Abstract:
We announce proofs of Macdonald's constant term
conjectures for the affine root systems $S( F_4 )$ and
$S ( F_4 )^\vee$. We also give an algorithm for deciding
the conjectures for the remaining root systems $S( E_6 )$,
$S( E_7 )$ and $S( E_8 )$ and prove that the constant
term in question can be indeed expressed in closed form.
Combined with previous work of Zeilberger-Bressoud, Kadell, and
Gustafson, our results imply that Macdonald's conjectures are
true in form for any root system, and the complete truth
of Macdonald's conjectures is a finite number of mips away.