Families of twisted $\mathcal D$-modules and arithmetic models of Harish-Chandra modules

$\mathrm{SO}(3)$-homogeneous decomposition of the flag scheme of $\mathrm{SL}_3$ over $\mathbb{Z}\left[1/2\right]$

“…Hence it is naturally defined over the real numbers by Galois descent. In fact, we realized this object as the real flag variety of SO (3) in [15] Section 5. More strongly, we gave an equivariant closed immersion of flag schemes over Z [ 1 2].…”

“…Philosophically, this result says that all irreducible admissible representations of SL 3 (R) (with regular infinitesimal character) are controlled by real algebro-geometric objects (cf. [15] Definition 3.5.1, Corollary 4.2.2).…”

“…The last phase says that we prove that a K ⊗ k k ′ -orbit admits a k-form by showing that it admits a Galois action; otherwise we get a k-scheme by joining K ⊗ k k ′ -orbits. For example, see [15] Example 5.2.21. We can regard that [15] Proposition 5.1.1 is the case that the Galois orbit has two elements…”

“…For example, see [15] Example 5.2.21. We can regard that [15] Proposition 5.1.1 is the case that the Galois orbit has two elements…”

“…Among those, Harris proposed to construct rational models of discrete series representations from the corresponding closed K-orbits in the flag variety and line bundles on them over the field C of complex numbers through the localization. In [15], we studied descent properties of rings of definition of certain closed K-orbits in the moduli scheme of parabolic subgroups of G for reductive group schemes K ⊂ G in the sense of [2] Définition 2.7. As a consequence, we established real and smaller arithmetic forms of A q (λ)-modules ([15] Section 6.2).…”