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In this paper, we investigate when weighted composition operators acting on Dirichlet spaces D(B N ) are complex symmetric with respect to some special conjugations, and provide some characterizations of Hermitian weighted composition operators on D(B N ). Furthermore, we give a sufficient and necessary condition for J-symmetric weighted composition operators on Hardy spaces H 2 (B N ) to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on H 2 (B N ) are obtained. We also discuss the normality of complex symmetric weighted composition operators on H 2 (B N ).2010 Mathematics Subject Classification. MSC: 47B33, 47B15, 47B38, 32A35, 32A37.

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