Download Automorphic Forms, Representations, and L-Functions by W Casselman, Armand Borel, W. Casselman PDF

By W Casselman, Armand Borel, W. Casselman

This used to be the convention on $L$-functions and automorphic varieties. the 2 volumes are actually classics.

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Extra resources for Automorphic Forms, Representations, and L-Functions (Proceedings of Symposia in Pure Mathematics)

Example text

Tubular extensions and coextensions of algebras 37 bound by the relations εβ = 0, εδ = 0, σμ = 0, and α = νγ. The component C3 is given by 1 1 0 1 0 00 1 1 1 | 0 0 1 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 11 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 1 1 10 0 0 0 0 0 0 1 1 11 0 0 0 1 1 0 1 0 00 1 1 1 | | | 1 1 0 2 0 00 1 1 0 0 0 0 1 0 00 0 0 0 1 1 0 2 0 00 1 1 1 1 1 0 1 0 00 1 1 0 1 1 0 2 1 00 1 1 0 .. 1 1 1 2 1 10 1 1 0 | | 1 1 0 2 0 00 1 1 0 | | .. 1 1 0 2 1 10 1 1 0 .. 1 1 0 2 1 11 1 1 0 .. 1 1 0 2 0 00 1 1 0 ..

Xi −−−−→ . . starting at X satisfies the following three conditions: (i) If M is an indecomposable module in C such that HomA (X, M ) = 0, then there exists i ≥ 0 such that M ∼ = Xi . (ii) If f : M −→ N is a homomorphism between indecomposable modules in C such that HomA (X, f ) = 0, then there exist i, j with i ≤ j such that M ∼ = Xj and f is a scalar multiple of the com= Xi , N ∼ posite homomorphism uj . . ui+1 : Xi −−−−→ Xj , if i < j, and f is a scalar multiple of the identity, if i = j.

Moreover, for each i ≥ 0, there exists an almost split sequence of the form 0 −−−−→ Xi −−−−−−→ Xi+1 ⊕ Zi,1 −−−−−−→ Zi+1,1 −−−−→ 0, in mod A , where Z0,1 = Z0,1 is a projective module and Zj,1 is indecomposable non-projective such that Zj,1 ∼ = Xj+1 , for any j ≥ 1. To see this, we note that if Xi −→ N is an irreducible morphism in mod A , with N indecomposable non-projective, then there is an irreducible morphism τA N −→ Xi in mod A , and the claim follows from the preceding description of the right almost split morphisms in mod A ending at Xi .

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