  By normalizing the space of commuting pairs of elements in a reductive Lie
group G, and the corresponding space for the Langlands dual group, we construct
pairs of hyperkahler orbifolds which satisfy the conditions to be mirror
partners in the sense of Strominger-Yau-Zaslow. The same holds true for
commuting quadruples in a compact Lie group. The Hodge numbers of the mirror
partners, or more precisely their orbifold E-polynomials, are shown to agree,
as predicted by mirror symmetry. These polynomials are explicitly calculated
when G is a quotient of SL(n).
