Dimensions of Affine Springer Fibers for Some $\mathrm{GL}_n$ Symmetric Spaces

TL;DR

This paper derives a formula for the dimensions of affine Springer fibers linked to symmetric spaces from $ ext{GL}_n$ embeddings, with applications to unitary symmetric spaces.

Contribution

It provides the first explicit dimension formula for affine Springer fibers in these symmetric spaces, extending previous understanding.

Findings

Derived a general dimension formula for affine Springer fibers in symmetric spaces.

Applied the formula to compute dimensions for fibers in certain unitary symmetric spaces.

Enhanced understanding of geometric structures related to affine Springer fibers.

Abstract

We determine a formula for the dimension of a family of affine Springer fibers associated to a symmetric space arising from the block diagonal embedding $\mathrm{GL}_n\times\mathrm{GL}_n\hookrightarrow\mathrm{GL}_{2n}$ . As an application, we determine the dimension of affine Springer fibers attached to certain unitary symmetric spaces.