A one-vertex framed quiver variety and its Hecke correspondence

Let $Q$ be the quiver with one vertex and no arrows. Its diagram is

For framing space $W=\mathbb{C}$ and dimension $v\in \{0,1\}$, put $V_v={\mathbb{C}}^v$. A framed datum is a pair of complex-linear maps $I:W\to V_v$ and $J:V_v\to W$, subject to the moment equation $I\circ J=0\in {\mathrm{End}}_{\mathbb{C}}(V_v)$. The two framing maps are drawn separately from the arrows of $Q$:

The group $G_v=\mathrm{GL}(V_v)$ of invertible complex-linear maps acts by $g\cdot (I,J)=(gI,J{g}^{-1})$. Fix the positive real stability parameter ${\zeta }_R>0$ and, for this one-vertex quiver, call a solution stable when $J$ is injective. Give the stable solution set its usual complex topology and define $\mathfrak{M}(v,1)$ to be its quotient topological space by $G_v$.

For $v=0$ the solution set is a point, with coordinate algebra $A_0=\mathbb{C}$; for $v=1$, writing $I(w)=iw$ and $J(z)=jz$, it is the affine algebraic set $ij=0$ with coordinate algebra $A_1=\mathbb{C}[i,j]\mathrm{/}(ij)$. These are $\mathbb{C}$-algebras in the sense of Algebras over a commutative ring, central structure maps, and algebra homomorphisms. The action on data induces an action on each $A_v$; write $A_v^{G_v}$ for its invariant subring of fixed elements. Define ${\mathfrak{M}}_0(v,1)=\mathrm{Spec}(A_v^{G_v})$ using The underlying space of an affine spectrum. An orbit of stable data maps to the point whose prime ideal consists of invariant functions vanishing on that orbit; this is the quotient map ${\pi }_v:\mathfrak{M}(v,1)\to {\mathfrak{M}}_0(v,1)$. Let $0$ denote the point induced by the zero datum and put $\mathcal{L}(v,1)={\pi }_v^{-1}(0)$.

Let $S$ be the unframed simple datum with one-dimensional vertex space and zero framing maps. A framed subdatum with vertex space $V^{\mathrm{\prime }}=0$ uses the same framing space $W$ and zero maps $I^{\mathrm{\prime }}:W\to 0$ and $J^{\mathrm{\prime }}:0\to W$. Its inclusion uses the identity on $W$ and the unique map $\xi :0\to V_1$, compatible with the data by $I=\xi I^{\mathrm{\prime }}$ and $J^{\mathrm{\prime }}=J\xi$; hence it forces $I=0$, while the restriction of $J$ to $0$ is automatically zero. The quotient has framing space $0$ and vertex space $V_1$, so its induced framing maps vanish and it is $S$. The Hecke correspondence $P_{0,1}$ is the locus of pairs in $\mathfrak{M}(0,1)\times \mathfrak{M}(1,1)$ obtained this way. Its two projection maps are the source and target maps of the correspondence.

The degree-zero homology of a point is $\mathbb{C}$ with its point class, using The singular chain complex and singular homology and Singular homology satisfies dimension and arbitrary additivity. When $P_{0,1}$ and both spaces are points and the projections are identities, define its fundamental-class correspondence map to send the source point class to the target point class; the reverse map uses the same rule. This is the point push-pull convention used here.

This definition fixes the no-loop $A_1$ case and Nakajima's convention $\mu =IJ$, with positive stability meaning $J$ is injective. It makes no claim about the stability or properness of correspondences for other quivers.