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Takuma Hayashi

Publications and source records attributed to Takuma Hayashi.

12 recordsLinked to original sources
arXiv2026

Rationality patterns

In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel--Tits' criterion for the existence of rational forms of representations of $\bar{F}\otimes_F G$ for a connected reductive algebraic group $G$ over a field $F$ of characteristic zero and its algebraic closure $\bar{F}$. We also discuss applications of these general formalisms to the theory of Harish-Chandra modules, specifically to classify irreducible Harish-Chandra modules over fields $F$ of characteristic zero and to identify smaller fields of definition of irreducible Harish-Chandra modules over $\bar{F}$, particularly in the case of cohomological irreducible essentially unitarizable modules.

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arXiv2024

Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms

In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the division superalgebras of their endomorphisms. Finally, we give numerical conclusions for quasi-reductive algebraic supergroups under certain conditions, based on Shibata's Borel--Weil theory for split quasi-reductive algebraic supergroups.

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arXiv2024

Uniform decomposition of the flag scheme by a symmetric subgroup action

In this paper, we establish a scheme-theoretic analog of the works of Matsuki and Richardson--Springer on the symmetric subgroup orbit decomposition of the flag variety under a certain assumption that asserts local constancy of their combinatorial description of the classification of orbits at geometric points (the local constancy hypothesis). We also prove that scheme-theoretic models of orbits are affinely imbedded into the flag scheme under the same assumption (Beilinson--Bernstein's affinity theorem). Finally, we compute some classical examples to give scheme-theoretic consequences of the geometric point free and descent phenomena of orbit decompositions.

math.AG↗
arXiv2024

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

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.

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arXiv2024

Algebraic approach to contraction families

In this paper, we give a purely algebraic approach to the contraction group scheme predicted by Bernstein--Higson--Subag and constructed by Barbasch--Higson--Subag. We also compare quotient schemes of contraction group schemes with other related schemes, equipped with actions of contraction group schemes in the cases of symmetric and $θ$-stable parabolic subgroups.

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arXiv2024

Filtrations on the globalization of twisted D-modules over Dedekind schemes

Fabian Januszewski and the author established the theory of twisted D-modules over general base schemes. In this short note, we construct a $K$-invariant positive exhaustive filtration on the globalization of the twisted D-module on a smooth quasi-compact $K$-scheme over a Dedekind scheme $S$ obtained by the direct image of a $K$-equivariant twisted integrable connection along a $K$-equivariant closed immersion from a smooth proper $K$-scheme $Y$ with $K$ a smooth $S$-affine group scheme, whose $p$th associated graded $\mathcal{O}_S$-module is locally free of finite rank for every integer $p$. In particular, the $k$-module of its global sections is projective if $S$ is affine with coordinate ring $k$.

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arXiv2023

Half-integrality of line bundles on partial flag schemes of classical Lie groups

In this paper, we develop a theory of Galois descent for equivariant line bundles on partial flag schemes. In particular, we study computational aspects of the classification of descent data of equivariant line bundles attached to characters of parabolic subgroups. As an application, we classify equivariant line bundles on partial flag schemes of the standard $\mathbb{Z}\left[1/2\right]$-forms of classical Lie groups.

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arXiv2020

Flat Base Change Formulas for $(\mathfrak{g},K)$-modules over Noetherian rings

The fucntor $I$ and its derived functor over the complex number field have been playing important roles in representation theory of real reductive Lie groups. In this paper, we discuss the flat base change formulas of the functor I and its derived functor over Noetherian rings. In particular, a flat base change theorem for $A_{\mathfrak{q}}(λ)$ is obtained.

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