The K-theory of maximal and reduced Roe algebras for Hecke pairs with equivariant coarse embeddings

In this paper, we generalize the Dirac-dual-Dirac method to Hecke pairs with equivariant coarse embeddings and establish the K-theoretic isomorphisms between the maximal and reduced equivariant Roe algebras. We also extend these results to prove the Baum--Connes conjecture in this context.