Cluster-randomized trials (CRTs) are used to evaluate group-level interventions and can collect multiple outcomes capturing complementary dimensions of benefit and risk. Investigators typically seek a single global summary of treatment effect, yet existing methods largely focus on single-outcome estimands or rely on model-based procedures with unclear causal interpretation or limited robustness. We develop a unified potential outcomes framework for generalized treatment effects with multiple outcomes in CRTs, accommodating both non-prioritized and prioritized outcome settings. The proposed cluster-pair and individual-pair causal estimands are defined through pairwise contrast functions and explicitly account for informative cluster size. We establish nonparametric estimation via weighted clustered U-statistics and derive efficient influence functions to construct covariate-adjusted estimators that integrate debiased machine learning with U-statistics. The resulting estimators are consistent and asymptotically normal, attain semiparametric efficiency bounds under mild regularity conditions, and admit analytically tractable variance estimators that are consistent under cross-fitting. Simulations and an application to a CRT for chronic pain management illustrate the practical utility of our proposed methods.

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