4. The algorithm Set-up: 4. The algorithm Set-up: 4. The algorithm Set-up: 4. The algorithm Relating Deodhar's theorem to D-modules: 4. The algorithm Relating Deodhar's theorem to D-modules: 4. The algorithm Relating Deodhar's theorem to D-modules: 4. The algorithm The punchline: 4. The algorithm The punchline: 4. The algorithm The punchline: 4. The algorithm The punchline: So the polynomials in Deodhar's algorithm compute composition multiplicities of Whittaker modules! Thank you! 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras 2. Relation to W-algebras Interlude: Finite W-algebras independent of choices made 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras Functors: 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 2. Relation to W-algebras 3. The geometric story Set-up: example:
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  1. title
  2. plan
  3. set-up 1
  4. set-up 2
  5. set-up 3
  6. set-up 4
  7. set-up 5
  8. set-up 6
  9. category of reps 1
  10. category of reps 2
  11. category of reps 3
  12. category of reps 4
  13. why Whittaker 1
  14. why Whittaker 2
  15. why Whittaker 3
  16. why Whittaker 4
  17. first question 1
  18. first question 2
  19. first question 3
  20. first question 4
  21. first question 5
  22. first question 6
  23. example 1
  24. example 2
  25. example 3
  26. example 4
  27. second question 1
  28. second question 2
  29. second question 3
  30. second question 4
  31. interlude 1
  32. interlude 2
  33. interlude 3
  34. interlude 4
  35. interlude 5
  36. interlude 6
  37. interlude 7
  38. interlude 8
  39. interlude 9
  40. interlude 10
  41. interlude 11
  42. skryabin 1
  43. skryabin 2
  44. skryabin 3
  45. Brundan 1
  46. Brundan 2
  47. Brundan 3
  48. Brundan 4
  49. Brundan 5
  50. Brundan 6
  51. Brundan 7
  52. Brundan 8
  53. Losev 1
  54. Losev 2
  55. Losev 3
  56. geometry setup 1
  57. geometry setup 2
  58. geometry setup 3
  59. geometry setup 4
  60. geometry setup 5
  61. bridge 1
  62. bridge 2
  63. bridge 3
  64. HC sheaves 1
  65. HC sheaves 2
  66. HC sheaves 3
  67. HC sheaves 4
  68. HC sheaves 5
  69. HC sheaves 6
  70. HC sheaves 7
  71. HC sheaves 8
  72. example 1
  73. example 2
  74. example 3
  75. example 3
  76. example 4
  77. algorithm setup 1
  78. algorithm setup 2
  79. algorithm setup 3
  80. deodhar 1
  81. deodhar 2
  82. deodhar 3
  83. punchline 1
  84. punchline 2
  85. punchline 3
  86. punchline 4
  87. thank you