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We investigate the holonomy group of singular K\"ahler-Einstein metrics onklt varieties with numerically trivial canonical divisor. Finiteness of thenumber of connected components, a Bochner principle for holomorphic tensors,and a connection between irreducibility of holonomy representations andstability of the tangent sheaf are established. As a consequence, knowndecompositions for tangent sheaves of varieties with trivial canonical divisorare refined. In particular, we show that up to finite quasi-\'etale covers,varieties with strongly stable tangent sheaf are either Calabi-Yau orirreducible holomorphic symplectic. These results form one building block forH\"oring-Peternell's recent proof of a singular version of theBeauville-Bogomolov Decomposition Theorem.

Klt varieties with trivial canonical class -- Holonomy, differential forms, and fundamental groups