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Cohomology of G / P for classical complex Lie supergroups G and characters of some atypical G -modules

Ivan PenkovVera Serganova — 1989

Annales de l'institut Fourier

We compute the unique nonzero cohomology group of a generic G 0 - linearized locally free 𝒪 -module, where G 0 is the identity component of a complex classical Lie supergroup G and P G 0 is an arbitrary parabolic subsupergroup. In particular we prove that for G ( m ) , S ( m ) this cohomology group is an irreducible G 0 -module. As an application we generalize the character formula of typical irreducible G 0 -modules to a natural class of atypical modules arising in this way.

On bounded generalized Harish-Chandra modules

Ivan PenkovVera Serganova — 2012

Annales de l’institut Fourier

Let 𝔤 be a complex reductive Lie algebra and 𝔨 𝔤 be any reductive in 𝔤 subalgebra. We call a ( 𝔤 , 𝔨 ) -module M bounded if the 𝔨 -multiplicities of M are uniformly bounded. In this paper we initiate a general study of simple bounded ( 𝔤 , 𝔨 ) -modules. We prove a strong necessary condition for a subalgebra 𝔨 to be bounded (Corollary 4.6), to admit an infinite-dimensional simple bounded ( 𝔤 , 𝔨 ) -module, and then establish a sufficient condition for a subalgebra 𝔨 to be bounded (Theorem 5.1). As a result we are able to classify...

Adjoint representation of E 8 and del Pezzo surfaces of degree 1

Vera V. SerganovaAlexei N. Skorobogatov — 2011

Annales de l’institut Fourier

Let X be a del Pezzo surface of degree 1 , and let G be the simple Lie group of type E 8 . We construct a locally closed embedding of a universal torsor over X into the G -orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Néron-Severi torus T of X identified with a maximal torus of G extended by the group of scalars. Moreover, the T -invariant hyperplane sections of the torsor defined by the roots of G are the inverse images...

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