Reduced symmetric powers of natural realizations of the groups \(SL_ m(P)\) and \(Sp_ m(P)\) and their restrictions to subgroups (Q1174001)

Let \(P\) be an algebraically closed field of characteristic \(p > 0\), \(GL_ m(P)\) and \(SL_ m(P)\) be the general linear group, and the special linear group of degree \(m\) over the field \(P\) respectively, \(Y_ m = P[y_ 1,\dots,y_ m]/ \langle y^ p_ 1,\dots,y^ p_ m \rangle\) be the ring of reduced polynomials in \(m\) variables, \(Y_{m,i}\) be the subspace of homogeneous polynomials of degree \(i\) in \(Y_ m\). The natural action of an arbitrary subgroup \(\Gamma \subset GL_ m(P)\) on \(Y_{m,1}\) is extended, in a standard way, to a representation of \(\Gamma\) in \(Y_{m,i}\). We call this representation \(\varphi_ i\) of the group \(SL_ m(P)\), realized on \(Y_{m,i}\), the \(i\)-th reduced symmetric power of the natural representation of the group \(SL_ m(P)\). In this paper, properties of the restrictions of the representations \(\varphi_ i\) to some subgroups are described.