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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:
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