Szego Kernels and Finite Group Actions

Szego kernels, Toeplitz operators, and equivariant fixed point formulae

Let $γ$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $γ$ induces an automorphism $γ_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the trace of $γ_k$ in terms of fixed point data. More generally, one may consider the composition of $γ_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,γ)$, one may consider induce…

Equivariant Asymptotics of Szegö kernels under Hamiltonian $SU(2)$-action

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties the equivariant projector associated to a weight $k \, \boldsymbol{ ν}$, when $\boldsymbol{ ν}$ is fixed and $k\rightarrow +\infty$.

Equivariant asymptotics of Szegö kernels under Hamiltonian $SU(2)\times S^1$-actions

Let $M$ be complex projective manifold and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian and holomorphic manner and that this action linearizes to $A$. Then, there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space $H(X)$. The standard circle action on $H(X)$ commutes with the action of $G$ and thus one has a decompositions labeled by $(k\,\boldsymbolν,\,k)$, where $k\in\mathbb{Z}$ and $\boldsymbol{ ν}\in \hat{G}$. We consider the local and global asymptotic properties of the correspo…

Equivariant Asymptotics of Szegö kernels under Hamiltonian $U(2)$ actions

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymb…

Scaling asymptotics of Szego kernels under commuting Hamiltonian actions

Let M be a connected d-dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature. Let G be a compact and connected Lie group of dimension d(G), and let T be a compact torus T of dimension d(T). Suppose that both G and T act on M in a holomorphic and Hamiltonian manner, that the actions commute, and linearize to A. If X is the principal circle-bundle associated to A, then this set-up determines commuting unitary representations of G and T on the Hardy space H(X) of X, which may then be decomposed over the irreducible repre…

Grauert管边界上Poisson核与Szegő核的等变渐近展开

Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries

本文研究紧致实解析黎曼流形上Grauert管边界处，等距群作用下复延拓特征函数与Hardy空间特征函数沿边界的等变渐近集中，并给出应用。

紧李群在复域中的特征函数渐近性

Eigenfunction asymptotics in the complex domain for a compact Lie group

本文利用 Grauert 管技巧研究紧李群复化中由不可约表示矩阵元定义的 Poisson 与 Szegő 核算子核在权空间射线趋于无穷时的近对角缩放渐近性。

无挠 CR 流形的量子化与约化

Quantization and reduction for torsion free CR manifolds

本文研究紧致无挠 CR 流形上的量子化与约化，证明在自然假设下群不变 Fourier-Szegő 投影算子有完全渐近展开，并当线丛张量幂足够大时量子化与约化可交换。

关于环面作用的锥形变换的渐近等距嵌入的评注

Remarks on asymptotic isometric embeddings of conic transforms for torus actions

本文研究 Hamilton 全纯环面作用下量子化线丛对偶圆丛的几何，利用等变 Szegő 投影算子构造与不可约权射线相关的锥形变换的渐近嵌入。