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

Abstract

Abstract For a variable Z = ( z i ⁢ j ) {Z=(z_{ij})} of the Siegel upper half space H n {H_{n}} of degree n, put ∂ Z = ( 1 + δ i ⁢ j 2 ⁢ ∂ ∂ ⁡ z i ⁢ 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 polynomial P ⁢ ( T ) {P(T)} in components of n × n {n\times n} symmetric matrix T, we have P ⁢ ( ∂ Z ) ⁢ det ⁡ ( Z ) s = det ⁡ ( Z ) s ⁢ Q ⁢ ( Z - 1 ) {P({\partial}_{Z})\det(Z)^{s}=\det(Z)^{s}Q(Z^{-1})} for some polynomial Q ⁢ ( T ) {Q(T)} . We show that the correspondence of P and Q are bijective for most s, and give a formula of P for any Q. In particular, when Q is a monomial, we show that such P corresponds exactly to the descending basis developed 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 operators 𝔻 {{\mathbb{D}}} such that for any Siegel modular forms F of weight k, the restriction Res H n 1 × H n 2 ⁢ ( 𝔻 ⁢ F ) {\mathrm{Res}_{H_{n_{1}}\times H_{n_{2}}}({\mathbb{D}}F)} to the diagonal blocks H n 1 × H n 2 ⊂ H n 1 + n 2 = H n {H_{n_{1}}\times H_{n_{2}}\subset H_{n_{1}+n_{2}}=H_{n}} is a vector-valued Siegel modular forms of weight det k ⁡ ρ {\det^{k}\rho} , where ρ is a fixed representation of GL n 1 ⁢ ( ℂ ) × GL n 2 ⁢ ( 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 of 𝔻 ⁢ E k n {{\mathbb{D}}E_{k}^{n}} to H n 1 × H n 2 {H_{n_{1}}\times H_{n_{2}}} for holomorphic Siegel Eisenstein series E k n {E_{k}^{n}} of weight k of degree n.