We study moduli spaces of O’Grady’s ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3[n] case, which are of dimension 19 and 20 respectively.

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Let$A$and$B$be positive numbers and$m$and$n$positive integers,$m<n$. Then there is for complex valued functions φ on$R$with sufficient differentiability and boundedness properties a representationwhere$v$_{$1$}and$v$_{$2$}are bounded Borel measures with$v$_{$1$}absolutely continuous, such that there exists a function φ with ∣φ^{(n)}∣ ⩽$A$and ∣φ∣ ⩽$A$on$R$and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.