Asymptotic $q,t$-Fuss--Catalan numbers for type $B$

Abstract

Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus {\mathfrak{h}}^{\ast }$, let $S=\mathbb{C}[\mathfrak{h}\oplus {\mathfrak{h}}^{\ast }]$, let $J\subset S$ be the ideal generated by the $W$-alternating polynomials and ${\mathfrak{m}}_S$ is the maximal ideal of the origin. For sufficiently large $m$ we compute $q,t$-Fuss-Catalan polynomial $Ca{t}^{(m)}(B_n;q,t):=Hilb(\frac{J^m}{{\mathfrak{m}}_S{J}^m}{)}_{det-part}$ and imply $Ca{t}^{(m)}(B_n;1,1)=\left(\genfrac{}{}{0ex}{}{n(m+1)}{n}\right)$. For proofs, we work with the $\Gamma$-equivariant Hilbert scheme $Y_n=n\Gamma$-$Hilb({\mathbb{C}}^2)$, $\Gamma ={\mu }_2$ and Haiman-type Koszul complex that defines the punctual locus of $Y_n$. Our formula for $Ca{t}^{(m)}(B_n;q,t)$ is derived from a localization computaion for the Haiman-type Koszul complex.

88 pages, comments are welcome; new section with examples and conjectures is added; some typos are corrected