Whittaker Modules for Classical Lie Superalgebras

Abstract

We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić–Soergel equivalence of a category of Whittaker modules and a category of Harish–Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories ${\mathcal {O}}$. As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras $\mathfrak {gl}(m|n)$ and the ortho-symplectic Lie superalgebras $\mathfrak {osp}(2|2n)$ can be computed via the Kazhdan–Lusztig combinatorics.

Appendix

1.1 Character formulae of $\mathfrak {gl}(2|1)$

The goal of this subsection is to give the list of character formulae of projective covers in the principal block of ${\mathcal {O}}$ of ${\mathfrak {g}}:=\mathfrak {gl}(2|1)$ computed in [27]. With the canonical isomorphic $\mathfrak {gl}(1|2)\cong \mathfrak {gl}(2|1)$ and the BGG reciprocity, the composition factors of Verma modules over $\mathfrak {gl}(1|2)$ can be read off from character formulae in this subsection.

We choose the Borel subalgebra ${\mathfrak {b}} :=\bigoplus _{1\le i\le j\le 3}{\mathbb {C}}E_{ij}$ and the standard Cartan subalgebra ${\mathfrak {h}}^*:=\oplus _{i=1}^3{\mathbb {C}}E_{ii}$. Set $\rho \in {\mathfrak {h}}^*$ to be the corresponding Weyl vector. For $\mu \in {\mathfrak {h}}^*$, we let $M^{{\mathfrak {a}}}(\mu )$ denote Verma module over $\mathfrak {gl}(2|1)$ with highest weight $\mu -\rho$. Also, we let $P^{{\mathfrak {a}}}(\mu )$ denote the projective cover of the simple quotient of $M^{{\mathfrak {a}}}(\mu )$. For any $a,b,c\in {\mathbb {C}}$, we set $(a|b,c):= a\epsilon _1+b\epsilon _2+c\epsilon _3$. We adopt the notation $P^{{\mathfrak {a}}}(\lambda ) = \sum _{\mu \in {\mathfrak {h}}^*}(P^{{\mathfrak {a}}}(\lambda ): M^{{\mathfrak {a}}}(\mu ))M^{{\mathfrak {a}}}(\mu )$ to record the Verma flag structure $P^{{\mathfrak {a}}}(\lambda ).$

Lemma 36

[27, Section 9]. We have the following character formulae:

$$\begin{aligned}&(1). ~P^{{\mathfrak {a}}}(0,0|0) = M^{{\mathfrak {a}}}(0,0|0)+M^{\mathfrak a}(0,1|-1)+M^{{\mathfrak {a}}}(1,0|-1). \\&(2). ~P^{{\mathfrak {a}}}(0,-1|1) = M^{\mathfrak a}(0,-1|1)+M^{{\mathfrak {a}}}(0,0|0)+M^{{\mathfrak {a}}}(1,0|-1).\\&(3). ~P^{{\mathfrak {a}}}(-1,0|1) = M^{\mathfrak a}(-1,0|1)+M^{{\mathfrak {a}}}(0,-1|1)+M^{{\mathfrak {a}}}(00|0).\\&(4). ~P^{{\mathfrak {a}}}(0,-k|k) = M^{\mathfrak a}(0,-k|k)+M^{{\mathfrak {a}}}(0,-(k-1)|(k-1)),~\text {for }k> 1.\\&(5). ~P^{{\mathfrak {a}}}(-k,0|k) = M^{\mathfrak a}(-k,0|k)+M^{{\mathfrak {a}}}(0,-k|k) +M^{\mathfrak a}(-(k-1),0|(k-1))+\\ {}&M^{{\mathfrak {a}}}(0,-(k-1)|(k-1)),~\text {for }k> 1.\\&(6). ~P^{{\mathfrak {a}}}(k,0|-k) = M^{\mathfrak a}(k,0|-k)+M^{{\mathfrak {a}}}(k+1,0|-(k+1)),~\text {for }k\ge 1.\\&(7). ~P^{{\mathfrak {a}}}(0,k|-k) = M^{\mathfrak a}(0,k|-k)+M^{{\mathfrak {a}}}(k,0|-k)+M^{\mathfrak a}(0,k+1|-(k+1))\\&\qquad +M^{{\mathfrak {a}}}(k+1,0|-(k+1)),~\text {for }k\ge 1. \end{aligned}$$