From "Hilbert series and Gelfand dualityHilbert series and Gelfand duality" by Vadim Schechtman (CiteSeerX, arXiv:0908.4533):
In their great work on spherical functions ... Berezin and Gelfand wrote:
”... there exists a deep duality between the function ... giving the law of multiplication in the center of the [infinitesimal] group ring [of a semisimple Lee group]and the function ... giving multiplication of representations.
... an analogous duality exists between matrix elements of ... an irreducible representation of the group SU(2) ... and the so-called ”Clebsch-Gordan coefficients”...
Another example of such a duality are the formulas of Gelfand-Tsetlin for matrix elements of irreducible representations of the algebra of complex matrices with trace 0 and the formulas for coordinates in the group of unitary matrices... In all of these cases the duality consists in the fact that functions of discrete arguments satisfy finite difference equations analogous to differential equations satisfied by functions of real variables that correspond to them.”
The second of the above examples may be expressed by saying that we have a duality between the classical orthogonal polynomials (Jacobi etc.) and their discrete analogues (Hahn etc.). (In fact, all of the above examples admit a similar reformulation.)
The main purpose of the present note is to propose an example illustrating that exactly this type of dual polynomials appears in certain Hilbert series.
