4 added 7 characters in body

When $\mathfrak g$ is a complex, simple, simply laced Lie algebra of rank r then the (specialized) character of the basic representation for the corresponding affine Lie algebra $\hat {\mathfrak g}$ is given by $\chi_{\hat {\mathfrak g}}(q)=\frac{\Theta _{\mathfrak g}(q)}{\eta (q)^r}$, where $\Theta _{\mathfrak g} (q)$ is the theta function for the root lattice of $\mathfrak g$. It is well known that $\Theta _{E_8} (q)$ is equal to the normalized Eisenstein series of weight 4, $E_4 (q)$. We also have another basic result that $E_4(q)=\frac{1}{2}(\theta_2 (0,q)^8 +\theta_3 (0,q)^8 + \theta _4 (0,q)^8)$, where $\theta_2(x,q)$, $\theta_3(x,q)$, $\theta_4(x,q)$ are the three classical even Jacobi theta functions. Being odd, the other Jacobi theta function $\theta_1(x,q)$ vanishes at $x=0$. My question is in regards to the unspecialized form of the character; that is, I'm wondering if evaluating the character on a non-zero element of the Cartan subalgebra produces something of the form $$\chi _{E_8}(x_1,...,x_8,q)=\frac{\frac{1}{2}(\prod_{i=1}^8\theta_2 {\hat {E_8}}(x_1,...,x_8,q)=\frac{\frac{1}{2}(\prod_{i=1}^8\theta_2 (x_i,q) + \prod_{i=1}^8\theta_3(x_i,q) + \prod_{i=1}^8\theta _4 (x_i,q)+\alpha \prod_{i=1}^8\theta_1 (x_i,q))}{\eta (q)^8}$$ for some $\alpha$? Moreover, in the case $\alpha = 0$, does it make sense that Taylor expanding the numerator of this expression and writing it in terms of elementary symmetric polynomials in $x_1^2,...,x_8^2$ one should find that everything below degree 8 can be written as quasimodular forms times only powers of $p_1(x_1,...,x_8)=x_1^2+...+x_8^2$? This is what I got when I plugged it into Mathematica and in fact the coefficients are similar to the Eisenstein-Jacobi series $E_{4,1}(z,q)$. What would probably be very relevant is if there's anything regarding Jacobi forms in several variables that has been studied that is similar to the "development coefficients" which are well studied in Eichler and Zagier "The Theory of Jacobi Forms".

I've tried looking through a good bit of the literature regarding both of these matters, but couldn't find what I was looking for. If anyone could point me in the right direction, I would be greatly appreciative.

3 added 19 characters in body; edited title

# Character of the Basic Representation for Affine E_8 in termsTerms of Jacobi Theta Functions

When $\mathfrak g$ is a complex, simple, simply laced Lie algebra of rank r then the (specialized) character of the basic representation for the corresponding affine Lie algebra $\hat {\mathfrak g}$ is given by $\chi_{\hat {\mathfrak g}}(q)=\frac{\Theta _{\mathfrak g}(q)}{\eta (q)^r}$, where $\Theta _{\mathfrak g} (q)$ is the theta function for the root lattice of $\mathfrak g$. It is well known that $\Theta _{E_8} (q)$ is equal to the normalized Eisenstein series of weitght weight 4, $E_4 (q)$. We also have another basic result that $E_4(q)=\frac{1}{2}(\theta_2 (0,q)^8 +\theta_3 (0,q)^8 + \theta _4 (0,q)^8)$, where $\theta_2(x,q)$, $\theta_3(x,q)$, $\theta_4(x,q)$ are the three classical even Jacobi theta functions. Being odd, the other Jacobi theta function $\theta_1(x,q)$ vanishes at $x=0$. My question is in regards to the unspecialized form of the character; that is, I'm wondering if evaluating the character on a non-zero element of the Cartan subalgebra produces something of the form $$\chi _{E_8}(x_1,...,x_8,q)=\frac{\frac{1}{2}(\prod_{i=1}^8\theta_2 (x_i,q) + \prod_{i=1}^8\theta_3(x_i,q) + \prod_{i=1}^8\theta _4 (x_i,q)+\alpha \prod_{i=1}^8\theta_1 (x_i,q))}{\eta (q)^8}$$ for some $\alpha$? Moreover, in the case $\alpha = 0$, does it make sense that Taylor expanding the numerator of this expression and writing it in terms of elementary symmetric polynomials in $x_1^2,...,x_8^2$ one should find that everything below degree 8 can be written as quasimodular forms times only powers of $p_1(x_1,...,x_8)=x_1^2+...+x_8^2$? This is what I got when I plugged it into Mathematica and in fact the coefficients are similar to the Eisenstein-Jacobi series $E_{4,1}(z,q)$. What would probably be very relevant is if there's anything regarding Jacobi forms in several variables that has been studied that is similar to the "development coefficients" which are well studied in Eichler and Zagier "The Theory of Jacobi Forms".

I've tried looking through a good bit of the literature regarding both of these matters, but couldn't find what I was looking for. If anyone could point me in the right direction, I would be greatly appreciative.

2 edited title

1