arXiv subjects

Atsumu Sasaki

Publications and source records attributed to Atsumu Sasaki.

5 recordsLinked to original sources

arXiv2026

Weyl group of semisimple symmetric space

This paper investigates a generalization of the notion of the Weyl group of a real reductive Lie group to a semisimple symmetric space $G/H$. First, we show that the group structure of the Weyl group of $G/H$ is independent of the choice of split Cartan subalgebras of $G/H$ and the choice of Cartan involutions of $G$ stabilizing $H$. After that, we can understand all elements of the Weyl group of $G/H$ by comparing the Weyl group of some reductive Lie group associated to $G/H$. With the aim of extending these results to reductive real spherical homogeneous spaces, this work serves as a first step in this direction.

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arXiv2021

Visible actions and criteria for multiplicity-freeness of representations of Heisenberg groups

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \operatorname{id}_S$. Let $G$ be the Heisenberg group and $H$ a non-trivial connected closed subgroup of $G$. We prove that any complex homogeneous space $D = G^{\mathbb{C}}/H^{\mathbb{C}}$ admits a strongly visible $L$-action, where $L$ stands for a connected closed subgroup of $G$ explicitly constructed through a co-exponential basis of $H$ in $G$. This leads in turn that $G$ itself acts strongly visibly on $D$. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic diffeomorphism and a totally real submanifold $S$, for which the dimension depends upon the dimensions of $G$ and $H$. As a direct application, our geometric results provide a proof of various multiplicity-free theorems on continuous representations on the space of holomorphic sections on $D$. Moreover, we also generate as a consequence, a geometric criterion for a quasi-regular representation of $G$ to be multiplicity-free.

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arXiv2020

A duality between non-compact semisimple symmetric pairs and commutative compact semisimple symmetric triads and its general theory

The present paper investigates a natural generalization of the duality between Riemannian symmetric pairs of compact type and those of non-compact type à la É. Cartan. The main result of this paper is to construct an explicit description of a one-to-one correspondence between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads, which is called the duality theorem. Further, we develop a general theory of the duality theorem.

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arXiv2018

A Cartan decomposition for non-symmetric reductive spherical pairs of rank-one type and its application to visible actions

A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some compact group on a non-symmetric reductive spherical homogeneous space of rank-one type is strongly visible.

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arXiv2015

Visible actions on spherical nilpotent orbits in complex simple Lie algebras

This paper studies nilpotent orbits in complex simple Lie algebras from the viewpoint of strongly visible actions in the sense of T. Kobayashi. We prove that the action of a maximal compact group consisting of inner automorphisms on a nilpotent orbit is strongly visible if and only if it is spherical, namely, admitting an open orbit of a Borel subgroup. Further, we find a concrete description of a slice in the strongly visible action. As a corollary, we clarify a relationship among different notions of complex nilpotent orbits: actions of Borel subgroups (sphericity); multiplicity-free representations in regular functions; momentum maps; and actions of compact subgroups (strongly visible actions).

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