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

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 ${\mathcal{W}}_{1+\mathrm{\infty }}$ 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 ${\mathrm{\Delta }}_{\mu }(\alpha )$ 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 ${\mathcal{W}}_{1+\mathrm{\infty }}$ algebra ${\mathbf{S}\mathbf{H}}^c$, equivalently in the affine Yangian of ${\mathfrak{g}\mathfrak{l}}_1$: in our normalization its first nontrivial Cartan mode is ${\psi }_3=3{\mathrm{\Delta }}_2(\alpha )+2(\alpha -1)E$, 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 ${\mathcal{W}}_{1+\mathrm{\infty }}$ representation used by Lascoux and the author.