Let $k$ be a field of characteristic $p$. Consider the algebra $A:=\mathcal{D}(k[x])^{S_2}$ consisting of Grothendieck differential operators invariant under the $S_2$ action $x\mapsto -x$. The algebra $A$ has a $k$-basis $x^i\partial^{(j)}: x^a\mapsto\binom{a}{j}x^{a-j+i}$ where $2|i+j$.
Let $A_r$ be the subalgebra of $A$ generated by operators whose order is less than $2p^r$. What is the center of $A_r$? What about the center of $A$?
I would appreciate any partial answers or references.
