We have 3 kinds of generators of the invariant ring for the elliptic Weyl groups:
(1) the fundamental characters of the affine Lie algebras,
(2) the (generalized weak) Jacobi forms,
(3) the flat invariants for the Frobenius structure.
For the D4 case, we could construct (2) by (1) by a determinant of a matrix whose entries are fundamental characters. By these Jacobi forms, we could give the explicit description of the Frobenius structure, since the Frobenius structure has the modular invariance. Then we could construct (3) by (2). So we could construct (3) by (1) (arxiv:1708.03875).
Afterwards we find a characterization of (3) by using the behavior on the fixed points of the modified Coxeter transformation for the elliptic root system.
Combining these results, we give some new properties of the fundamental characters and the Weyl denominator for the affine Lie algebra of type D4.

The seminar will take place at RIMS room 006.