series. This is the so-called “equal rank case” because it occurs if and only if the groups $G({{\mathbb{R}}})$ and $K_\infty$ have the same rank. Then the $L^2$-cohomology $H^*_{(2)}(X_K)$ is finite-dimensional, and Matsushima’s formula, proved in this generality by Borel and Casselman [@Borel_Casselman], gives a description of this cohomology and of its Hecke algebra action in terms of discrete automorphic representations of $G$ whose infinite component is a cohomological representation of $G({{\mathbb{R}}})$, and in particular, either a discrete series or a special type of non-tempered representation.

Another cohomology of interest in this case is the intersection cohomology $IH^*(\overline{X}_K)$ of the minimal Satake compactification $\overline{X}_K$ of $X_K$. In order to study this cohomology, Goresky, Harder and MacPherson introduced in [@GHM] a family of cohomology theories called “weighted cohomologies” and showed that the two middle weighted cohomologies agree with $IH^*(\overline{X}_K)$ if $X_K$ has the structure of a complex algebraic variety. This result was later generalized by Saper in [@Saper].

All the cohomology theories that we discussed have actions of the Hecke algebra, and the isomorphism of the previous paragraph is equivariant for this action. Zucker conjectured that there should be a Hecke-equivariant isomorphism between $H^*_{(2)}(X_K)$ and $IH(\overline{X}_K)$. This conjecture was proved by Looijenga 