Projective lines in the affine flag manifold with given tangent root vector

TL;DR

This paper characterizes the tangent space at a point in the affine flag manifold of a simple algebraic group, constructs projective lines tangent to specific root vectors, and identifies their placement within Schubert varieties.

Contribution

It provides a detailed description of the tangent space and constructs explicit projective lines tangent to imaginary root vectors in the affine flag manifold.

Findings

Explicit description of the tangent space at the distinguished point.

Construction of projective lines tangent to imaginary root vectors.

Identification of Schubert varieties containing these lines.

Abstract

We first describe the tangent space to the affine flag manifold associated to a simple algebraic group over $\mathbb{C}$ at the distinguished point starting from standard definitions. We then construct projective lines in the affine flag manifold tangent to given root vectors associated to imaginary roots of the corresponding affine Kac-Moody algebra and describe in which Schubert varieties they lie.