Let G be a connected reductive group over an algebraically closed field k, and let $\operatorname {Fl}$ be the affine flag variety of G. For every regular semisimple element $\gamma $ of $G(k((t)))$, the affine Springer fiber $\operatorname {Fl}_\gamma $ can be presented as a union of closed subvarieties $\operatorname {Fl}^{\leq w}_{\gamma }$, defined as the intersection of $\operatorname {Fl}_{\gamma }$ with an affine Schubert variety $\operatorname {Fl}^{\leq w}$.

The main result of this paper asserts that if elements $w_1,\ldots ,w_n$ are sufficiently regular, then the natural map $H_i(\bigcup _{j=1}^n \operatorname {Fl}^{\leq w_j}_{\gamma })\to H_i(\operatorname {Fl}_{\gamma })$ is injective for every $i\in \mathbb Z$. It plays an important role in our work [BV], where our result is used to construct good filtrations of $H_i(\operatorname {Fl}_{\gamma })$. Along the way, we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.

The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.