Steffen Koenig: Vorlesung Halbeinfache komplexe Lie-Algebren und Darstellungstheorie II (SS 2013)

Inhalt der Vorlesung:

Section 1: Category O: basic properties.
Tuesday 9.4.: BGG category O. Highest weight modules. Verma modules.
Thursday 11.4.: Duality. Tensoring with finite dimensional modules. Tensor identity.
Tuesday 16.4.: Verma flags. Existence of enough projectives. Basic properties of Verma modules.
Thursday 18.4.: Orthogonality. Filtration multiplicities. The category of Verma filtered modules. Highest weight categories and quasi-hereditary algebras. Decomposition numbers.
Tuesday 23.4.: Examples.

Section 2: Translation functors.
Tuesday 23.4.: Dominant, antidominant, compatible. Translation functors.
Thursday 25.4.: The big projective module. Homomorphisms between Verma modules. The socle of a Verma module. Injective envelopes.
Tuesday 30.4.: Dominant dimension. Double centraliser properties. Soergel's 'Struktursatz' for category O.
Thursday 2.5.: Left and right adjointness of translation functors. Facets, closures, upper and lower closures, chambers and walls. Examples.
Tuesday 7.5.: Lemma on linkage. Translating Verma modules and simple modules. Grothendieck groups. Morita equivalences.
Tuesday 28.5.: Proofs. More results on translation functors.

Section 3: Characters and decomposition numbers.
Tuesday 28.5.: Formal characters. Kostant's partition function.
Tuesday 4.6.: Various functions. Weyl's character formula, Weyl's denominator formula and Kostant's weight multiplicity theorem. Remark on BGG resolution. Remark on Weyl's dimension formula.
Thursday 6.6.: Statement of Kazhdan-Lusztig conjecture. Discussion of proof.
Tuesday 11.6.: Discussion of Soergel's approach. Generic algebras and Hecke algebras. Bruhat ordering.
Thursday 13.6.: R-polynomials. Inverting T_w.
Tuesday 18.6.: Involution. Involution on R-polynomials. Kazhdan-Lusztig polynomials.
Thursday 20.6.: Existence of Kazhdan-Lusztig basis.
Tuesday 25.6.: Example dihedral groups.

Section 4: The coinvariant algebra.
Tuesday 25.6.: Statement of Soergel's 'Endomorphismensatz'. Trace of matrices. Relative trace.
Thursday 27.6.: Trace formula. Trace in categories. Endomorphisms of the identity functor.
Tuesday 2.7.: Trace morphism and Bernstein's trace formula.
Thursday 4.7.: Proof of the endomorphisms theorem.

Section 5: Further properties of the algebra of category O.
Tuesday 9.7.: Koszul algebras and linear resolutions. Quadratic algebras. Examples. Pure and weight. Characterisation of Koszul algebras.
Thursday 11.7.: Koszul implies quadratic. Yoneda extension algebra. Parabolic singular duality. Remark on derived equivalences.
Tuesday 16.7.: Numerical Koszul criterion. Abstract Kazhdan-Lusztig theory.
Thursday 18.7.: Characteristic tilting modules and Ringel duality. Even global dimension. Ringel self-duality of category O.

Termine für mündliche Prüfungen können ab sofort im Sekretariat bei Frau Gangl vereinbart werden.

References:

On Lie algebras and representation theory:
James Humphreys, Introduction to Lie algebras and representation theory.
Karin Erdmann and Mark Wildon, Introduction to Lie algebras.
William Fulton and Joe Harris, Representation theory - a first course.

On category O:
Jacques Dixmier, Enveloping algebras.
Jens Carsten Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren.

Main source for the course:
James Humphreys, Representations of semisimple Lie algebras in the BGG category O.

Articles discussed in the course:
A.Beilinson, V.Ginzburg and W.Soergel, Koszul duality in representation theory.
J.Bernstein, Trace in categories.
S.Koenig, I.Slungard and C.Xi, Double centraliser properties, dominant dimension and tilting modules.
V.Mazorchuk and S.Ovsienko, Finitistic dimension of properly stratified algebras.
C.M.Ringel, The category of modules with good filtration over a quasi-hereditary algebra has almost split sequences.
W.Soergel, Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe.