The restriction of admissible modules to parabolic subalgebras

Abstract:

This paper studies algebraic versions of Casselman’s subrepresentation theorem. Let $\mathfrak {g}$ be a semisimple Lie algebra over an algebraically closed field $F$ of characteristic zero and $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}$ be an Iwasawa decomposition for $\mathfrak {g}$. Then $(\mathfrak {g},\mathfrak {k})$ is said to satisfy property $(\mathfrak {n})$ if $M \ne M$ for every admissible $(\mathfrak {g},\mathfrak {k})$-module $M$. We prove that, if $(\mathfrak {g},\mathfrak {k})$ satisfies property $(\mathfrak {n})$, then $\mathfrak {n}N \ne N$ whenever $N$ is a $(\mathfrak {g},\mathfrak {k})$-module with $\dim N < \operatorname {card} F$. This is then used to show (purely algebraically) that $(\mathfrak {s}l(n,F),\mathfrak {s}o(n,F))$ satisfies property $(\mathfrak {n})$. The subrepresentation theorem for $\mathfrak {s}l(n)$ is an easy consequence of this.