Jack Content Operators and the Deformed ${\mathcal W}_{1+\infty}$ Algebra

Abstract

Frenkel and Wang obtained a representation of the Virasoro algebra by commuting Goulden's cut-and-join operator with the Heisenberg generators. A vertex-operator construction by Lascoux and the author extends this representation to $\mathcal W_{1+\infty}$ by means of differential operators whose eigenvalues are the power sums of the contents of a Young diagram. We develop a Jack deformation in the spherical degenerate double affine Hecke algebra and its stable limit. Starting from the Heckman--Polychronakos integrals, we isolate operators whose eigenvalues are the power sums of the $\alpha$-contents. Because the Goulden--Jackson product is defined in the convention dual to the usual Calogero--Sutherland Hamiltonians, the multiplication operators $\Delta_\mu(\alpha)$ are obtained by taking Hall adjoints. This gives conceptual derivations of the Jack cut-and-join operator and of the stable $3$-cycle operator. A normal-ordering construction due to Sergeev and Veselov makes the latter calculation explicit and suggests an integral form over $\mathbb Z[\alpha]$. The commutators of the cut-and-join operator contain one half of the usual Feigin--Fuchs realization, but this Virasoro completion is not the deformation of the Frenkel--Wang construction. The latter takes place in the deformed $\mathcal W_{1+\infty}$ algebra $\mathbf{SH}^c$, equivalently in the affine Yangian of $\mathfrak{gl}_1$: in our normalization its first nontrivial Cartan mode is $\psi_3=3\Delta_2(\alpha)+2(\alpha-1)E$, and its commutators with the first raising and lowering modes recursively generate the remaining currents. At $\alpha=1$ these relations specialize to the central-charge-one $\mathcal W_{1+\infty}$ representation used by Lascoux and the author.