On large cardinals and generalized Baire spaces

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $\kappa$. We show the consistency of $E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{\lambda^{++}}_\lambda$ in the space $(\lambda^{++})^{\lambda^{++}}$, being continuously reducible to $E^{2,\lambda^{++}}_{\lambda^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{\lambda^{++}}_{\lambda^+}$ in the space $2^{\lambda^{++}}$. Then we show the consistency of $E^{2,\kappa}_{reg}$, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space $2^{\kappa}$, being $\Sigma_1^1$-complete. We finish by showing, for $\Pi_2^1$-indescribable $\kappa$, that the isomorphism relation between dense linear orders of cardinality $\kappa$ is $\Sigma_1^1$-complete.