Questions tagged [verma-modules]

Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight $ \lambda $, where $ \lambda $ is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

Let $\mathfrak {g}$ be a semisimple Lie algebra (over $ \mathbb {C} $, for simplicity). Let $\mathfrak {h}$ be a fixed Cartan subalgebra of $\mathfrak {g}$ and let $R$ be the associated root system. Let $R^{+}$ be a fixed set of positive roots. For each $\alpha \in R^{+}$, choose a nonzero element $X_{\alpha }$ for the corresponding root space $ \mathfrak {g} _ \alpha $ and a nonzero element $Y_{\alpha }$ in the root space $ \mathfrak {g} _ {-\alpha }$. We think of the $X_{\alpha }$'s as "raising operators" and the $Y_{\alpha }$'s as "lowering operators."

Now let $\lambda \in {\mathfrak {h}}^{*}$ be an arbitrary linear functional, not necessarily dominant or integral. Our goal is to construct a representation $W_{\lambda }$ of $\mathfrak {g}$ with highest weight $\lambda$ that is generated by a single nonzero vector $v$ with weight $\lambda$. The Verma module is one particular such highest-weight module, one that is maximal in the sense that every other highest-weight module with highest weight $\lambda$ is a quotient of the Verma module. It will turn out that Verma modules are always infinite dimensional; if $\lambda$ is dominant integral, however, one can construct a finite-dimensional quotient module of the Verma module. Thus, Verma modules play an important role in the classification of finite-dimensional representations of $\mathfrak {g}$. Specifically, they are an important tool in the hard part of the theorem of the highest weight, namely showing that every dominant integral element actually arises as the highest weight of a finite-dimensional irreducible representation of $\mathfrak {g}$.

Source: Wikipedia