Verma Modules

Verma modules are universal highest weight modules that provide a systematic approach to constructing and studying irreducible representations.

DefinitionVerma Module

For a dominant integral weight $\lambda \in \mathfrak{h}^*$ , the Verma module $M(\lambda)$ is the induced module: $M(\lambda) = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$ where $\mathbb{C}_\lambda$ is the one-dimensional $\mathfrak{b}$ -module on which $\mathfrak{h}$ acts by $\lambda$ and $\mathfrak{n}$ acts trivially.

Equivalently, $M(\lambda) = U(\mathfrak{n}^-) \otimes \mathbb{C}_\lambda$ where $\mathfrak{n}^-$ is the negative nilradical.

TheoremUniversal Property

$M(\lambda)$ has a unique (up to scalar) highest weight vector $v_\lambda$ of weight $\lambda$ . Moreover, $M(\lambda)$ is universal among highest weight modules:

For any highest weight module $V$ with highest weight $\lambda$ and highest weight vector $v$ , there exists a unique surjective homomorphism $\phi: M(\lambda) \to V$ with $\phi(v_\lambda) = v$ .

ExampleVerma Module for $\mathfrak{sl}_2$

For $\mathfrak{sl}_2$ with highest weight $\lambda = n$ , the Verma module $M(n)$ has basis: $\{v_n, F \cdot v_n, F^2 \cdot v_n, F^3 \cdot v_n, \ldots\}$ where $v_n$ is the highest weight vector with $H \cdot v_n = n \cdot v_n$ and $E \cdot v_n = 0$ .

For non-negative integer $n$ , the element $F^{n+1} \cdot v_n$ generates a proper submodule, and the irreducible quotient $L(n) = M(n) / \langle F^{n+1} \cdot v_n \rangle$ has dimension $n+1$ .

TheoremStructure of Verma Modules

$M(\lambda)$ has a unique maximal proper submodule $N(\lambda)$
The quotient $L(\lambda) = M(\lambda) / N(\lambda)$ is the unique finite-dimensional irreducible representation with highest weight $\lambda$
Every finite-dimensional irreducible arises this way

The submodule $N(\lambda)$ is generated by singular vectors (weight vectors killed by positive root spaces but not highest weight).

Remark

Verma modules are crucial for:

Existence: Prove that irreducible representations exist for all dominant integral weights
Characters: The Weyl character formula can be derived via Verma module resolutions
BGG category $\mathcal{O}$ : Verma modules form building blocks for a rich categorical structure
Kazhdan-Lusztig theory: Multiplicities in composition series connect to intersection cohomology

Though usually infinite-dimensional, Verma modules provide a systematic framework for studying finite-dimensional representations through universal properties and exact sequences.