Harmonic analysis, the trace formula, and Shimura varieties: by James Arthur

By James Arthur

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Additional info for Harmonic analysis, the trace formula, and Shimura varieties: proceedings of the Clay Mathematics Institute, 2003 Summer School, the Fields Institute, Toronto, Canada, June 2-27, 2003

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En . The simple coroots form the basis ∨ ∆∨ 0 = {βi = ei − ei+1 : 1 ≤ i ≤ n − 1} of the subspace n aG 0 = {u ∈ R : ui = 0}. The simple weights give the dual basis ∆0 = { i : 1 ≤ i ≤ n − 1}, where n−i i (u1 + · · · + ui ) − (ui+1 + · · · + un ). n n The Weyl group W0 of the root system for GL(n) is the symmetric group Sn , acting by permutation of the coordinates of vectors in the space a0 ∼ = Rn . The dot product n on R give a W -invariant inner product ·, · on both a0 and a∗0 . It is obvious that i (u) = βi , βj ≤ 0, i = j.

0 be the subspace of vectors φ ∈ HP that are K-finite, in the sense that Let HP the subset {IP (λ, k)φ : k ∈ K} of HP spans a finite dimensional space, and that lie in a finite sum of irreducible subspaces of HP under the action IP (λ) of G(A). The two conditions do not depend on the choice of λ. Taken together, they are equivalent to the requirement that the function φ(x∞ xfin ), x∞ ∈ G(R), xfin ∈ G(Afin ), be locally constant in xfin , and smooth, KR -finite and Z∞ -finite in x∞ , where Z∞ 0 denotes the algebra of bi-invariant differential operators on G(R).

The resulting distributions are again new linear forms in f , known as weighted characters. This will set the stage for Part II, where one of the main tasks will be to write the entire geometric expansion in terms of weighted orbital integrals, and the entire spectral expansion in terms of weighted characters. There is a common thread to Part I. 1. 1. 4, as well as their geometric analogues in §10 and their spectral analogues in §14. We have tried to emphasize this pattern in order to give the reader some overview of the techniques.

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