The reduced polynomial algebra as a module for \(\text{SO}_{2n}(k)\) (Q1210078)

The submodule structure of the Weyl modules for groups of type \(A_n\) over an algebraically closed field \(k\) of characteristic \(p > 0\) was determined by Doty, provided that the highest weight is a multiple of the first fundamental weight. These modules are the homogeneous parts \(\text{S}^m V\) of the symmetric algebra \(\text{S}(V)\) of the defining representation \(V\) of \(\text{SL} (V)\). As discovered by Krop a key object of study in understanding the symmetric powers is the so-called reduced polynomial algebra \(Q\) that is a quotient algebra of \(\text{S}(V)\) obtained by dividing out the ideal \(I\) generated by \(p\)th powers in \(\text{S}(V)\). Using this idea, some computations involving the hyperalgebra, and the theory of contravariant forms developed by Wong we managed to determine the composition factors of the \(\text{Sp} (V)\)- modules \(\text{S}^m V\) [J. Algebra 140, No. 2, 415-425 (1991; Zbl 0795.20025)]. In this note we extend some of the above methods to the case of the orthogonal group \(G = \text{SO}_{2n} (k)\) of even rank.