Whittaker categories and the minimal nilpotent finite $W$-algebra for $\mathfrak{sl}_{n+1}$

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