Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules

Details Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules

2009-03-25

arxiv.org/abs/0903.4239v1

Mathematics

Representation Theory

22pages; This is a reformulation of our earlier manuscript "Partial Differential Equations for Singular Vectors of sl(n)" (arX

Scientific paper

Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group $S_n$ on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by $\{\sgm(1)\mid \sgm\in S_n\}$. Moreover, the singular vectors of $sl(n,\mbb{C})$ in the Verma module are given by those $\sgm(1)$ that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\mbb{C})$ are naturally included in our almost elementary approach of partial differential equations.

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