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.
Categories: math.RT math.RA