Contents

# Contents

## Idea

Macdonald polynomials are a generalization of a Schur functions; they unify a theory of Hall-Littlewood and Jack polynomials. They form a family of orthogonal polynomials? which are symmetric functions in $x_1,\ldots,x_n$ with coefficients which are rational functions of two additional variables $q$ and $t$.

Given a partition $\lambda$, one defines a shift operator $T_{q,x_i}$ which maps $f = f(x_1,\ldots, x_n)$ to $f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n)$ and the operators $D_r$, $r = 0, 1, \ldots, n$ via

$D_r = t^{\frac{r(r-1)}{2}} \sum_{I\subset \{1,\ldots,n\}, |I| = r} \prod_{i\in I, j\notin I} \frac{t x_i-x_j}{x_i-x_j}\prod_{i\in I} T_{q, x_i},$

and the corresponding generating series $D := \sum_{r=0}^n D_r u^r$.

The Macdonald polynomial $P_\lambda(x;q,t)$ is an eigenfunction of $D$ with the eigenvalue

$\prod_{i=1}^n (1 + u t^{n-i} q^{\lambda_i})$

In the case $q = t$ we get the Schur function $P_\lambda(x; t,t) = s_\lambda(t)$. Similarly, shifted Macdonald polynomials generalize shifted Schur functions.

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A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with Pan. Haiman’s geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.