# Nakajima quiver varieties for ADE quiver with one dimensional framing

Let $$Q$$ be a quiver of type $$ADE$$, $$I$$ is the set of vertices of $$Q$$. Let $$\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$$ be a Nakajima quiver variety for such quiver (here $${\mathbf{v}}=(v_i)_{i \in I}$$ is the dimension vector and $${\mathbf{w}}=(w_i)_{i \in I}$$ is framing). We can associate to $$\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$$ a pair of weights $$\lambda:=\sum_i w_i\omega_i$$, $$\mu:=\lambda-\sum_i v_i\alpha_i$$.

We assume that $$\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$$ is nonempty (so $$\mu$$ is a weight of the irreducible representation $$L_{\lambda}$$). Assume also that $$w_i=\delta_{i,k}$$ for some $$k \in I$$ (i.e. all $$w_i$$ are zero except $$w_k$$ which is one). In other words we assume that $$\lambda=\omega_k$$ for some $$k$$.

If $$\mu$$ is conjugate to $$\omega_k$$ via the action of the Weyl group (this for example always holds if $$\omega_k$$ is minuscule) then the quiver variety $$\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$$ is one point.

If $$\mu$$ is not conjugate to $$\omega_k$$ then $$\mathfrak{M}({\mathbf{v}},\mathbf{w})$$ has dimension at least two.

Question: do we have a good (explicit) description of the variety $$\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$$ when $$\lambda=\omega_k$$ (i.e. $$w_{i}=\delta_{i,k}$$)?

We, at least, have the explicit answer for their Betti numbers. For an example, let $$Q$$ be type $$E_8$$, and $$k$$ be the triplet vertex. Take $$\mu = 0$$. The normalized Poincare polynomial of $$\mathfrak M(\mathbf v,\mathbf w)$$ was computed in https://arxiv.org/pdf/math/0606637.pdf as $$1357104 + 2232771t^2 + 2002423t^4 + 1317308t^6 + 716312t^8 + 342421t^{10} + 148512t^{12} + 59490t^{14} + 22162t^{16} + 7687t^{18} + 2463t^{20} + 726t^{22} + 192t^{24} + 44t^{26} + 8t^{28} + t^{30}.$$ Here the constant term corresponds to the middle degree, while the highest term corresponds to the degree 0.