Components of Springer fibers for the exceptional groups G2 and F4

Let G be the complex connected simply connected simple Lie group of type G2 or F4. Let K denote the fixed point subgroup relative to an involution of G that is lifted from a Cartan involution. This article gives a description of certain components of Springer fibers associated to closed K-orbits contained in the flag variety of G. These components allow us to describe certain multiplicity polynomials associated to discrete series representations of the real form G22 of G2 and the two real forms F44 and F4−20 of F4. The goals for this paper are motivated by the descriptions of Springer fiber components and the associated multiplicity polynomials for type SU(p,q) described in a paper of Barchini and Zierau.