Differential operators, exact pullback formulas of Eisenstein series, and Laplace transforms

Abstract

AbstractFor a variableZ=(zi⁢j){Z=(z_{ij})}of the Siegel upper half spaceHn{H_{n}}of degreen, put∂Z=(1+δi⁢j2⁢∂∂⁡zi⁢j)1≤i,j≤n{{\partial}_{Z}=(\frac{1+\delta_{ij}}{2}\frac{{\partial}}{{\partial}z_{ij}})_{% 1\leq i,j\leq n}}. For a polynomialP⁢(T){P(T)}in components ofn×n{n\times n}symmetric matrixT, we haveP⁢(∂Z)⁢det⁡(Z)s=det⁡(Z)s⁢Q⁢(Z-1){P({\partial}_{Z})\det(Z)^{s}=\det(Z)^{s}Q(Z^{-1})}for some polynomialQ⁢(T){Q(T)}. We show that the correspondence ofPandQare bijective for mosts, and give a formula ofPfor anyQ. In particular, whenQis a monomial, we show that suchPcorresponds exactly to thedescending basisdeveloped in a joint work with D, Zagier, for which an explicit generating series is known. By using the above results and the generating series, we give an exact formula for differential operators𝔻{{\mathbb{D}}}such that for any Siegel modular formsFof weightk, the restrictionResHn1×Hn2⁢(𝔻⁢F){\mathrm{Res}_{H_{n_{1}}\times H_{n_{2}}}({\mathbb{D}}F)}to the diagonal blocksHn1×Hn2⊂Hn1+n2=Hn{H_{n_{1}}\times H_{n_{2}}\subset H_{n_{1}+n_{2}}=H_{n}}is a vector-valued Siegel modular forms of weightdetk⁡ρ{\det^{k}\rho}, where ρ is a fixed representation ofGLn1⁢(ℂ)×GLn2⁢(ℂ){\mathrm{GL}_{n_{1}}({\mathbb{C}})\times\mathrm{GL}_{n_{2}}({\mathbb{C}})}. These results are applied to give an exact Garrett–Böcherer-type pullback formula for any ρ that describes the restriction of𝔻⁢Ekn{{\mathbb{D}}E_{k}^{n}}toHn1×Hn2{H_{n_{1}}\times H_{n_{2}}}for holomorphic Siegel Eisenstein seriesEkn{E_{k}^{n}}of weightkof degreen.