Siegel theta series and theta series with pluriharmonics (awaiting review)

Given a positive definite quadratic $m\times m$ positive definite even quadratic form $Q$, and given a degree $g$, define $$\vartheta_Q^{(g)}(\Omega) = \sum_{N\in\Bbb Z^{m \times g}} \exp(\pi i N' Q N \Omega ) $$

If $m$ is a multiple of 4, this is a Siegel modular form of degree $g$ of weight $m/2$ with respect to the subgroup $\Gamma_0(\ell)$ where $\ell$ is such that $\ell Q^{-1}$ is an even quadratic form.

We say $P$ is pluriharmonic (spherical function) in the matrix variable $X=(x_{r,s})$ for $1\le r\le m$, $1\le s\le g$, if

• for each $1\le i,j\le n$, we have $\displaystyle{\sum_{t=1}^g\frac{\partial^2 P}{\partial_{i,t}\partial_{j,t}}=0}$

Define the theta series with pluriharmonic polynomial $P$ homogeneous of degree $\nu$ to be $$\vartheta_{Q,P}^{(g)}(\Omega)= \sum_{N\in\Bbb Z^{m \times g}} P(N) \exp(\pi i N' Q N \Omega ) $$

If $m$ is a multiple of 4, this is a Siegel cusp form of weight $m/2+\nu$ in degree $g$ with respect to the subgroup $\Gamma_0(\ell)$.

When $m$ is not a multiple of $4$, one gets a Siegel modular form with a character.