  Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying
the Weyl orbit $W(\rho)$ of its Weyl vector $\rho$. A remarkable fact for
Affine Lie algebras is that the number of elements of $W(\rho)$ is finite at
each and every depth level though totally it has infinite number of elements.
This allows us to look at $W(\rho)$ as a manifold graded by depths of its
elements and hence a new kind of Poincare Polynomial is defined. We give these
polynomials for all Affine Kac-Moody Lie algebras, non-twisted or twisted. The
remarkable fact is however that, on the contrary to the ones which are
classically defined,these new kind of Poincare polynomials have modular
properties, namely they all are expressed in the form of eta-quotients. When
one recalls Weyl-Kac character formula for irreducible characters, it is
natural to think that this modularity properties could be directly related with
Kac-Peterson theorem which says affine characters have modular properties.
Another point to emphasize is the relation between these modular Poincare
Polynomials and the Permutation Weights which we previously introduced for
Finite and also Affine Lie algebras. By the aid of permutation weights, we have
shown that Weyl orbits of an Affine Lie algebra are decomposed in the form of
direct sum of Weyl orbits of its horizontal Lie algebra and this new kind of
Poincare Polynomials count exactly these permutation weights at each and every
level of weight depths.
