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We verify the self-duality of Green-Schwarz supercoset sigma models on AdS$_d\times S^d $ backgrounds (d=2,3,5) under combined bosonic and fermionicT-dualities without gauge fixing kappa symmetry. We also prove this propertyfor superstrings on AdS$_d \times S^d \times S^d$ (d=2,3) described bysupercoset sigma models with the isometries governed by the exceptional Liesupergroups $D(2,1;\alpha)$ (d=2) and $D(2,1;\alpha)\times D(2,1;\alpha)$(d=3), which requires an additional T-dualisation along one of the spheres.Then, by taking into account the contribution of non-supercoset fermionic modes(up to the second order), we provide evidence for the T-self-duality of thecomplete type IIA and IIB Green-Schwarz superstring theory on AdS$_d\times S^d\times T^{10-2d}$ (d=2,3) backgrounds with Ramond-Ramond fluxes. Finally,applying the Buscher-like rules to T-dualising supergravity fields, we provethe T-self-duality of the whole class of the AdS$_d\times S^d \times M^{10-2d}$superbackgrounds with Ramond-Ramond fluxes in the context of supergravity.

T-Duality of Green-Schwarz Superstrings on AdS(d) x S(d) x M(10-2d)