dominant weight

The weight space of a representation \(V\) of \({\mathfrak{g}}\) with weight \(\lambda\) is the subspace \(V_{\lambda}\) given by \begin{align*} V_{\lambda}:=\{v \in V{~\mathrel{\Big\vert}~}\forall H \in \mathfrak{h}, \quad H \cdot v=\lambda(H) v\} . \end{align*} where \(\lambda: {\mathfrak{h}} {}^{ \vee }\to {\mathbf{C}}\).

A weight is a \(\lambda\) as above such that \(V_\lambda \neq 0\); elements of \(V_\lambda\) are weight vectors, which are simultaneous eigenvectors for the action \({\mathfrak{h}}\curvearrowright V\).

If \(V\) is the adjoint representation of \({\mathfrak{g}}\), then the weights are roots and form a root system.

A weight is integral iff \begin{align*} \left\langle\lambda, H_{\alpha}\right\rangle=2 \frac{\langle\lambda, \alpha\rangle}{\langle\alpha, \alpha\rangle} \in \mathbf{Z} \end{align*} for every coroot \(H_{\alpha}=2 \frac{\alpha}{\langle\alpha, \alpha\rangle}\) associated to a root \(\alpha\).

Recall that a positive root system is a subset \(\Phi^+ \subseteq \Phi\) which is closed under sums such that for \(\alpha\in \Phi\), exactly one of \(\pm\alpha \in \Phi^+\).

An integral weight is a dominant weight iff \({\left\langle {\lambda},~{\gamma} \right\rangle} \geq 0\) for every positive root \(\gamma \in \Phi^+\).

#why-care Dominant integral weights parametrize irreducible finite dimensional representations.

From Humphreys

Write \(\Delta=\left\{\alpha_{1}, \ldots, \alpha_{\ell}\right\}\) for a simple system Then \(\Lambda\) is a free abelian group of rank \(\ell\), with a basis consisting of fundamental weights \begin{align*}\varpi_{1}, \ldots, \varpi_{\ell} \quad \text{ satisfying}\quad \left\langle\varpi_{i}, \alpha_{j}^{\vee}\right\rangle=\delta_{i j} .\end{align*} The subset \(\Lambda^{+}:=\mathbb{Z}^{+} \varpi_{1}+\cdots+\mathbb{Z}^{+} \varpi_{\ell}\) is called the set of dominant integral weights. From the fact that \(\left\langle\beta, \alpha^{\vee}\right\rangle=\beta\left(h_{\alpha}\right)\) when \(\beta \in \Phi\), one shows easily that \(\left\langle\lambda, \alpha^{\vee}\right\rangle=\lambda\left(h_{\alpha}\right)\) for all \(\lambda \in \Lambda\).

There is a special weight \begin{align*} \rho \coloneqq\varpi_{1}+\cdots+\varpi_{\ell} \in \Lambda^{+} = {1\over 2}\sum_{\alpha\in \Phi^+} \alpha ,\end{align*} i.e. the sum of fundamental weights or the half-sum of positive roots. It satisfies \begin{align*} \left\langle\rho, \alpha^{\vee}\right\rangle=1 \qquad\text{ and }\qquad s_{\alpha} \rho=\rho-\alpha \qquad \forall \alpha\in \Delta .\end{align*} It is the smallest regular dominant weight fixed by no nontrivial element of \(W\), and the associated line bundle on the flag variety \(G/B\) is ample, and is in fact a square root of the canonical bundle.

Examples

For \({\mathfrak{sl}}_3({\mathbf{C}})\):

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