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Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. Thesymmetric spaces associated to these G's are the classical bounded symmetricdomains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Usingthe correspondence between representations of fundamental groups of K\"{a}hlermanifolds and Higgs bundles we study representations of uniform lattices ofSU(m,1), m>1, into G. We prove that the Toledo invariant associated to such arepresentation satisfies a Milnor-Wood type inequality and that in case ofequality necessarily G=SU(p,2) with p>=2m and the representation is reductive,faithful, discrete, and stabilizes a copy of complex hyperbolic space (ofmaximal possible induced holomorphic sectional curvature) holomorphically andtotally geodesically embedded in the Hermitian symmetric spaceSU(p,2)/S(U(p)xU(2)), on which it acts cocompactly.

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type