Dixmier’s sixth problem for an algebra \(A\) over a field \(k\) of characteristic zero is reduced to the following one. Let \(a\in A\) and \(f(t)\in k[t]\), \(\deg f(t)>1\). Is it the case that \(\text{ad\,}f(a)\) has a nonzero eigenvalue?
The author considers a filtered \(k\)-algebra \(B=\bigcup_{i\geqslant 0}B_i\) such that \(\text{gr\,}B\) is a commutative domain. Then for any \(a\in B\) and any \(f(t)\in k[t]\), \(\deg f(t)>1\), the eigenvalues of \(\text{ad\,}f(a)\) are equal to zero.
As an immediate consequence it is shown that we can take as \(B\) the ring of differential operators \(\mathcal D(X)\) on a smooth irreducible affine algebraic variety. Moreover, if \(X\) is an algebraic curve and \(a\in\mathcal D(X)\) then the set of eigenvalues of \(\text{ad\,}a\) forms an additive cyclic group \(\mathbb{Z}_\rho\) for some \(\rho\in k\). This result solves Dixmier’s problem 3 for the ring \(\mathcal D(X)\).