Harish-Chandra Modules over ℤ

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 ????ℤ. On the group scheme G/ℤ there is a Cartan involution ???? that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under ???? is a flat group scheme ????/ℤ. A Harish-Chandra module over ℤ is a ℤ-module ???? that comes with an action of the Lie algebra ????ℤ, an action of the group scheme ????, and some compatibility conditions is required between these two actions. Finally, ????-finiteness is also required, which is that ???? is a union of finitely generated ℤ modules ????I that are ????-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.