Paths and root operators in representation theory. (English) Zbl 0858.17023

Littelmann, Peter

Ann. Math. (2) 142, No. 3, 499-525 (1995).

Let \(X\) be the weight lattice of a complex symmetrizable Kac-Moody algebra \({\mathfrak g}\), and \(\pi\) the set of all piecewise linear paths \(\pi: [0,1] \to X_\mathbb{Q}\) starting at 0. For a simple root \(\alpha\), the author defines linear operators \(e_\alpha\) and \(f_\alpha\) on the \(\mathbb{Z}\)-module \(\mathbb{Z}\pi\) spanned by \(\pi\). Let \(A\subset \text{End}_\mathbb{Z} \mathbb{Z} \pi\) be the subalgebra generated by these operators. Let \(P^+\) be the set of all paths \(\pi\) such that the image is contained in the dominant Weyl chamber and \(\pi(1) \in X\). Let \(\pi \in P^+\) and \(M_\pi\) be the \(A\)-module \(A\pi\) and \(B_\pi\) the set of paths contained in \(M_\pi\). Let \(\pi(1) = \lambda\), and \(V(\lambda)\) the irreducible integrable \({\mathfrak g}\)-module with highest weight \(\lambda\). The author shows that \(\text{char} V(\lambda) = \sum_{\eta \in B_\pi} e^{\eta(1)}\). Denoting by \(*\) the “concatariation” of paths, the author shows that \(M_{\pi_1}* M_{\pi_2} = \oplus_\pi M_\pi\), where \(\pi_1, \pi_2 \in P^+\) and \(\pi\) runs over all paths in \(P^+\) of the form \(\pi = \pi_1* \eta\), for some \(\eta \in B_{\pi_2}\). For a Levi subalgebra \({\mathfrak p}\) of \({\mathfrak g}\), denoting by \(A_{\mathfrak p}\) the subalgebra generated by those \(e_\alpha\), \(f_\alpha\), such that \(\alpha\) is a simple root of \({\mathfrak p}\), the author shows that \(M_\pi = \oplus_\eta N_\eta\) (as \(A_{\mathfrak p}\)-modules) where \(\pi \in P^+\), and \(\eta\) runs over all paths in \(B_\pi\) which are contained in \(P^+_{\mathfrak p}\). (Here \(P^+_{\mathfrak p}\) denotes the set of paths contained in the Weyl chamber corresponding to \({\mathfrak p}\), and for \(\eta \in P^+_{\mathfrak p}\), \(N_\eta\) denotes the \(A_{\mathfrak p}\)-module \(A_{\mathfrak p} \eta\).
As an application the author obtains a tensor product decomposition rule and restriction rule for Kac-Moody algebras.
This paper makes an important contribution to representation theory.

Reviewer: V.Lakshmibai (Boston)

MSC:

17B67

Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Keywords:

root operators; character formula; symmetrizable Kac-Moody algebra; piecewise linear paths; tensor product decomposition rule; restriction rule