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We construct reflection functors for Yetter-Drinfeld modules over Nicholssystems and discuss their fundamental properties. We will obtain propertiesabout the geometry of the support of Nichols systems and their Yetter-Drinfeldmodules, by looking at iterated reflections.  We will also study the maximal subobject of Yetter-Drinfeld modules overNichols systems and find a special morphism, that we name Shapovalov morphism,whose kernel coincides with this maximal subobject. Moreover, we will use thismorphism to characterize properties about the reflections of theYetter-Drinfeld modules.  We calculate an explicit formula of the Shapovalov morphism in the case wherethe Nichols system is of group type. We will use the formula to calculate itskernel in the components of degree $2$ and to ascribe the theory of reflectionsof Yetter-Drinfeld modules over Nichols systems of diagonal type with thereflection theory of Dynkin diagrams.  We will also apply and specify the theory to the Yetter-Drinfeld modules overNichols systems that are obtained by inducing comodules of the Nichols systems,a construction that is reminiscent of Verma modules in the representationtheory of Lie algebras. For Nichols systems of diagonal type we will obtain theresult, that such an induced objects irreducibility can be characterized by apolynomial that is given by the positive roots of its Nichols system. Thispolynomial appeared in previous works and is known as Shapovalov determinant.  Finally, we will apply the theory to some specific examples of Nicholsalgebras of nonabelian group type and to some examples of Nichols algebras ofdiagonal type.

Yetter-Drinfeld modules over Nichols systems and their reflections