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New unitary representations of loop groups. (English) Zbl 0603.17012
Let \({\mathfrak g}\) be a finite dimensional complex simple Lie algebra with a Cartan subalgebra \({\mathfrak h}\) and a fixed positive root-system for \(({\mathfrak g,h})\). Let \(\bar L({\mathfrak g})\) denote the loop algebra \(L({\mathfrak g})=L\otimes {\mathfrak g}\) \((L={\mathbb C}[t,t^{-1}])\) together with a derivation \(d\). Now fix \(k>0\); a sequence \(a=(a_ 1,...,a_ k)\) of distinct complex numbers; and representations \(W_ 1,...,W_ k\) of \({\mathfrak g}\). Then \(L(W_ 1\otimes...\otimes W_ k)\) can be given a ’natural’ structure of \(\bar L({\mathfrak g})\)-module as follows: \[ (f\cdot \Omega)(z)=(\sum^{k}_{i=1}1\otimes... \otimes f(a_ iz)\otimes... \otimes 1)\Omega (z),\quad (d\Omega)(z)=z d\Omega /dz, \] for all \(f\in L({\mathfrak g})\), \(\Omega\in L(\otimes^{k}_{i=1}W_ i)\) and \(z\in {\mathbb C}^*\). If we take now for \(W_ 1,...,W_ k\) the irreducible representations \(V(\lambda_ 1),...,V(\lambda_ k)\) of highest weights \((\lambda_ 1,...,\lambda_ k)\) resp., then \(L(W_ 1\otimes... \otimes W_ k)\) with the above module structure is denoted by \(V(\lambda,a)\) and is called the ‘loop module’. Further, to any pair \((\lambda,a)\) as above, the authors associate a character \(\chi_{(\lambda,a)}: \cup (L(h)\to L\) and show that the image of \(\chi_{(\lambda,a)}\) is a Laurent subring \({\mathbb C}[t^ r,t^{-r}]\), for some \(r\geq 0.\)
The aim of the paper under review is to give an explicit realization of irreducible modules in the category \({\mathcal I}_{\text{fin}}\) studied earlier by the first author [Invent. Math. 85, 317–335 (1986; Zbl 0603.17011)]. More specifically some of the principal results are:
(a) \(V(\lambda,a)\) is the direct sum of \(r\) irreducible representations of \(\bar L({\mathfrak g})\), which are all mutually \(L({\mathfrak g})\)-isomorphic (but not \(\bar L({\mathfrak g})\)-isomorphic).
(b) Assume further that all the \(\lambda_ i\) are dominant integral. Then \((b_ 1)\) \(V(\lambda,a)\) is an integrable \(\bar L({\mathfrak g})\)- module. \((b_ 2)\) \(V(\lambda,a)\) is unitarizable with respect to the standard ’compact’ real form of \(\bar L({\mathfrak g})\) iff \(| a_ i| =| a_ j|\), for all \(i\) and \(j\).
For any fixed \(b\in {\mathbb C}\), one can twist the action of d on \(V(\lambda,a)\) by defining \((d\Omega)(z)=b\Omega(z)+z(d\Omega/dz)\). Denote \(V(\lambda,a)\) with the twisted \(d\)-action by \(V(\lambda,a,b)\). Now any integrable module \(V(\lambda,I)\) (notation as in the first author’s paper) is isomorphic with an irreducible piece of \(V(\lambda,a,b)\), for suitable choices of \(\lambda\), \(a\), and \(b\).
Some of the results of this paper have recently been extended by S. Eswara Rao [J. Algebra 275, No. 1, 59–74 (2004; Zbl 1138.17009)].
Reviewer: S. Kumar

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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References:
[1] Chari, V.: Integrable representations of affine Lie algebras. Invent. Math.85, 317-335 (1986) · Zbl 0603.17011 · doi:10.1007/BF01389093
[2] Garland, H.: The arithmetic theory of loop algebras. J. Algebra53, 480-551 (1978) · Zbl 0383.17012 · doi:10.1016/0021-8693(78)90294-6
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