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

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’s method for constructing eigenvarieties is successful in cohomological dimension 1. For real rank 2 groups, we show that a slight modification of Emerton’s method allows one to construct eigenvarieties in cohomological dimension 2. Throughout this paper we shall use the following standard notation: • k is an algebraic number field, fixed throughout. • p, q denote finite primes of k, and kp, kq the corresponding local fields. • k∞ = k ⊗Q R is the product of the archimedean completions of k. • A is the adèle ring of k. • Af is the ring of finite adèles of k. • For a finite set S of places of k, we let