4 problems

Let $W(E_8)$ be the Weyl group of $E_8$, and consider all $W(E_8)$-invariant weak Jacobi forms of integral weight and integral index. Let $M_*(\operatorname{SL}_2(\mathbb Z))$ deno…

Let $H(t)$ be the dimension of the space of $W(E_8)$-invariant holomorphic Jacobi forms of weight $4$ and positive index $t$. Let $N(t)$ be the number of distinct Weyl orbits of ve…

For each index $t$, let $d_{k,t}$ be the number of weight-$k$ generators of the free module $J_{*,E_8,t}^{\operatorname{w},W(E_8)}$ of $W(E_8)$-invariant weak Jacobi forms. Stabili…

Let $t$ be an index, and let $W(E_8)$ denote the Weyl group of the $E_8$ root system. A weak Jacobi form is allowed to have Fourier terms with negative discriminant, and its weight…