Alphabeta MathProjective Functors and the Soergel Functor in Category O
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1 · Prerequisites
Abstract Algebra
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Set Theory
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Topology
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- Separation Axioms: the Hierarchy
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- The Fundamental Group
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- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
The coinvariant algebra of a finite Weyl group
Definition
The regular integral block and its antidominant weight
Lemma
The sum of basic Weyl invariant degrees minus one counts positive roots
Lemma
Adjoint isotypic modules have the zero-weight rank
Definition
Projective functors on a regular integral block
Lemma
The coinvariant algebra is a complete intersection with one-dimensional socle
Lemma
The harmonic complement of the coinvariant algebra
Definition
The deformed category O family over the Cartan symmetric algebra
Lemma
The nilpotent central fiber is reduced and integral
Definition
The wall-crossing functor and its unit and counit
Theorem
The coinvariant algebra is the graded regular representation of W
Definition
The antidominant projective and the Soergel functor
Theorem
Duflo–Joseph g-finite endomorphisms of every Verma module
Lemma
Wall-crossing acts on standard modules by a two-step extension
Proposition
Base change, freeness and specialization in the deformed family
Lemma
The wall-crossing action on the Grothendieck group is multiplication by 1+s
Lemma
Wall-crossing compositions produce the projective with the top standard label
Theorem
Lifting of indecomposable projectives to the deformed family
Theorem
Indecomposable projective functors are indexed by the Weyl group
Lemma
The central action on the deformed antidominant projective
Theorem
The endomorphisms of the antidominant projective form the coinvariant algebra
Theorem
Soergel's Struktursatz: the functor V is fully faithful on projectives
Theorem
Wall-crossing under the Soergel functor is tensoring with the coinvariant bimodule
Theorem
Indecomposable projectives correspond to indecomposable Soergel modules
Proposition
Verma-flag multiplicities are standard-filtration ranks under V
Sources
- Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46 full text)
- Arthur Garnier, The classical theory of Soergel bimodules and the link with Lusztig's conjecture, Oberwolfach Arbeitsgemeinschaft note (2022)
- Pavel Etingof, Representations of Lie Groups, MIT18.757 full lecture notes (2023)
- J. N. Bernstein, S. I. Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980) 245-285 (Numdam full text)
- Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J. reine angew. Math. 429 (1992) 49-74 (full text)
- Jonathan Brundan, Representations of the general linear Lie superalgebra in the BGG category O (survey; Section 4 with Soergel's Endomorphismensatz and Struktursatz)
- Wolfgang Soergel, Universelle versus relative Einhuelle (Math. Ann. 284 (1989) 177-198)
- Wolfgang Soergel, Universelle versus relative Einhuelle: Eine geometrische Untersuchung von Quotienten von universellen Einhüllenden halbeinfacher Lie-Algebren, Math. Ann. 284 (1989), 177-198
- Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules (1992)