Propagation of multiplicity-freeness property for holomorphic vector bundles

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

In this paper we compute explicitly the norm of the vector-valued holomorphic discrete series representations, when its $K$-type is "almost multiplicity-free." As an application, we discuss the properties of highest weight modules, such as unitarizability, reducibility and composition series.

Hilbert series of nearly holomorphic sections on generalized flag manifolds

Let X=G/P be a complex flag manifold and E->X be a G-homogeneous holomorphic vector bundle. Fix a U-invariant Kaehler metric on X with U in G maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U-finite smooth sections in E. The degree of nearly holomorphic sections defines a U-invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out to be given in terms of Lusztig's q-analog of Kostant's weight multiplici…

模 Gelfand 对与无重数表示

Modular Gelfand pairs and multiplicity-free representations

本文推广 Gelfand 判据，证明任意特征代数闭域上 Gelfand 对与无重数表示定理，并用于 Whittaker 模型和三线性型唯一性。

Stein manifolds and multiplicity-free representations of compact Lie groups

The paper is a survey of recent results in geometric representation theory describing group actions which induce multiplicity-free representations in the spaces of holomorphic functions. For connected compact Lie groups of automorphisms of Stein manifolds we characterize such actions in terms of antiholomorphic involutions. Some proofs are given and some results are new. For example, spherical complex spaces are defined for arbitrary real forms of complex reductive groups. Their properties, which we prove here, were known only for compact real forms. We also show that a complex compact homogen…

Singular conformally invariant trilinear forms, I Multiplicity one results

A normalized holomorphic family (depending on $\boldsymbol λ\in \mathbb C^3$) of conformally invariant trilinear forms on the sphere is studied. Its zero set $Z$ is described. For $\boldsymbol λ\notin Z$, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1, except perhaps for a denumerable subset.

Free reflection multiarrangements and quasi-invariants

To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vecto…

Multiplicity-free homogeneous operators in the Cowen-Douglas class

In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\mathbb D$ which are {\em homogeneous} with respect to the action of the group Möb -- the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. The {\em associated representation} for each of these operators is {\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the…

Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)

A fundamental question, first raised by Langlands, is to know whether the Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose standard L-function equals L^*(s, f x g) after removing the ramified and archimedean factors. The first main result of this paper is to answer it in the affirmative, in fact with the base field Q replaced by any number field F. Our proof uses a mixture of converse theorems, base change and descent, and it also appeals to the local regularity properties of Eisenstein…

Visible actions on symmetric spaces

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 $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-fre…

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ of $Ξ$ with G/H as Shilov boundary. Let $π$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $π$, which admit a holomorphic extension to $Ξ_H$, is one dimensional. Furthermore we give…

有限域上对$(\mathrm{GL}_{2}\times\mathrm{GL}_{2},\mathrm{diag}(\mathrm{GL}_{2}))$与$(\mathrm{SL}_{3},\mathrm{GL}_{2})$的无重数诱导

Multiplicity free induction for the pairs $(\mathrm{GL}_{2}\times\mathrm{GL}_{2},\mathrm{diag}(\mathrm{GL}_{2}))$ and $(\mathrm{SL}_{3},\mathrm{GL}_{2})$ over finite fields

本文分类了有限域上$\mathrm{GL}_{2}(q)$在对角嵌入下诱导无重数分解的不可约表示，并证明$\mathrm{SL}_{3}(q)$情形下不存在此类表示，揭示了与复数域情形的本质差异。

Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces and New Estimates for the Heckman-Opdam Hypergeometric Functions

Paley-Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. Several authors have studied Paley-Wiener type theorems for Euclidean spaces, Riemannian symmetric spaces of compact or non-compact type as well as affine Riemannian symmetric spaces. In this article we prove a Paley-Wiener theorem for homogeneous line bundles over a compact symmetric space $U/K$. The Paley-Wiener theorem characterizes f with sufficiently small support in terms of holomorphic …