In this paper, we study the irreducible decomposition of a (a",[X];G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over a",. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the 'fiber' M/(x (0))M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M. Our main result is a generalization of earlier work by F. SatAi in [17]. We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.

If a locally finite rational representation V of a connected reductive algebraic group G has uniformly bounded multiplicities, the multiplicities may have good properties such as stability. Let X be a quasi-affine spherical G-variety, and M be a (C[X], G)-module. In this paper, we show that the decomposition of M as a G-representation can be controlled by the decomposition of the fiber M/m(x(0))M with respect to some reductive subgroup L subset of G for sufficiently large parameters. As an application, we apply this result to branching laws for simple real Lie groups of Hermitian type. We show that the sufficient condition on multiplicity-freeness given by the theory of visible actions is also a necessary condition for holomorphic discrete series representations and symmetric pairs of holomorphic type. We also show that two branching laws of a holomorphic discrete series representation with respect to two symmetric pairs of holomorphic type coincide for sufficiently large parameters if two subgroups are in the same epsilon-family.

In the representation theory of real reductive Lie groups, many objects have
finiteness properties. For example, the lengths of Verma modules and principal
series representations are finite, and more precisely, they are bounded. In
this paper, we introduce a notion of uniformly bounded families of holonomic
$\mathscr{D}$-modules to explain and find such boundedness properties.
A uniform bounded family has good properties. For instance, the lengths of
modules in the family are bounded and the uniform boundedness is preserved by
direct images and inverse images. By the Beilinson--Bernstein correspondence,
we can deduce several boundedness results about the representation theory of
complex reductive Lie algebras from corresponding results of uniformly bounded
families of $\mathscr{D}$-modules. In this paper, we concentrate on proving
fundamental properties of uniformly bounded families, and preparing abstract
results for applications to the branching problem and harmonic analysis.

Let $G_{\mathbb{R } }$ be a real reductive Lie group and $G'_{\mathbb{R } }$ a
reductive subgroup of $G_{\mathbb{R } }$ such that $\mathfrak{g'}$ is algebraic
in $\mathfrak{g}$. In this paper, we consider restrictions of irreducible
representations of $G_{\mathbb{R } }$ to $G'_{\mathbb{R } }$ and induced
representations of irreducible representations of $G'_{\mathbb{R } }$ to
$G_{\mathbb{R } }$. Our main concern is when such a representation has uniformly
bounded multiplicities, i.e. the multiplicities in the representation are
(essentially) bounded. We give characterizations of the uniform boundedness by
polynomial identities and coisotropic actions.
For the restriction of (cohomologically) parabolically induced
representations, we find a sufficient condition for the uniform boundedness by
spherical actions and some fiber condition. This result gives an affirmative
answer to a conjecture by T. Kobayashi.
Our results can be applied to $(\mathfrak{g}, K)$-modules, Casselman--Wallach
representations, unitary representations and objects in the BGG category
$\mathcal{O}$. We also treat with an upper bound of cohomological
multiplicities.

リー群の分岐則が、具体的な不変量によってどのように統制されるかについて研究を行った。分岐則が扱いやすい状況として、離散的に分解するという場合が存在する。本研究において、表現が離散的に分解するための条件を、 wave front set と呼ばれる表現の不変量を用いて与えた。さらに、離散的に分解したうえで重複度が有限になるという許容的になるための条件も、同様の手法により与えた。これらの結果は実簡約リー群の場合には、小林俊行氏の1994,98年における論文の一般化となっている。この研究結果は、2019年度の表現論シンポジウムにおいて発表された。