Littlewood-Richardson coefficient, Springer fibers and the annihilator varieties of induced representations

Abstract

For G=GL(n,C) and a parabolic subgroup P=LN with a two-block Levi subgroup L=GL(n1)×GL(n2), the space G⋅(O+n), where O is a nilpotent orbit of l, is a union of nilpotent orbits of g. In the first part of our main theorem, we use the geometric Sakate equivalence to prove that O′⊂G⋅(O+n) if and only if some Littlewood-Richardson coefficients do not vanish. The second part of our main theorem describes the geometry of the space O∩p, which is an important space to study for the Whittaker supports and annihilator varieties of representations of G.