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Lowest weight representations of the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$graded superalgebra introduced by Rittenberg and Wyler are investigated. Wegive a explicit construction of Verma modules over the ${\mathbb Z}_2 \otimes{\mathbb Z}_2$ graded superalgebra and show their reducibility by usingsingular vectors. The explicit formula of singular vectors are given and areused to derive partial differential equations invariant under the colorsupergroup generated by the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ gradedsuperalgebra.

Verma Modules over a ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ Graded Superalgebra and Invariant Differential Equations