Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 2

Use a triangular decomposition of a Lie algebra $\mathfrak{g}={\mathfrak{n}}_{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}_{+}$. The highest weight space of $M^{\chi }$ has dimension one. If $z$ is central, $z{m}_{\chi }$ is still of representation weight $\chi -\rho$ and is killed by ${\mathfrak{n}}_{+}$, soSince the module is generated by $m_{\chi }$ and $z$ commutes with $U(\mathfrak{g})$, the same scalar acts on every vector. Addition, multiplication and the unit show that $z\mapsto {\theta }_{\chi }(z)$ is an algebra homomorphism to $\mathbb{C}$.

For the polynomial assertion, the relevant variable for an element of $S(\mathfrak{h})$ is $\chi$. The dependence on $z$ itself is linear, as already follows from the scalar action. Apply the Poincaré-Birkhoff-Witt theorem ordering $U({\mathfrak{n}}_{-})U(\mathfrak{h})U({\mathfrak{n}}_{+})$. Because $z$ has adjoint representation weight zero, its surviving action on the highest vector is precisely the pure $U(\mathfrak{h})$ component $p_z$; terms with a positive root factor kill that vector, and a term with only a nontrivial negative root factor could not have representation weight zero. Since $U(\mathfrak{h})=S(\mathfrak{h})$,This specifies the intended polynomial despite the printed variable mismatch. The unadorned module symbol in that sentence is understood as $M^{\chi }$.

Write $h_{\alpha }$ for the simple coroot and choose the corresponding normalized ${\mathfrak{sl}}_2$ triple. The order condition meansThe highest vector has $h_{\alpha }$-representation weight $⟨\chi -\rho ,{\alpha }^{\vee }⟩=k-1$. The raising-lowering identity givesOther simple positive root vectors commute with $E_{-\alpha }$ and also kill this vector. Thus it is a highest vector of representation weightThe universal property of the Verma module gives the required homomorphism. It is injective: under the Poincaré-Birkhoff-Witt theorem identification of each Verma module with the free rank-one $U({\mathfrak{n}}_{-})$ module, it is right multiplication by $E_{-\alpha }^k$. The algebra $U({\mathfrak{n}}_{-})$ has no zero divisors, because its associated graded algebra is the polynomial domain $S({\mathfrak{n}}_{-})$. HenceFor $k=0$ this is the identity map, so the equality case of the order is covered.

For every $\chi$ on a hyperplane $⟨\chi ,{\alpha }^{\vee }⟩=k$, with $k\in {\mathbb{Z}}_{\ge 0}$, part (b) embeds one Verma module into the other. A central element acts on the nonzero image by both scalars, so $j_z(\chi )=j_z(s_{\alpha }\chi )$ there.

The difference of these two expressions is a polynomial on ${\mathfrak{h}}^{\ast }$. Fix all coordinates parallel to those hyperplanes and vary the remaining coordinate. It vanishes at every nonnegative integer, so as a polynomial in that coordinate it is identically zero. Its coefficient polynomials therefore vanish as well. Consequently the equality holds for all $\chi$. Since the simple reflections generate the Weyl group,

In the shifted normalization of part (a), the Harish-Chandra isomorphism is the algebra isomorphismThus the centre is identified with the Weyl-invariant polynomials on ${\mathfrak{h}}^{\ast }$. Central characters satisfy ${\theta }_{\chi }={\theta }_{\xi }$ exactly when $\chi$ and $\xi$ belong to the same Weyl orbit. The $\rho$ shift is already incorporated in $M^{\chi }$ having highest weight $\chi -\rho$; without that shift the corresponding Weyl action is the dot action.

The centre commutes with $\mathfrak{h}$, so every nonzero representation weight component of the common eigenvector $m$ has the same central character $\theta$. Pick such a component $m_{\lambda }$. In Category O, its $U({\mathfrak{n}}_{+})$-span is finite-dimensional and has finitely many weights. Choose a maximal representation weight in that span and a nonzero vector $v$ of that representation weight, say $\nu$. Every positive root operator kills $v$, since it would raise the representation weight within the same finite span. Centrality ensures $zv=\theta (z)v$.

The universal property supplies a nonzero map $M^{\nu +\rho }\to M$ taking its highest vector to $v$. The centre acts on that image both by $\theta$ and by ${\theta }_{\nu +\rho }$. Hence

Use the standard defining properties of Category O: finite generation over $U(\mathfrak{g})$, a direct sum of finite-dimensional weight spaces, and locally finite positive root action. In particular the weights lie in a finite union of downward root cones. Submodules and quotients retain the representation weight bounds and finite weight spaces; finite generation of submodules follows from the Poincaré-Birkhoff-Witt theorem and the noetherian property of $U(\mathfrak{g})$.

Choose finitely many weight vector generators. The sum $T$ of their full weight spaces is finite-dimensional and preserved by the centre. The kernel of its central action has finite codimension in $Z(U(\mathfrak{g}))$ and kills the generators, hence, by centrality, all of $M$. Thus only finitely many central characters ${\theta }_1,\dots ,{\theta }_r$ can occur in any subquotient: they are the characters of a fixed finite-dimensional commutative quotient algebra of the centre.

By the Harish-Chandra isomorphism each ${\theta }_i$ determines one finite Weyl orbit of shifted weights. Let $S$ be the finite union of these orbits after subtracting $\rho$. Every nonzero subquotient has a nonzero highest-weight vector, using the bounded-above weights. Within its finite-dimensional highest weight space, the commuting central operators have a common eigenvector. Part (e) and the central-character restriction show that its highest weight belongs to $S$.

Define for any subquotient $N$It is finite, additive in short exact sequences, and positive whenever $N\ne 0$. Hence each strict step of a submodule chain consumes at least one unit of $d(M)$. Refining a nonsimple factor into a proper submodule and quotient cannot continue more than $d(M)$ times. It ends with simple factors, provingThe argument proves the assertion for every object of $\mathcal{O}$, rather than just for a cyclic highest weight object.