Whittaker categories and the minimal nilpotent finite W-algebras for \(\mathfrak {sl}_{n+1}\)

For any \({\textbf{a}}=(a_1,\dots ,a_n)\in {\mathbb {C}}^n\) $\mathbf{a}=(a_1,\cdots ,a_n)\in {\mathbb{C}}^n$ , we introduce a Whittaker category \({\mathcal {H}}_{{\textbf{a}}}\) ${\mathcal{H}}_{\mathbf{a}}$ whose objects are \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ -modules M such that \(e_{0i}-a_i\) $e_{0i}-a_i$ acts locally nilpotently on M for all \(i \in \{1,\dots ,n\}\) $i\in \{1,\cdots ,n\}$ , and the subspace \(\textrm{wh}_{{\textbf{a}}}(M)=\{v\in M \mid e_{0i} v=a_iv, \ i=1,\dots ,n\}\) ${\text{wh}}_{\mathbf{a}}(M)=\{v\in M\mid e_{0i}v=a_{i}v,i=1,\cdots ,n\}$ is finite dimensional. In this paper, we first give a tensor product decomposition \(U_S=W\otimes B\) $U_S=W\otimes B$ of the localization \(U_S\) $U_S$ of \(U(\mathfrak {sl}_{n+1})\) $U({\mathfrak{sl}}_{n+1})$ with respect to the Ore subset S generated by \(e_{01},\dots , e_{0n}\) $e_{01},\cdots ,e_{0n}$ . We show that the associative algebra W is isomorphic to the type \(A_n\) $A_n$ finite W-algebra W(e) defined by a minimal nilpotent element e in \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ . Then using W-modules as a bridge, we show that each block with a generalized central character of \({\mathcal {H}}_{{\textbf{1}}}\) ${\mathcal{H}}_{\mathbf{1}}$ is equivalent to the corresponding block of the cuspidal category \({\mathcal {C}}\) $\mathcal{C}$ , which is completely characterized by Grantcharov and Serganova. As a consequence, each regular integral block of \({\mathcal {H}}_{{\textbf{1}}}\) ${\mathcal{H}}_{\mathbf{1}}$ and the category of finite dimensional modules over W(e) can be described by a well-studied quiver with certain quadratic relations.