By Oregon State University, 1977 Symposium in Pure Mathematics

Includes sections on Automorphic representations and L-functions in addition to Arithmetical algebraic geometry and L-functions

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**Additional info for Automorphic Forms, Representations and L-Functions, Part 2**

**Sample text**

The coordinates that are left over from τ are all equal, so the remaining yj , xi are uniquely determined. 5. Unipotent representations with Dirac cohomology In this section we give an exposition of unipotent representations, and compute Dirac cohomology for many examples. 1. Langlands Homomorphisms. In order to explain the parameters of unipotent representations we recast the classiﬁcation of (g, K)-modules in terms of Langlands homomorphisms. First some notation: For the ﬁeld of reals the Weil group is WR := C× · {1, j}, j 2 = −1 ∈ C× , jzj −1 = z, ˇ C ´ DAN BARBASCH AND PAVLE PANDZI 16 where Γ := Gal(C/R) is the Galois group.

2. Homological complexes associated to the stack. Let us start by ﬁxing some notation. Let A be an associative unital algebra over a unital ring k. Set Cp (A, A) = Cp (A) = A⊗(p+1) . We denote by b : Cp (A) → Cp−1 (A) and B : Cp (A) → Cp+1 (A) the standard diﬀerentials from the Hochschild and cyclic homology theory (cf. [21]). The Hochschild chain complex is by deﬁnition (C• (A) , b). Let u be a formal variable of degree −2. Deﬁne CC− • (A) = (C• (A) [[u]] , b + uB) ; per CC• (A) = C• (A) u, u−1 , b + uB ; CC• (A) = C• (A) u, u−1 / (uC• (A) [[u]]) , b + uB .

Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. [BV] D. Barbasch, D. Vogan, Unipotent representations of complex semisimple groups, Ann. of Math. 121 (1985), 41–110. [BW] A. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI, 2000. [E] T. Enright, Relative Lie algebra cohomology and unitary representations of complex Lie groups, Duke Math.