Consider an arbitrary derivation δ=hddxδ=hddx, where hh is a smooth function of xx. What is the general rule for computing the iterations δk(f)δk(f), formally? How is this related to representation theory? For example:
δ(f)δ2(f)δ3(f)δ3(f)===f(1)hf(2)h2+f(1)h(1)hf(3)h3+3f(2)h(1)h2+f(1)h(2)h2+f(1)(h(1))2hhas 7 summands with coefficients 1,6,4,7,1,4,1.δ(f)=f(1)hδ2(f)=f(2)h2+f(1)h(1)hδ3(f)=f(3)h3+3f(2)h(1)h2+f(1)h(2)h2+f(1)(h(1))2hδ3(f)has 7 summands with coefficients 1,6,4,7,1,4,1.
Together with the multiplication by xx operator, δδ generates a noncommutative algebra AhAh whose elements can be written as differential operators with coefficients in F[x]F[x], where FF is a field of arbitrary characteristic. I will talk about some features of this algebra related to Hochschild cohomology and representation theory, also addressing the combinatorial problem described above.
Parts of this talk are based on joint work with G. Benkart and M. Ondrus, and work in progress with A. Solotar.
