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

Abstract

For any ${\textbf{a}}=(a_1,\dots ,a_n)\in {\mathbb {C}}^n$, we introduce a Whittaker category ${\mathcal {H}}_{{\textbf{a}}}$ 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 $\textrm{wh}_{{\textbf{a}}}(M)=\{v\in M \mid e_{0i} v=a_iv, \ i=1,\dots ,n\}$ 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 ${\mathcal {H}}_{{\textbf{1}}}$ 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 ${\mathcal {H}}_{{\textbf{1}}}$ and the category of finite dimensional modules over W(e) can be described by a well-studied quiver with certain quadratic relations.