Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 1 / b

The relation $[h,e]=2e$ forces $e{w}_i$ to have weight $a+2(i+1)$, sofor scalars $c_i$. The relation $[e,f]=h$ becomesWith $c_0=b$, this recurrence has the unique solutionfor every $i\in \mathbb{Z}$. Direct substitution also verifies $[h,f]=-2f$ and $[h,e]=2e$, so these formulas define the unique required sl2 Lie algebra action. They form an Intermediate-series sl2 module.

Let $0\ne U\subseteq W$ be a subrepresentation and choosewith finite support. The $h$-eigenvalues $a+2r$ are pairwise distinct. By Lagrange interpolation, there is a polynomial $P$ which is one at one chosen eigenvalue appearing in $u$ and zero at all the others. Then $P(h)u$ is a nonzero scalar multiple of one basis vector $w_i$. Since $U$ is invariant under $h$, it contains $w_i$.

Ifthen the coefficient $c_j=b-ja-j(j+1)$ vanishes. Consequentlyis stable under $h$, $f$, and $e$: the only raising operation that could leave it is $e{w}_j$, and that is zero. Thus $W$ is reducible.

Conversely, let $U$ be a nonzero subrepresentation. By part ii it contains some $w_i$, and repeated application of $f$ gives every $w_r$ with $r\le i$. If every $c_r$ is nonzero, repeated application of $e$ also gives every $w_r$ with $r>i$, so $U=W$. Hence a proper nonzero subrepresentation exists exactly when some $c_j=0$, or