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Working paper

Interpolation Macdonald polynomials and Cauchy-type identities

Let Sym denote the algebra of symmetric functions and Pμ(⋅;q,t) and Qμ(⋅;q,t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q,t)-Cauchy identity ∑μPμ(x1,x2,…;q,t)Qμ(y1,y2,…;q,t)=∏i,j=1∞(xiyjt;q)∞(xiyj;q)∞ expresses the fact that the Pμ(⋅;q,t)'s form an orthogonal basis in Sym with respect to a special scalar product ⟨⋅,⋅⟩q,t. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric functions Iμ(x1,x2,…;q,t)=Pμ(x1,x2,…;q,t)+lower degree terms. These functions come from the N-variate interpolation Macdonald polynomials, extensively studied in the 90's by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions Jμ(⋅;q,t) with the biorthogonality property ⟨Iμ(⋅;q,t),Jν(⋅;q,t)⟩q,t=δμν. These new functions live in a natural completion of the algebra Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit is also described.