Hikita surjectivity for $\mathcal N /// T$

Abstract

The Hamiltonian reduction $\mathcal N///T$ of the nilpotent cone in $\mathfrak{sl}_n$ by the torus of diagonal matrices is a Nakajima quiver variety which admits a symplectic resolution $\widetilde{\mathcal N///T}$, and the corresponding BFN Coulomb branch is the affine closure $\overline{T^*(G/U)}$ of the cotangent bundle of the base affine space. We construct a surjective map $\mathbb C\left[\overline{T^*(G/U)}^{T\times B/U}\right] \twoheadrightarrow H^*\left(\widetilde{\mathcal N /// T}\right)$ of graded algebras, which the Hikita conjecture predicts to be an isomorphism. Our map is inherited from a related case of the Hikita conjecture and factors through Kirwan surjectivity for quiver varieties. We conjecture that many other Hikita maps can be inherited from that of a related dual pair.

Figures (5)

• Figure 1: Left: The bouquet quiver $Q_4$, with vertices labelled. Right: The abundant bouquet quiver $Q_4^+$, with vertices labelled.
• Figure 2: An element of the representation space $\mathbb M(Q_4,\mathbf v,\mathbf e_{s_3})$. (The boxed $\mathbb C$ is the framing associated with the vertex $s_3$.)
• Figure 3: Left: The $A_{n-1}$ Dynkin quiver defining the Slodowy slice $X$. Right: The affine $\widetilde{D}_n$ Dynkin quiver, with dimension vector $\mathbf v'$, defining $X/\!\!/\!\!/ T = \mathbb C^2/D_n$.
• Figure 4: Left: The quiver defining the nilpotent orbit $X$ in the case $n = 8$. Right: The star quiver $Q_n^{\mathrm{star}}$, with dimension vector $\mathbf v"$, defining $X/\!\!/\!\!/ T$, also for $n = 8$.
• Figure 5: The variety $\mathcal{M}_C$, for $n = 8$, as a Hamiltonian reduction of the form $(T^*\mathop{\mathrm{Hom}}\nolimits(\mathbb C^2,\mathbb C^2))^6/\!\!/\!\!/ (\mathrm{SL}_2)^5$.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.3: broer93, cf. also gr15
• proof
• Proposition 2.5
• proof
• Proposition 2.6
• proof