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Genkai Zhang

Publications and source records attributed to Genkai Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Quasi-convexity of energy functions along Teichm\"uller geodesics

Hyperbolic length functions are among the most fundamental ones on Teichm\"uller space, and they are quasi-convex along Teichm\"uller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichm\"uller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichm\"uller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichm\"uller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

math.DG↗
arXiv2026

Fourier positivity for spherical functions I: split tori and spherical principal series

We prove Fourier positivity for spherical functions on a semisimple linear algebraic group $G$ over a local field restricted to its split tori $A$ for unitary principal series parameters of $G$. For ${\rm SL}_n(F)$, where $F$ is a local field, we obtain an explicit recursive formula for the Fourier transform on the diagonal split torus in terms of local Rankin--Selberg factors for ${\rm GL}_n\times {\rm GL}_{n-1}$, together with uniform exponential lower bounds in the spectral parameters. The main input is a Plancherel expansion for the restriction of a ${\rm GL}_n(F)$-spherical function to ${\rm GL}_{n-1}(F)$. Its coefficients are spherical periods computed by Rankin--Selberg theory. Positivity of the Fourier transform for general semisimple groups with unitary principal series parameters is obtained by reduction to full-rank subgroups of type A. The results are motivated by variance non-vanishing problems for mixing abelian actions on homogeneous spaces.

math.RT↗
arXiv2026

Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces

Let $G/K$ be a Hermitian symmetric space and $V_\tau$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_\tau)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, \tau)} V_\tau$ over $G/K$ defined by a finite-dimensional representation $(V_\tau, \tau)$ of $K$. We classify irreducible representations $(V_\tau, \tau)$ such that $\mathcal D^G(G/K, V_\tau)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$.

math.RT↗
arXiv2026

On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup

We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two.

math.RT↗
arXiv2024

Wehrl inequalities for matrix coefficients of holomorphic discrete series

We prove Wehrl-type $L^2(G)-L^{p}(G)$ inequalities for matrix coefficients of vector-valued holomorphic discrete series of $G$, for even integers $p=2n$. The optimal constant is expressed in terms of Harish-Chandra formal degrees for the discrete series. We prove the maximizers are precisely the reproducing kernels.

math.RT↗
arXiv2023

Casimir energy of hyperbolic orbifolds with conical singularities

In this article, we obtain the explicit expression of the Casimir energy for 2-dimensional Clifford-Klein space forms in terms of the geometrical data of the underlying spacetime with the help of zeta-regularization techniques. The spacetime is geometrically expressed as a compact hyperbolic orbifold surface that may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption (known to hold asymptotically) on the growth of the lengths of primitive closed geodesics of the $(2, 3, 7)$-triangle group orbifold its Casimir energy is positive (repulsive).

math.SP↗
arXiv2023

Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups

Let $G/K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation $\pi_\nu=\text{Ind}_P^G(1\otimes e^\nu\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K/L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K/L_0\to K/L$ and a circle bundle $K/L_0\to K/L_1$ over a compact Hermitian symmetric space $K/L_1$. We compute the Lie algebra $\mathfrak g$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations.

math.RT↗
arXiv2023

Cartan--Helgason theorem for quaternionic symmetric and twistor spaces

Let $(\mathfrak{g}, \mathfrak{k})$ be a complex quaternionic symmetric pair with $\mathfrak{k}$ having an ideal $\mathfrak{sl}(2, \mathbb{C})$, $\mathfrak{k}=\mathfrak{sl}(2, \mathbb{C})+\mathfrak{m}_c$. Consider the representation $S^m(\mathbb{C}^2)=\mathbb{C}^{m+1}$ of $\mathfrak{k}$ via the projection onto the ideal $\mathfrak{k}\to \mathfrak{sl}(2, \mathbb{C})$. We study the finite dimensional irreducible representations $V(\lambda)$ of $\mathfrak{g}$ which contain $S^m(\mathbb{C}^2)$ under $\mathfrak{k}\subseteq \mathfrak{g}$. We give a characterization of all such representations $V(\lambda)$ and find the corresponding multiplicity $m(\lambda,m)=\dim \operatorname{Hom} (V(\lambda)|_\mathfrak{k},S^m(\mathbb{C}^2)).$ We consider also the branching problem of $V(\lambda)$ under $\mathfrak{l}=\mathfrak{u}(1)_{\mathbb{C}} + \mathfrak{m}_c\subseteq \mathfrak{k}$ and find the multiplicities. Geometrically the Lie subalgebra $\mathfrak{l}\subseteq \mathfrak{k}$ defines a twistor space over the compact symmetric space of the compact real form $G_c$ of $G_{\mathbb{C}}$, $\text{Lie}(G_{\mathbb{C}})=\mathfrak{g}$, and our results give the decomposition for the $L^2$-spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan--Helgason's theorem for symmetric spaces $(\mathfrak{g}, \mathfrak{k})$ and Schlichtkrull's theorem for Hermitian symmetric spaces where one-dimensional representations of $\mathfrak{k}$ are considered.

math.RT↗
arXiv2023

Heisenberg parabolically induced representations of Hermitian Lie groups, Part II: Next-to-minimal representations and branching rules

Every simple Hermitian Lie group has a unique family of spherical representations induced from a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. For most Hermitian groups, this family contains a complementary series, and at its endpoint sits a proper unitarizable subrepresentation. We show that this subrepresentation is next-to-minimal in the sense that its associated variety is a next-to-minimal nilpotent coadjoint orbit. Moreover, for the Hermitian groups $\operatorname{SO}_0(2,n)$ and $E_{6(-14)}$ we study some branching problems of these next-to-minimal representations.

math.RT↗
arXiv2023

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of ``invariantly-polyanalytic'' functions -- a~generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In~both cases, the~kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are~given.

math.CV↗
arXiv2022

Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Unitarity and subrepresentations

For a Hermitian Lie group $G$, we study the family of representations induced from a character of the maximal parabolic subgroup $P=MAN$ whose unipotent radical $N$ is a Heisenberg group. Realizing these representations in the non-compact picture on a space $I(\nu)$ of functions on the opposite unipotent radical $\bar{N}$, we apply the Heisenberg group Fourier transform mapping functions on $\bar N$ to operators on Fock spaces. The main result is an explicit expression for the Knapp-Stein intertwining operators $I(\nu)\to I(-\nu)$ on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein-Sato identity for the Knapp-Stein kernel on $\bar{N}$ and the decomposition of the metaplectic representation under the non-compact group $M$.

math.RT↗
arXiv2022

Curvature of the total space of a Griffiths negative vector bundle and quasi-Fuchsian space

For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over K\"ahler manifold, then there is a K\"ahler metric on its total space $E$, and we calculate the curvature and prove the non-positivity of the curvature along the tautological direction. The Nakano positivity can be formulated as a positivity for the Nakano curvature operator and we give estimate the Nakano curvature operator associated with a Nakano positive direct image bundle. As applications we construct a mapping class group invariant K\"ahler metric on the quasi-Fuchsian space QF$(S)$, which extends the Weil-Petersson metric on the Teichm\"uller space $\mathcal{T}(S)\subset {\rm QF}(S)$, and we obtain estimates for the Nakano curvature operator for the dual Weil-Petersson metric on the holomorphic cotangent bundle of Teichm\"uller space.

math.DG↗
arXiv2021

Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

Let $G$ be an irreducible Hermitian Lie group and $D=G/K$ its bounded symmetric domain in $\mathbb C^d$ of rank $r$. Each $γ$ of the Harish-Chandra strongly orthogonal roots $\{γ_1, \cdots, γ_r\}$ defines a Heisenberg parabolic subgroup $P=MAN$ of $G$. We study the principal series representations $\Ind_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from $P$. We find the complementary series, reduction points, and unitary subrepresentations in this family of representations.

math.RT↗
arXiv2021

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for large $k$. As an application we prove that the analytic torsion $τ_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(τ_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.

math.DG↗
arXiv2019

Convexity of energy function associated to the harmonic maps between surfaces

For a fixed smooth map $u_0$ between two Riemann surfaces $Σ$ and $S$ with non-zero degree, we consider the energy function on Teichmüller space $\mc{T}$ of $Σ$ that assigns to a complex structure $t\in \mc{T}$ on $Σ$ the energy of the harmonic map $u_t:Σ_t:=(Σ,t) \to S$ homotopic to $u_0$. We prove that the energy function is convex at its critical points. If $t_0\in\mc{T}$ is a critical point such that $du_{t_0}$ is never zero, then the energy function is strictly convex at this point. As an application, in the case that $u_0$ is a covering map, we prove that there exists a unique critical point $t_0\in \mc{T}$ minimizing the energy function. Moreover, the energy density satisfies $\frac{1}{2}|du|^2(t_0)\equiv 1$ and the Hessian of the energy function is positive definite at this point.

math.DG↗
arXiv2019

Second variation of Selberg zeta functions and curvature asymptotics

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmüller space. We then use this formula to determine the asymptotic behavior as $\text{Re} (s) \to \infty$ of the second variation. As a consequence, for $m \in \mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, \mathcal K_t)\to t\in \mathcal T$ of holomorphic m-differentials over the Teichmüller space $\mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.

math.SP↗
arXiv2019

Norm estimates and asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact Kähler manifolds and in [10] using skein theory.

math.DG↗