[Schur functors](http://en.wikipedia.org/wiki/Schur_functor) are functors from the category of vector spaces to itself.
If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get [Schur polynomial](http://en.wikipedia.org/wiki/Schur_polynomial) in the eigenvalues of $M$.


**Question** Can one generalize (deform) Schur functors, such that 
$Tr(M^{\Lambda})$ will give polynomials which generalize (deform) Schur polynomials e.g. 
[Hall-Littlewood](http://en.wikipedia.org/wiki/Hall-Littlewood_polynomial)  polynomial, or [Jack](http://en.wikipedia.org/wiki/Jack_polynomial) polynomial and most generally 
[Macdonald](http://en.wikipedia.org/wiki/Macdonald_polynomials) polynomials ? 



Schur functors generalization to "Jack", "Hall-Littlewood", "Macdonald" functors ? 
It seems to me that this is answered, perhaps in a boring way, by [Haiman's work](http://arxiv.org/abs/math.AG/0010246) on the $n!$-conjecture (now a theorem due to Haiman). For any partition $\lambda$, Haiman constructs a finite dimensional graded module $C\_{\lambda}$ for $\mathbb{C}[x,y][S\_n]$. (The group elements commute with $x$ and $y$, and $x$ and $y$ commute with each other.) The doubly graded Frobenius character of $C\_{\lambda}$ is the $\lambda$-Macdonald polynomial.


Now just use Schur-Weyl duality: Define the functor $F\_{\lambda}$ from vector spaces to vector spaces by
$$V \mapsto V^{\otimes |\lambda|} \otimes\_{\mathbb{C}[S\_n]} C\_{\lambda}.$$
The result is a doubly graded $\mathbb{C}[x,y]$ module which is the sum of Schur functors corresponding to the Macdonald polynomial.


