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Let G be the group A_4 or Z_2xZ_2. We compute the integral of \lambda_g onthe Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1having monodromy group G. We compute the generating functions for theseintegrals and write them as a trigonometric expression summed over the positiveroots of the E_6 and D_4 root systems respectively. As an application, we provethe Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and[C^3/(Z_2xZ_2)].

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture