Let $(G,H)$ be a reductive spherical pair and $P\subset H$ a parabolic subgroup such that $(G,P)$ is spherical. The triples $(G,H,P)$ with this property are called multiplicity free systems and they are classified in this paper. Denote by $\pi^{H}_{\mu}=\mathrm{ind}_{P}^{H}\mu$ the Borel-Weil realization of the irreducible $H$-representation of highest weight $\mu\in P^{+}_{H}$ and consider the induced representation $\mathrm{ind}_{P}^{G}\chi_{\mu}=\mathrm{ind}_{H}^{G}\pi^{H}_{\mu}$, a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discussed. For three multiplicity free systems the spectra are calculated explicitly. The spectra give rise to families of multi-variable orthogonal polynomials which generalize families of Jacobi polynomials: they are simultaneous eigenfunctions of a commutative algebra of differential operators, they satisfy recurrence relations and they are orthogonal with respect to integrating against a matrix weight on a compact subset. We discuss some difficulties in describing the theory for these families of polynomials in the generality of the classification.

Multiplicity free induced representations and orthogonal polynomials

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).

Visible actions on spherical nilpotent orbits in complex simple Lie algebras

We give a generalization of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma$ is a Chevalley--Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma}H$ holds if and only if either Case I: both $H$ and $L$ are maximal and of type A, or Case II: $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switch of $H$ and $L$. This classification gives a visible action of $L$ on the generalized flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit `slice' $B$ with dim $B=$rank $G$ in Case I, and dim $B=2$ or $3$ in Case II such that a generalized Cartan decomposition $G=LBH$ holds. An application to multiplicity-free theorems of representations is also discussed.

 10.1017/S0004972712000615

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/90F5C1397964B87E51823DC964FEFE24/S0004972712000615a.pdf/div-class-title-visible-actions-on-flag-varieties-of-type-b-and-a-generalisation-of-the-cartan-decomposition-div.pdf

Visible actions on flag varieties of type b and a generalisation of the cartan decomposition

We give a generalization of the Cartan decomposition for connected compact Lie groups of type C motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Let $G$ be a compact simple Lie group of type C, $K$ a Chevalley--Weyl involution-fixed point subgroup and $L,H$ Levi subgroups. We firstly show that $G=LKH$ holds if and only if either Case I: $(G,H)$ and $(G,L)$ are both symmetric pairs or Case II: $L$ is a Levi subgroup of maximal dimension and $H$ is an arbitrary maximal Levi subgroup up to switch of $L,H$. This classification gives a visible action of $L$ on the generalized flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on the direct product of $G/L$ and $G/H$. Secondly, we find a generalized Cartan decomposition $G=LBH$ explicitly, where $B$ is a subset of $K$. An application to multiplicity-free theorems of representations is also discussed.

/pdf/visible-actions-on-flag-varieties-of-type-c-and-a-47d62w4s84.pdf

Visible actions on flag varieties of type c and a generalization of the cartan decomposition

Visible actions on flag varieties of type D and a generalizaiton of the Cartan decomposition

We give a generalization of the Cartan decomposition for connected compact Lie groups motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A group. This paper extends his results to type D group. First, we classify a pair of Levi subgroups $(L,H)$ of a simple compact Lie group $G$ of type D such that $G=LG^{\sigma}H$ where $\sigma$ is a Chevalley–Weyl involution. This gives the visibility of the $L$-action on the generalized flag variety $G/H$ as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find a generalized Cartan decomposition $G=LBH$ with $B$ in $G^{\sigma}$ by using the herringbone stitch method which was introduced by Kobayashi in his 2007 paper. Applications to multiplicity-free theorems of representations are also discussed.

/pdf/visible-actions-on-flag-varieties-of-type-d-and-a-2xd9r32tuq.pdf

Visible actions on flag varieties of type D and a generalization of the Cartan decomposition

For compact simple groups G$G$ of type B or D, we find the pairs (V1,V2)$(V_{1},V_{2})$ of irreducible representations of G$G$ such that the tensor product representation V1?V2$V_{1}\otimes V_{2}$ is multiplicity-free by a geometric consideration based on the notion of visible actions on complex manifolds, introduced by T. Kobayashi. The pairs that we find exhaust all the multiplicity-free pairs (V1,V2)$(V_{1},V_{2})$ in comparison with the classification that was obtained earlier by Stembridge by combinatorial methods.

Geometry of Multiplicity-Free Representations of SO(N)$\mathrm {SO}(N)$ and Visible Actions

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible. 

Yuichiro Tanaka

Papers

Visible actions on flag varieties of type b and a generalisation of the cartan decomposition

Visible actions on flag varieties of type c and a generalization of the cartan decomposition

Visible actions on flag varieties of type D and a generalization of the Cartan decomposition

Geometry of Multiplicity-Free Representations of SO(N)$\mathrm {SO}(N)$ and Visible Actions

Visible actions of compact Lie groups on complex spherical varieties