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4. An algebraic incarnation
A categorification of the Lusztig-Vogan module
Anna RomanovUniversity of Sydney
Categorification Learning SeminarOctober 11, 2020
Plan:
1. Setting the scene2. The Lusztig-Vogan module3. A catgory of equivariant sheaves4. An algebraic incarnation5. Putting it all together
1. Setting the scene
1. Setting the scene
1. Setting the scene
1. Setting the scene
1. Setting the scene
1. Setting the scene
2. The LV module
2. The LV module
2. The LV module
3. A category of sheaves
3. A category of sheaves
3. A category of sheaves
3. A category of sheaves
4. An algebraic incarnation
5. Putting it all together
Today's main character:
today's running example
Guiding philosophy:
Harish-Chandra's approach (1950's): recast the problem algebraically
"admissibility"
complexified Lie algebra
complexification of maximal compact subgroup
This motivates the setting we'll work in today:
realreductive Lie group
maximal compactsubgroup
complexifications
(have structure of complex algebraic groups)
complexified Lie algebras
Example:
=
not usuallyaffine!
"Harish-Chandramodules"
fixed infinitesimal character
Example:
projectivize
two closedorbits
What about character theory?
Question: What are these multiplicities?
Answer: Given by the Lusztig-Vogan module of the Hecke algebra
Example:
Idea: imitate Kazhdan-Lusztig theory
!
Kazhdan-Lusztig-Vogan polynomials
But I still haven't told you how the Hecke algebra acts on ...
First step: need a categorical upgrade of the Hecke algebra
split Grothendieck group
- monoidal category under - categorifies the Hecke algebra:
- convolution gives a right action- define- take split Grothendieck group
Second step: a categorical upgrade of