# Euler factorization of global integrals (2005)(en)(63s) by Garrett P.

By Garrett P.

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For the orbit with isotropy group Q∗ : −1 s 1/2 χ1 (a)χ2 (d)|ad|−s |a/d|1/2 χ−1 =1 1 (d)χ2 (d )|dd | |d/d | which is equivalent to χ1 = | ∗ |s−1/2 from a χ1 = χ2 from d χ2 = | ∗ |s−1/2 from d Unless χ1 = χ2 this system has no solutions s. Even when this condition is satisfied, there is only one s which satisfies it. So generically s does not satisfy this system. ) The family of representations C∞ c (E) ⊗ (Iχ )−s ⊗ (Iχ−1 )s considered above is a parametrized family of representations, over the ring O = C[z, z −1 ] where the pointwise representations are recovered by z → q −s .

Done. 3Exactness of some functors Now we begin preparations for proof of the orbit criterion for locally strong meromorphy. Here we prove the exactness of some relatively elementary functors on smooth representation spaces. Some of this occurs in the unpublished notes [Casselman 1976] in a different form. Let O be a commutative ring with identity, and A an associative O-algebra. Say that A is an idempotented algebra if for any finite collection η1 , . . , ηn of elements of A there is an idempotent e ∈ A so that eηi = ηi = ηi e for all i.

Thus, δ(mj ⊗ 1) = 0, proving uniqueness. Now we can return to discussion of parametrized families of smooth representations, in the setting of linear systems and parametrized linear systems. Let {ti } be a countable set of O-generators for V . Let {gj } be a countable dense subset of G. The condition that an O-linear map λ : M ⊗O M → M be an O-parametrized family of intertwining operators is that λ(π(gj )ti ) = λ(ti ) for all indices i, j, since the isotropy group of each ti is open (by smoothness).