By Armand Borel

Includes sections on Reductive teams, representations, Automorphic kinds and representations

**Read Online or Download Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1 PDF**

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**Additional resources for Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1**

**Sample text**

We shall often need the following elementary lemma, which enables us to identify Λ(G) with a sub-algebra of Λ(G). Let J = {1, ι} = Gal(F∞ /F∞ ). If M is any Zp [J ]-module, then since p is odd, there is the decomposition M = M + ⊕ M − , where M + = 1+ι 1−ι M, M − = M. 2 2 In particular, we have Λ(G) = Λ(G)+ ⊕ Λ(G)− . 1. The restriction to Λ(G)+ of the natural surjection from Λ(G) onto Λ(G) induces an isomorphism Λ(G)+ Λ(G). 2) Proof. 3) and write Gn = Gal(Fn /Q), Gn = Gal(Fn /Q). Let πn : Zp [Gn ] −→ Zp [Gn ] denote the natural surjection.

We note that, since the norm map from Kn to Kn−1 induces the identity map on the residue ﬁelds, we have 1 1 U∞ = µp−1 × U∞ , U∞ = µp−1 × U∞ . 11) can be rewritten as an exact sequence L1 1 −→ Λ(G) −→ Tp (µ) −→ 0, 0 −→ Tp (µ) −→ U∞ 1 . As p is odd, the above sequence with L1 being the restriction of L to U∞ remains exact after taking invariants under J = {1, ι}. Since Tp (µ)J = 0, there is a canonical Λ(G)-isomorphism 1 L1 : U∞ Λ(G). b, C∞ so that 1 ) = Λ(G)L1 (b). 4 shows that L1 (b) = ζ p θ+ (e, 1) where θ+ (e, 1) denotes the image of θ(e, 1) in Λ(G).

6), to conclude that there exists w in W with ∆(w) = h. By construction and the commutativity of the diagram, we have g = D(f ) = DL(w). Hence f = L(w) by the injectivity of D. Thus f belongs to the image of L and the proof of the theorem is complete. 2 to deﬁne the higher logarithmic derivatives of elements of U∞ . We study these maps and show by a mysterious, but elementary, calculation going back to Kummer, that the values of the Riemann zeta function at the odd negative integers arise as the higher logarithmic derivatives of cyclotomic units.