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

Takuma Hayashi, Fabian Januszewski

Abstract

We develop a theory of tdos and twisted $\mathcal D$-modules over general base schemes with a focus on functorial aspects. In particular, we introduce a flat base change functor and establish its compatibility with globalization and direct image functors. We also study forms of closed $K$-orbits of $θ$-stable parabolic subgroups in the total flag variety. We apply these two developed theories to give a geometric construction of half-integral models of cohomologically induced modules. With a view towards arithmetic applications, we further demonstrate desirable properties of the constructed half-integral models, such as projectivity over the base and torsion-free relative Lie algebra cohomology.

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

Abstract

We develop a theory of tdos and twisted -modules over general base schemes with a focus on functorial aspects. In particular, we introduce a flat base change functor and establish its compatibility with globalization and direct image functors. We also study forms of closed -orbits of -stable parabolic subgroups in the total flag variety. We apply these two developed theories to give a geometric construction of half-integral models of cohomologically induced modules. With a view towards arithmetic applications, we further demonstrate desirable properties of the constructed half-integral models, such as projectivity over the base and torsion-free relative Lie algebra cohomology.

Paper Structure

This paper contains 66 sections, 203 theorems, 391 equations.

Table of Contents

  1. Arithmetic background
  2. Twisted $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules
  3. Background
  4. Towards a general theory of twisted $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules
  5. Tdos and twisted $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules over schemes
  6. Twisted $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules over Dedekind schemes
  7. Cohomologically induced modules and their geometric construction
  8. Models of orbits and orbit decompositions
  9. Future directions
  10. Results and organization of this paper
  11. Algebras of twisted differential operators and Picard algebroids
  12. Twisted differential operators
  13. Picard algebroids and $\Omega^{\geq 1}$-torsors
  14. Functorial constructions
  15. Equivalence of $\mathop{\mathrm{TDO}}\nolimits$ and $\mathop{\mathrm{PA}}\nolimits$
  16. ...and 51 more sections

Key Result

Theorem 2.1

Let $K$ be a smooth affine group scheme over a Dedekind scheme $S$, $i:Y\hookrightarrow X$ be a $K$-equivariant closed immersion of smooth $K$-schemes over $S$, and ${\mathcal{A}}$ be a $K$-equivariant tdo on $X$. Write $x:X\to S$ for the structure morphism. Suppose that the following conditions are Let ${\mathcal{M}}$ be a $K$-equivariant left $i^\cdot{\mathcal{A}}$-module which is locally free o

Theorems & Definitions (491)

  • Theorem 2.1: hayashifil
  • Remark 3.1: voganzuckerman1984, knappvogan
  • Remark 3.2
  • Theorem 3.3: hechtetal, kitchen2012, oshima2013, see also Corollary \ref{['cor:duality']}
  • Theorem 6.1: Theorem \ref{['thm:basechangevspullbackfortdo']}
  • Theorem 6.2: Theorems \ref{['thm:qcohpreservation']}, \ref{['thm:compositionlaw']}
  • Theorem 6.3: Theorem \ref{['thm:xbc']}
  • Corollary 6.4: Theorem \ref{['thm:inductionofequivtwistedDmod']}
  • Theorem 6.5: Theorem \ref{['thm:flatbasechangeofglobalsectionI']}, \ref{['thm:qcohpreservation']}
  • Theorem 6.6: Theorems \ref{['thm:inductionofequivtwistedDmod']}, \ref{['thm:globalization']}
  • ...and 481 more