this paper we shall consider a natural class ofinfinite dimensional Lie algebras and correspondin... more this paper we shall consider a natural class ofinfinite dimensional Lie algebras and corresponding Banach Lie groups, namelythe automorphism groups of symmetric domains in Hilbert spaces, according tothe classification by Kaup as Cartan domains of type I - IV (cf. [Ka83], see also[Up85] and [GW72] for more details on these domains).First we give a classification of the algebraic category of
We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geo... more We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to such a geodesic cycle can be expanded into eigenfunctions of the Laplacian on the geodesic cycle. We prove exponential decay for the coefficients in this expansion.
To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal gr... more To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $\lambda_1,\lambda_2,\lambda_3$, we construct a trilinear form on $\mathcal C^\infty(S^{n-1})\times\mathcal C^\infty(S^{n-1})\times \mathcal C^\infty(S^{n-1})$, which is invariant by $\pi_{\lambda_1}\otimes \pi_{\lambda_2}\otimes \pi_{\lambda_3}$. The trilinear form, first defined for $(\lambda_1, \lambda_2,\lambda_3)$ in an open set of $\mathbb C^3$ is extended
The proof of Theorem 3.1, as presented in the article needs more explanation and does require a l... more The proof of Theorem 3.1, as presented in the article needs more explanation and does require a little more work.
We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is... more We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein–Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures. 1 Triple product integral formula We consider the symplectic form [ , ] on R 2n = R n ⊕ R n given by [(x, ξ), (y, η)]: = −〈x, η 〉 + 〈y, ξ〉. (1.1) In this paper we prove a closed formula for the following triple integral: Theorem 1.1. Let dσ be the Euclidean measure on the sphere S2n−1. Then, ∣ α−n ∣ β−n ∣ γ−n
0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplici... more 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com-ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form ω. This real ...
Several authors have studied the case where $G_0/K_0$ is a quaternionic symmetric space and the i... more Several authors have studied the case where $G_0/K_0$ is a quaternionic symmetric space and the inducing holomorphic vector bundle is a line bundle. That is the case where $\mu$ is orthogonal to the compact simple roots and the inducing representation is 1--dimensional.
We present a method to calculate intertwining operators between the underlying Harish-Chandra mod... more We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a reductive Lie group $G$ and a reductive subgroup $G'$. Our method decribes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to a system of scalar equations. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators in the orthogonal case. This establishes the compact picture of the recently studied symmetry breaking operators by Kobayashi-Speh. Applications of our classification for orthogonal groups include the construction of discrete components in the restriction of certain unitary representations, a Funk-Hecke type formula and the computation of the spectrum of Juhl's conformally...
We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is... more We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein–Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures. 1 Triple product integral formula We consider the symplectic form [ , ] on R 2n = R n ⊕ R n given by [(x, ξ), (y, η)]: = −〈x, η 〉 + 〈y, ξ〉. (1.1) In this paper we prove a closed formula for the following triple integral: Theorem 1.1. Let dσ be the Euclidean measure on the sphere S2n−1. Then, ∣ α−n ∣ β−n ∣ γ−n
A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The... more A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The proof uses their relation with the Knapp-Stein intertwining operators for the spinorial principal series. We also treat the compact picture, i.e. the corresponding operators on the sphere, where certain polynomials of the Dirac operator appear. This gives a new representation-theoretic framework for earlier results.
Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic t... more Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic triangle in D, we compute explicitly the integral R � !, generalizing previous results (see (D-T)). As a consequence, if X is a manifold which admits D as universal cover, we calculate the Gromov norm of (!) 2 H2(X, R). The formula for R � ! is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see (C-Ø)). 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com- ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form !. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to m...
Let 7) be a Hermitian symmetric space of tube type, S its Shilov boundary and G the neutral compo... more Let 7) be a Hermitian symmetric space of tube type, S its Shilov boundary and G the neutral component of the group of bi-holomorphic diffeomorphisms of 7). In the model situation 7) is the Siegel disc, S is the manifold of Lagrangian subspaces and G is the symplectic group. We introduce a notion of transversality for pairs of elements in S, and then study the action of G on the set of triples of mutually transversal points in S. We show that there is a finite number of G-orbits, and to each orbit we associate an integer, thus generalizing the Maslov index. Using the scalar automorphy kernel of 7), we construct a C*-valued, G-invariant kernel on 7) • 7) • 7). Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a Z-valued, G-invariant kernel for triples of mutually transversal points in S. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.
ABSTRACT The proof of Theorem 3.1, as presented in the article needs more explanation and does re... more ABSTRACT The proof of Theorem 3.1, as presented in the article needs more explanation and does require a little more work.
this paper we shall consider a natural class ofinfinite dimensional Lie algebras and correspondin... more this paper we shall consider a natural class ofinfinite dimensional Lie algebras and corresponding Banach Lie groups, namelythe automorphism groups of symmetric domains in Hilbert spaces, according tothe classification by Kaup as Cartan domains of type I - IV (cf. [Ka83], see also[Up85] and [GW72] for more details on these domains).First we give a classification of the algebraic category of
We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geo... more We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to such a geodesic cycle can be expanded into eigenfunctions of the Laplacian on the geodesic cycle. We prove exponential decay for the coefficients in this expansion.
To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal gr... more To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $\lambda_1,\lambda_2,\lambda_3$, we construct a trilinear form on $\mathcal C^\infty(S^{n-1})\times\mathcal C^\infty(S^{n-1})\times \mathcal C^\infty(S^{n-1})$, which is invariant by $\pi_{\lambda_1}\otimes \pi_{\lambda_2}\otimes \pi_{\lambda_3}$. The trilinear form, first defined for $(\lambda_1, \lambda_2,\lambda_3)$ in an open set of $\mathbb C^3$ is extended
The proof of Theorem 3.1, as presented in the article needs more explanation and does require a l... more The proof of Theorem 3.1, as presented in the article needs more explanation and does require a little more work.
We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is... more We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein–Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures. 1 Triple product integral formula We consider the symplectic form [ , ] on R 2n = R n ⊕ R n given by [(x, ξ), (y, η)]: = −〈x, η 〉 + 〈y, ξ〉. (1.1) In this paper we prove a closed formula for the following triple integral: Theorem 1.1. Let dσ be the Euclidean measure on the sphere S2n−1. Then, ∣ α−n ∣ β−n ∣ γ−n
0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplici... more 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com-ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form ω. This real ...
Several authors have studied the case where $G_0/K_0$ is a quaternionic symmetric space and the i... more Several authors have studied the case where $G_0/K_0$ is a quaternionic symmetric space and the inducing holomorphic vector bundle is a line bundle. That is the case where $\mu$ is orthogonal to the compact simple roots and the inducing representation is 1--dimensional.
We present a method to calculate intertwining operators between the underlying Harish-Chandra mod... more We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a reductive Lie group $G$ and a reductive subgroup $G'$. Our method decribes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to a system of scalar equations. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators in the orthogonal case. This establishes the compact picture of the recently studied symmetry breaking operators by Kobayashi-Speh. Applications of our classification for orthogonal groups include the construction of discrete components in the restriction of certain unitary representations, a Funk-Hecke type formula and the computation of the spectrum of Juhl's conformally...
We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is... more We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein–Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures. 1 Triple product integral formula We consider the symplectic form [ , ] on R 2n = R n ⊕ R n given by [(x, ξ), (y, η)]: = −〈x, η 〉 + 〈y, ξ〉. (1.1) In this paper we prove a closed formula for the following triple integral: Theorem 1.1. Let dσ be the Euclidean measure on the sphere S2n−1. Then, ∣ α−n ∣ β−n ∣ γ−n
A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The... more A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The proof uses their relation with the Knapp-Stein intertwining operators for the spinorial principal series. We also treat the compact picture, i.e. the corresponding operators on the sphere, where certain polynomials of the Dirac operator appear. This gives a new representation-theoretic framework for earlier results.
Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic t... more Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic triangle in D, we compute explicitly the integral R � !, generalizing previous results (see (D-T)). As a consequence, if X is a manifold which admits D as universal cover, we calculate the Gromov norm of (!) 2 H2(X, R). The formula for R � ! is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see (C-Ø)). 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com- ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form !. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to m...
Let 7) be a Hermitian symmetric space of tube type, S its Shilov boundary and G the neutral compo... more Let 7) be a Hermitian symmetric space of tube type, S its Shilov boundary and G the neutral component of the group of bi-holomorphic diffeomorphisms of 7). In the model situation 7) is the Siegel disc, S is the manifold of Lagrangian subspaces and G is the symplectic group. We introduce a notion of transversality for pairs of elements in S, and then study the action of G on the set of triples of mutually transversal points in S. We show that there is a finite number of G-orbits, and to each orbit we associate an integer, thus generalizing the Maslov index. Using the scalar automorphy kernel of 7), we construct a C*-valued, G-invariant kernel on 7) • 7) • 7). Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a Z-valued, G-invariant kernel for triples of mutually transversal points in S. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.
ABSTRACT The proof of Theorem 3.1, as presented in the article needs more explanation and does re... more ABSTRACT The proof of Theorem 3.1, as presented in the article needs more explanation and does require a little more work.
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