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

Abstract

Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus\mathfrak{h}^*$, let $S=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]$, 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 $Cat^{(m)}(B_n;q,t):=Hilb(\frac{J^m}{\mathfrak{m}_S J^m})_{det-part}$ and imply $Cat^{(m)}(B_n;1,1)=\binom{n(m+1)}{n}$. 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 $Cat^{(m)}(B_n;q,t)$ is derived from a localization computaion for the Haiman-type Koszul complex.