Relative Lie algebra cohomology

Definition

Let $\mathfrak g$ be a Lie algebra, let $\mathfrak k\subseteq\mathfrak g$ be a subalgebra, and let $V$ be a module over $\mathfrak g$ with a compatible action of a group $K$ having Lie algebra $\mathfrak k$ . The following cochain complex is the relative Lie algebra cochain complex:

C^q(\mathfrak g,K;V)= \operatorname{Hom}_K \!\left(\bigwedge^q(\mathfrak g/\mathfrak k),V\right),

with the Chevalley–Eilenberg differential. Its cohomology is $H^q(\mathfrak g,K;V)$ , the relative Lie algebra cohomology or $(\mathfrak g,K)$ -cohomology of $V$ .

For a disconnected $K$ , using $K$ -equivariant rather than merely $\mathfrak k$ -equivariant cochains retains the component-group action.

Geometric interpretation

When $G$ is a real reductive group with maximal compact subgroup $K$ , the complex models $G$ -invariant differential forms on the symmetric space $G/K$ with coefficients in the local system determined by $V$ . This connects representation theory with the cohomology of locally symmetric spaces.

Automorphic use

An automorphic representation is cohomological when its archimedean Harish–Chandra module, after tensoring with a finite-dimensional algebraic coefficient representation, has nonzero relative Lie algebra cohomology.

References

Armand Borel and Nolan Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, second edition, Mathematical Surveys and Monographs 67, AMS, 2000, Chapter I.
David A. Vogan Jr. and Gregg J. Zuckerman, “Unitary representations with nonzero cohomology,” Compositio Mathematica 53 (1984), 51–90. Numdam.