On the Cohomology of Congruence Subgroups of GL₃ over the Eisenstein Integers

Let F be the imaginary quadratic field of discriminant −3 and (Formula presented.) its ring of integers. Let Γ be the arithmetic group (Formula presented.) and for any ideal (Formula presented.) let (Formula presented.) be the congruence subgroup of level (Formula presented.) consisting of matrices with bottom row (Formula presented.) In this paper we compute the cohomology spaces (Formula presented.) as a Hecke module for various levels (Formula presented.) where ν is the virtual cohomological dimension of Γ. This represents the first attempt at such computations for GL₃ over an imaginary quadratic field, and complements work of Grunewald–Helling–Mennicke and Cremona, who computed the cohomology of (Formula presented.) over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on (Formula presented.).