image/svg+xml Plan: Part 1: The objectsPart 2: The philosophyPart 3: The construction Part 1: The objects This talk is about nilpotent Gelfand pairs. What are Gelfand pairs? Why are Gelfand pairs useful? How? What are nilpotent Gelfand pairs? Nilpotent Gelfand pairs have been classified. (Vinberg, Yakimova) Why? In certain settings it works beautifully: (Kirillov '61,'62) N simply connected nilpotent Lie group unitary dual Part 3: The construction Lipsman, Pukanszky ('78,'80,'82): finite-to-one map Benson-Jenkins-Ratcliff, Nishihara ('99,'01,'08): A caricature: This gives a bijection between the set of K-sphericalorbits and K-spherical representations. Three sets in bijection: Gelfand space K-spherical rep'ns K-spherical orbits ANALYTIC OBJECT ALGEBRAIC OBJECT GEOMETRIC OBJECT Each of these sets has a natural structure of a topological space. Gelfand space K-spherical rep'ns K-spherical orbits compact open topology Fell topology subset topology Natural question: Do these set-theoretic bijections preservethe topological structure? Conjecture (Benson-Ratcliff '08): Theorem: Thank you for listening! An orbit model for the spectra of nilpotent Gelfand pairsAnna RomanovUniversity of SydneyJoint work with H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, B. Strasser, B. WesselProject initiated during an American Mathematical Society Mathematics Research Community workshop in Snowbird, Utah, June 2016
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  1. title
  2. plan
  3. part 1
  4. What are Gelfand pairs?
  5. definition
  6. why
  7. harmonic analysis
  8. how?
  9. bijections
  10. nilpotent Gelfand pairs
  11. definition
  12. example
  13. classified
  14. goal
  15. why?
  16. part 2
  17. part 2
  18. Kirillov
  19. works for n.G.p.'s too
  20. part 3
  21. Lipsman Pukanszky
  22. intersection of orbits
  23. bijection
  24. three bijections
  25. topological structure
  26. three topologies
  27. natural question
  28. conjecture
  29. theorem
  30. thank you