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

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 ${\mathfrak{s}\mathfrak{l}}_{n+1}$-modules $M$ such that $e_{0i}-a_i$ acts locally nilpotently on $M$ for all $i\in \{1,\dots ,n\}$, and the subspace ${\mathrm{w}\mathrm{h}}_{\mathbf{a}}(M)=\{v\in M\mid e_{0i}v=a_{i}v,i=1,\dots ,n\}$ is finite dimensional. In this paper, we first give a tensor product decomposition $U_S=W\otimes B$ of the localization $U_S$ of $U({\mathfrak{s}\mathfrak{l}}_{n+1})$ 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 ${\mathfrak{s}\mathfrak{l}}_{n+1}$. 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.