Sheaf-theoretic approach to the convolution algebras on quiver varieties (Q2856640)

Let \(V=\left( V_{i}\right) _{i\in I}\) and \(W=\left( W_{i}\right) _{i\in I}\) be finite dimensional complex vector spaces attached to the vertices of a given quiver with vertex set \(I.\) Let \(\mathbf{v}\) and \(\mathbf{w}\) be the dimension vectors of \(V\) and \(W\) respectively. Let \(M\left( \mathbf{v,w} \right) =\bigoplus_{h\in H}\text{Hom}\left( V_{\mathrm{out}\left( h\right) },V_{\mathrm{in}\left( h\right) }\right) \oplus\bigoplus_{i\in I}\text{Hom} \left( W_{k},V_{k}\right) \oplus\bigoplus_{i\in I}\text{Hom}\left( V_{k},W_{k}\right) ,\) where \(H\) is the set of ordered pairs corresponding to the arrows. The group \(G_{\mathbf{v}}:=\prod_{k\in I}\text{GL}\left( V_{k}\right) \) acts on \(M\left( \mathbf{v,w}\right) \) and there is a certain map \(\mu:M\left( \mathbf{v,w}\right) \rightarrow\bigoplus_{k\in I}\text{End}\left( V_{k}\right) .\) Let \(\mathfrak{M}_{0}\left( \mathbf{v,w}\right) \) be the affine algebro-geometric quotient \(\mu ^{-1}\left( 0\right) //G_{\mathbf{v}}.\) A quiver variety \(\mathfrak{M} \left( \mathbf{v,w}\right) \) is defined to be the GIT quotient \(\mu ^{-1}\left( 0\right) //_{\chi}G_{\mathbf{v}},\) where \(\chi\left( g\right) =\prod_{k\in I}\det\left( g_{k}^{-1}\right) .\) There is a natural map \(\pi_{\mathbf{v}}:\mathfrak{M}\left( \mathbf{v,w}\right) \rightarrow \mathfrak{M}_{0}\left( \mathbf{v,w}\right) .\)NEWLINENEWLINE\textit{H. Nakajima} [Duke Math. J. 91, No. 3, 515--560 (1998; Zbl 0970.17017)] used quiver varieties to geometrically construct the finite dimensional irreducible representations of \(\mathfrak{U}\left( \mathfrak{g}\right)\), the universal enveloping algebra of a symmetric Kac-Moody algebra. Suppose the underlying quivers are of finite ADE type. Then one can write \(\mathfrak{M}_{0}\left( \mathbf{v,w}\right) =\bigsqcup_{\mathbf{v}^{\prime}\leq\mathbf{v}}\mathfrak{M}_{0}^{\mathrm{reg}}\left( \mathbf{v}^{\prime},\mathbf{w}\right) \) and the Beilinson-Bernstein-Deligne decomposition allows for a decomposition of \(\pi_{\ast}\mathcal{C}\) where \(\mathcal{C}\) is a certain constant perverse sheaf on \(\sqcup_{\mathbf{v} }\mathfrak{M}_{0}^{\mathrm{reg}}\left( \mathbf{v}^{\prime},\mathbf{w}\right) .\)NEWLINENEWLINEThe work under review is a sheaf-theoretic interpretation of the work of Nakajima, allowing for a refined form of the decomposition for \(\pi_{\ast}\mathcal{C}.\)