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We study low order terms of Emerton's spectral sequence for simply connected,simple groups. As a result, for real rank 1 groups, we show that Emerton'smethod for constructing eigenvarieties is successful in cohomological dimension1. For real rank 2 groups, we show that a slight modification of Emerton'smethod allows one to construct eigenvarieties in cohomological dimension 2.

Construction of eigenvarieties in small cohomological dimensions for semi-simple, simply connected groups