BGG Seminar at CRM

Welcome! This site is the homepage of the BGG seminar at CRM in Fall 2020.

The BGG seminar will be held on Mondays from 9:30 -- 11:30.

References:

[BGG] Bernstein, Gelfand, Gelfand: Differential operators on the base affine space and a study of g-modules

[CE] Cartan, Eilenberg: Homological Algebra

[F] Faltings: On the cohomology of locally symmetric Hermitian spaces

[FC] Faltings, Chai: Degeneration of Abelian varieties, Chapter 6

[H] Humphrey: Representations of semisimple Lie algebras in the BGG category \mathcal{O}

[LP] Lan, Polo: Dual BGG complexes for automorphic bundles

[S] Serre: Complex semisimple Lie algebras

[T] Tilouine: Fornes compagnons et complexes BGG pour GSp4

Description:

Let g be a semisimple Lie algebra with a Cartan subalgebra h. By choosing the set of positive roots, we fix a Borel subalgebra b of g containing h and nilpotent n+ and n−. If V is a representation of g on a finite dimensional C-vector space, Bott described the n− cohomology groups of V by using a standard resolution of V by U(g)-Verma modules, where U(g) is the universal enveloping algebra of g.

The main object of the second part of the article [BGG] in the list above is to construct a much simpler resolution of V obtained by means of representation theory and which allows in a simpler way the calculations of the dimensions of H^i(n−,V). This resolution is now called the BGG complex of V.

Later Faltings and Faltings--Chai [FC] studied the following situation: let us fix a pair (G, Q) of a reductive algebraic group G and a parabolic subgroup of it Q, for example, G = GSp_{2g} and Q is a Siegel parabolic. Faltings--Chai use the ideas of [BGG] to construct a ''simple'' subcomplex, called dual BGG complex, of the de Rham complex of sheaves on a Shimura variety of some appropriate level, associated to some G-representation V , which is quasi-isomorphic to the de Rham complex.

As a result, they are able to describe the de Rham cohomology of V in terms of modular forms. Then they show that there are dual BGG complexes for other cohomology theories on the Shimura variety like \ell-adic, crystalline, log-crystalline, which, as a result, can be described in terms of modular forms.