Cohomological automorphic representation

Let $G$ be a connected reductive group over a number field $F$ , and write $\pi=\bigotimes_v'\pi_v$ for an automorphic representation. It is cohomological if, for every archimedean place $v$ , there is a finite-dimensional algebraic representation $E_v$ of $G(F_v)$ and an integer $q\geq 0$ such that

H^q(\mathfrak g_v,K_v;\pi_v\otimes E_v)\neq 0.

Here $\mathfrak g_v$ is the complexified Lie algebra and $K_v$ is a maximal compact subgroup, with the usual modification for a disconnected $K_v$ .

Geometric role

Relative Lie algebra cohomology connects automorphic representations to the cohomology of locally symmetric spaces. With suitable level $K_f$ , a cohomological $\pi$ can contribute to the cohomology of

G(F)\backslash G(\mathbb A_F)/(K_\infty K_f)

with the local system determined by the algebraic coefficient representation.

Algebraicity

Every cohomological automorphic representation is $C$ -algebraic. The converse requires additional hypotheses and is not true as a bare statement for arbitrary reductive groups. For regular algebraic representations of $\operatorname{GL}_n$ , the two notions are closely related, up to familiar twists.

Coefficient convention

Some authors put $E_v^\vee$ , rather than $E_v$ , in the cohomology group. This changes the recorded highest weight but not the substantive existence condition once the convention is stated. Cohomological degree and infinitesimal-character normalization must also be tracked in explicit formulas.

References

A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, second edition, AMS, 2000. AMS.
Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” §7.2. arXiv.