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Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions(AMS-203)$
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Anantharam Raghuram and Günter Harder

Print publication date: 2019

Print ISBN-13: 9780691197890

Published to Princeton Scholarship Online: September 2020

DOI: 10.23943/princeton/9780691197890.001.0001

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Harish-Chandra Modules over ℤ

Harish-Chandra Modules over ℤ

Chapter:
(p.126) Chapter Eight Harish-Chandra Modules over ℤ
Source:
Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions
Author(s):

Günter Harder

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691197890.003.0008

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by tt−1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

Keywords:   Harish-Chandra modules, Lie algebra, Cartan involution, induction, Lie group

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