My question is prompted by Ben Webster's [answer][1] to [this question][2]. **Is there a notion of tensor product for representations of a finite W-algebra?** I thought about this question years ago in the context of the infinite-dimensional W-algebras in conformal field theory, but failed to reach a satisfactory conclusion. I was hoping that now that the professional representation theorists have started to study finite W-algebras an answer might be forthcoming. [1]: http://mathoverflow.net/questions/10409/introduction-to-w-algebras-why-w-algebras/10552#10552 [2]: http://mathoverflow.net/questions/10409/introduction-to-w-algebras-why-w-algebras