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

Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL3(O-F), and for any ideal n subset of O-F let Gamma(0)(n) be the congruence subgroup of level n consisting of matrices with bottom row (0, 0, *)mod n. In this paper we compute the cohomology spaces H nu-1 (Gamma(0)(n); C) as a Hecke module for various levels n; where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL(3) over an imaginary quadratic field, and complements work of Grunewald-Helling-Mennicke and Cremona, who computed the cohomology of GL(2) over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on GL(3)/F.