Homogeneous Line Bundles Research Articles

Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains

Let Ω=G/K be a bounded symmetric domain of non-compact type. In this paper the image of the Poisson transform on the degenerate principal series representations of G attached to the Shilov boundary of Ω is considered. We characterize the images in terms of the third-order Hua operators Uν and Wν. When Ω is the exceptional domain of type V, we give the explicit formulas for the operators Uν and Wν.

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

Lp-Poisson integral representations for solutions of the generalized Hua system on line bundles over SU(n,n)/S(U(n)×U(n))

Let τν (ν∈Z) be a character of K=S(U(n)×U(n)), and SU(n,n)×KC the associated homogeneous line bundle over D={Z∈M(n,C):I−ZZ⁎>0}. Let Hν be the Hua operator on the sections of SU(n,n)×KC. Identifying sections of SU(n,n)×KC with functions on D we transfer the operator Hν to an equivalent matrix-valued operator H˜ν which acts on D. Then for a given C-valued function F on D satisfying H˜νF=−14(λ2+(n−ν)2)F.(I00−I) we prove that F is the Poisson transform by Pλ,ν of some f∈Lp(S), when 1<p<∞ or F=Pλ,νμ for some Borel measure μ on the Shilov boundary S, when p=1 if and only ifsup0≤r<1⁡(1−r2)−n(n−ν−ℜ(iλ))2(∫S|F(rU)|pdU)1p<∞, provided that the complex parameter λ satisfies iλ∉2Z−+n−2±ν and ℜ(iλ)>n−1.This generalizes the result in [1] which corresponds to τν the trivial representation.

SPHERICAL ACTIONS ON ISOTROPIC FLAG VARIETIES AND RELATED BRANCHING RULES

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to H of all irreducible representations of G realized in spaces of sections of homogeneous line bundles on X.

Branching Rules Related to Spherical Actions on Flag Varieties

Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where $H$ is a Levi subgroup of $G$) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples $(G,H,X)$ satisfying one of the following two conditions: (1) $G$ is simple and $H$ is a symmetric subgroup of $G$; (2) $G = \mathrm{SL}_n$.

Irreducible Unitary Representations Concerning Homogeneous Holomorphic Line Bundles Over Elliptic Orbits

In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.

Paley–Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman–Opdam hypergeometric functions

AbstractPaley–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. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.

ALGEBRAIC ANALYSIS OF SCALAR GENERALIZED VERMA MODULES OF HEISENBERG PARABOLIC TYPE I: A n -SERIES

In the present article, we combine some techniques in harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ( $$ \mathcal{D} $$ -modules), and reformulate the composition series and branching problems for objects in the Bernstein–Gelfand–Gelfand parabolic category $$ {\mathcal{O}}^{\mathfrak{p}} $$ geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold G / P for G = SL(n + 2, ℂ) and P the Heisenberg parabolic subgroup; and algebraic analysis gives a complete classification of $$ {\mathfrak{g}}_r^{\prime } $$ -singular vectors for all $$ {\mathfrak{g}}_r^{\prime }=\mathfrak{s}\mathfrak{l}\left(n-r+2,\mathbb{C}\right)\kern1em \subset \mathfrak{g}=\mathfrak{s}\mathfrak{l}\left(n+2,\mathbb{C}\right),n-r>21 $$ . A consequence of our results is that we classify SL(n − r + 2, ℂ) -covariant differential operators acting on homogeneous line bundles over the complexification of the odd-dimensional CR-sphere S 2n+1 and valued in homogeneous vector bundles over the complexification of the CR-subspheres S 2(n-r)+1.

Dimensions of some locally analytic representations

Let G G be the group of points of a split reductive group over a finite extension of Q p \mathbb {Q}_p . In this paper, we compute the dimensions of certain classes of locally analytic G G -representations. This includes principal series representations and certain representations coming from homogeneous line bundles on p p -adic symmetric spaces. As an application, we compute the dimensions of the unitary GL 2 ( Q p ) \textrm {GL}_2(\mathbb {Q}_p) -representations appearing in Colmez’ p p -adic local Langlands correspondence.

COHOMOLOGY GROUPS OF SECTIONS OF HOMOGENEOUS LINE BUNDLES OVER A TOROIDAL GROUP

We completely determine cohomology groups of sections of homogeneous line bundles over a toroidal group.

Special values for conformally invariant systems associated to maximal parabolics of quasi-Heisenberg type

In this paper we construct conformally invariant systems of first order and second order differential operators associated to a homogeneous line bundle $\Cal{L}_{s} \to G_0/Q_0$ with $Q_0$ a maximal parabolic subgroup of quasi-Heisenberg type. This generalizes the results by Barchini, Kable, and Zierau.To do so we use techniques different from ones used by them.

Conformally invariant systems of differential operators associated to maximal parabolics of quasi-Heisenberg type

Let $G_{0}$ be a simple Lie group and $Q_{0}$ a maximal parabolic subgroup of quasi-Heisenberg type. In this paper we construct conformally invariant systems of differential operators associated to a homogeneous line bundle $\mathcal{L}_{s} \to G_{0}/Q_{0}$. The systems that we construct yield explicit homomorphisms between appropriate generalized Verma modules. We also determine whether or not these homomorphisms are standard.

Harmonic analysis on spherical homogeneous spaces with solvable stabilizer

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H.

The Penrose transform in split signature

A version of the Penrose transform is introduced in split signature. It relates cohomological data on CP3∖RP3 and the kernel of differential operators on M, the (real) Grassmannian of 2-planes in R4. As an example we derive the following cohomological interpretation of the so-called X-ray transformHc1(CP3∖RP3,O(−2))→≅ker(□2,2:Γω(M,ε[−1]˜)→Γω(M,ε[−3]˜)) where Γω(M,ε[−1]˜) and Γω(M,ε[−3]˜) are real analytic sections of certain (homogeneous) line bundles on M, c stands for cohomology with compact support and □2,2 is the ultrahyperbolic operator. Furthermore, this gives a cohomological realization of the so-called “minimal” representation of SL(4,R). We also present the split Penrose transform in split instanton backgrounds.

Hua operators, Poisson transform and relative discrete series on line bundles over bounded symmetric domains

For bounded symmetric domains Ω=G/K of tube type and general domains of type I, we consider the action of G on sections of a homogeneous line bundle over Ω and the corresponding eigenspaces of G-invariant differential operators. The Poisson transform maps hyperfunctions on the Shilov boundary S=K/L to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type whose image is in a relative discrete series; we compute the corresponding elements in the discrete series.

Borel–Weil theory for groups over commutative Banach algebras

Let be a commutative unital Banach algebra, 𝔤 be a semisimple complex Lie algebra and be the 1-connected Banach–Lie group with Lie algebra . Then there is a natural concept of a parabolic subgroup of and we obtain generalizations of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over with non-zero holomorphic sections. In particular, we show that all these line bundles are tensor products of pullbacks of line bundles over X(ℂ) by evaluation maps.

Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

The extended weight semigroup of a homogeneous space of a connected semisimple algebraic group characterizes the spectra of the representations of on spaces of regular sections of homogeneous line bundles over , including the space of regular functions on . We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces , where is a simply connected non-simple semisimple complex algebraic group and is a connected closed subgroup of . In all cases we also find the highest-weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.

A Method for Weight Multiplicity Computation Based on Berezin Quantization

Let $G$ be a compact semisimple Lie group and $T$ be a maximal torus of $G$. We describe a method for weight multiplicity computation in unitary irreducible representations of $G$, based on the theory of Berezin quantization on $G/T$. Let $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle $\mathcal{L}^{\lambda}$ over $G/T$ associated with the highest weight $\lambda$ of the irreducible representation $\pi_{\lambda}$ of $G$. The multiplicity of a weight $m$ in $\pi_{\lambda}$ is computed from functional analytical structure of the Berezin symbol of the projector in $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ onto subspace of weight $m$. We describe a method of the construction of this symbol and the evaluation of the weight multiplicity as a rank of a Hermitian form. The application of this method is described in a number of examples.

Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains

We develop a group-theoretic method to generalize the Laplace-Beltrami operators on the classical domains. In K. Okamoto, "Harmonic analysis on homogeneous vector bundles," Lecture Notes in Mathematics, Springer-Verlag, 266 (1971), 255–271, inspired by Helgason's paper, "A duality for symmetric spaces with applications to group representations," Advan. Math. 5 (1970), 1–154, we defined the "Poisson transforms" for homogeneous vector bundles over symmetric spaces. In K. Okamoto, M. Tsukamoto and K. Yokota, "Generalized Poisson and Cauchy kernel functions on classical domains," Japan. J. Math. 26 No. 1 (2000), 51–103., we defined the generalized Poisson-Cauchy transforms for homogeneous holomorphic line bundles over hermitian symmetric spaces and computed explicitly the kernel functions for each type of the classical domains. In E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori, "Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical domains," Proc. Japan Acad., 82, Ser. A (2006), 167–172., making use of the Casimir operator, we defined the "generalized Laplacians" on homogeneous holomorphic line bundles over hermitian symmetric spaces and showed that the generalized Poisson-Cauchy transforms give rise to eigenfunctions of the "generalized Laplacians". In this paper, using the canonical coordinates for each type of the classical domains, we carry out the direct computation to obtain the explicit formulas of (line bundle valued) invariant differential operators which we call the generalized Laplacians and compute their eigenvalues evaluated at the generalized Poisson-Cauchy kernel functions

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szego kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in $${\mathbb C^n}$$ for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szego kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.