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Genkai Zhang

Publications and source records attributed to Genkai Zhang.

54 records · Page 3Linked to original sources
arXiv2014

Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary series representations of $G$ and $G_1$ on Sobolev spaces on the nilpotent radicals $N$ and $N_1$ of the minimal parabolics in $G$ and $G_1$, respectively. The groups $N$ and $N_1$ are of H-type and we construct explicitly invariant differential operators between $N$ and $N_1$. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as nilpotent radical of a parabolic subgroup in a semisimple group.

math.RT↗
arXiv2013

Radon, cosine and sine transforms on Grassmannian manifolds

Let $G_{n,r}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $\mathcal R_{r^\prime, r}$, $\mathcal C_{r^\prime, r}$ and $\mathcal S_{r^\prime, r}$, from the $L^2$ space $L^2(G_{n,r}(\bbK))$ to the space $L^2(G_{n,r^\prime}(\bbK))$, for $r, r^\prime \le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $\mathbb R^r/{2πQ^\vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $\mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein's complementary series in the compact picture.

math.RT↗
arXiv2013

Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

We consider the spherical complementary series of rank one Lie groups $H_n=\SO_0(n, 1; \mathbb F)$ for $\mathbb F=\mathbb R, \mathbb C, \mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\SO_0(n-1, 1; \mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup $G_{n-1}$.

math.RT↗
arXiv2013

Discrete components in restriction of unitary representations of rank one semisimple Lie groups

We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $π_ν$ of $G$ for $ν$ in an interval on $\mathbb R^+$, the so-called complementary series, and subquotient or subrepresentations of $G$ for $ν$ being negative integers. We consider the restriction of $(π_ν, G)$ under the subgroup $H=SO(n-1, 1; \mathbb K)$. We prove the appearing of discrete components. The corresponding results for the exceptional Lie group $F_{4(-20)}$ and its subgroup $Spin(8,1)$ are also obtained.

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arXiv2012

Principal representations of SO(p,p+1)

For p odd, the Lie group SO_0(p+1,p+1) has a family of unitary degenerate principal series representations realized on the space of real (p+1) by (p+1) skew symmetric matrices, similar to the Stein's complementary series for SL(2n,C) or Speh's representation for SL(2n,R). We consider their restriction on the subgroup G= SO(p+1,p) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of G, and also irreducible under a maximal parabolic subgroup of G.

math.RT↗
arXiv2011

Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.

math.FA↗
arXiv2011

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

Let $Ω=G/K$ be a bounded symmetric domain and $S=K/L$ its Shilov boundary. 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 $S$ to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.

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arXiv2008

Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities

Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \text{div}_X (|\na_{X} u|^{p-2} \na_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$, for $k=2$ and $\mathbb G$ being the Heisenberg group they are introduced by Greiner \cite{Greiner-cjm79}. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.

math.AP↗
arXiv2007

Biorthogonal Expansion of Non-Symmetric Jack Functions

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

math.CA↗
arXiv2007

Radon transform on symmetric matrix domains

Let $\bbK=\mathbb R, \mathbb C, \mathbb H$ be the field of real, complex or quaternionic numbers and $M_{p, q}(\bbK)$ the vector space of all $p\times q$-matrices. Let $X$ be the matrix unit ball in $M_{n-r, r}(\bbK)$ consisting of contractive matrices. As a symmetric space, $X=G/K=O(n-r, r)/O(n-r)\times O(r)$, $U(n-r, r)/U(n-r)\times U(r)$ and respectively $Sp(n-r, r)/Sp(n-r)\times Sp(r)$. The matrix unit ball $y_0$ in $M_{r^\prime-r, r}$ with $r^\prime \le n-1$ is a totally geodesic submanifold of $X$ and let $Y$ be the set of all $G$-translations of the submanifold $y_0$. The set $Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $\mathcal Rf(y)$ for functions $f\in C_0^\infty(X)$ defined by integration of $f$ over the subset $y$, and the dual transform $\mathcal R^t F(x), x\in X$ for functions $F(y)$ on $Y$. We find inversion formulas by constructing explicit certain invariant differential operators.

math.FA↗
arXiv2007

Degenerate principal series representations and their holomorphic extensions

Let $X=H/L$ be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain $D=G/K$. The intersection $S$ of the Shilov boundary of $D$ with $X$ defines a distinguished subset of the topological boundary of $X$ and is invariant under $H$ and can also be realized as $S=H/P$ for certain parabolic subgroup $P$ of $H$. We study the spherical representations $Ind_P^H(\lam)$ of $H$ induced from $P$. We find formulas for the spherical functions in terms of the Macdonald ${}_2F_1$ hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces $D$. We consider a class of $H$-invariant integral intertwining operators from the representations $Ind_P^H(\lam)$ on $L^2(S)$ to the holomorphic representations of $G$ on $D$ restricted to $H$. We construct a new class of complementary series for the groups $H=SO(n, m)$, $SU(n, m)$ (with $n-m >2$) and $Sp(n, m)$ (with $n-m>1$). We realize them as a discrete component in the branching rule of the analytic continuation of the holomorphic discrete series of $G=SU(n, m)$, $SU(n, m)\times SU(n, m)$ and $SU(2n, 2m)$ respectively.

math.RT↗
arXiv2006

Radon transform on real, complex and quaternionic Grassmannians

Let $G_{n,k}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. For $1\le k < k^\prime \le n-1$ we define the Radon transform $(\mathcal R f)(η)$, $η\in G_{n,k^\prime}(\bbK)$, for functions $f(ξ)$ on $G_{n,k}(\bbK)$ as an integration over all $ξ\subset η$. When $k+k^\prime \le n$ we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive $k\times k$ matrices over $\bbK$. This generalizes the recent results of Grinberg-Rubin for real Grassmannians.

math.FA↗
arXiv2005

Non-symmetric Jacobi and Wilson type polynomials

Consider a root system of type $BC_1$ on the real line $\mathbb R$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb R$ to a $L^2$-space of $\mathbb C^2$-valued functions on $\mathbb R^+$ with the Harish-Chandra measure $|c(\lam)|^{-2}d\lam$. By introducing a weight function of the form $\cosh^{-\sig}(t)\tanh^{2k} t$ on $\mathbb R$ we find an orthogonal basis for the $L^2$-space on $\mathbb R$ consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed $\sig$ and $k$). We find a Rodrigues type formula for the functions in terms of the Cherednik operator. We compute explicitly their Cherednik-Opdam transforms. We discover thus a new family of $\mathbb C^2$-valued orthogonal polynomials. In the special case when $k=0$ the even polynomials become Wilson polynomials, and the corresponding result was proved earlier by Koornwinder.

math.CA↗
arXiv2005

Spherical transform and Jacobi polynomials on root systems of type BC

Let $R$ be a root system of type BC in $\mathfrak a=\mathbb R^r$ of general positive multiplicity. We introduce certain canonical weight function on $\mathbb R^r$ which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and the author. Associated to the weight functions there are Heckman-Opdam orthogonal polynomials of Jacobi type on the compact torus, after a change of variables they form an orthogonal system on the non-compact space $\mathfrak a$. We consider their spherical transform and prove that they are the Macdonald-Koornwinder polynomials multiplied by the spherical transform of the canonical weight function. For rank one case this was proved earlier by Koornwinder.

math.RT↗
arXiv2004

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for $L^2$-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex, a quotient of the tangential Cauchy-Riemann complex on the unit sphere in $C^n$. We prove that Szegö map is a unitary operator from a subspace of $(p, q-1)$-forms on the sphere defined by the Tanaka operators and the normal vector field onto the space of $L^2$-harmonic $(p, q)$-forms on the unit ball. Our results generalize earlier result of Folland.

math.RT↗
arXiv2002

Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials

Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to $\mathcal{D}$ of the spherical functions on $\mathcal{D}_{\mathbb{R}}$ and find the expansion in terms of the $L$-spherical polynomials on $\mathcal{D}$, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an $L^2$-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on $\mathcal D$.

math.RT↗
arXiv2001

Branching coefficients of holomorphic representations and Segal-Bargmann transform

Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the holomorphic discrete series on $\mathbb D$ forms a family of interesting representations of $G$. We consider the restriction on $D$ of the scalar holomorphic representations of $G$, as a representation of $H$. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group $L$ is a spherical subgroup of $K$ and we find a canonical basis of $L$-invariant polynomials in components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transform of those $L$-invariant polynomials are, under the spherical transform on $D$, multi-variable Wilson type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on $D$, when extended to a neighborhood in $\mathbb D$, in terms of the $L$-spherical holomorphic polynomials on $\mathbb D$, the coefficients being the Wilson polynomials.

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