A new class of unitary representations for affine Lie algebras
The author is concerned with constructing irreducible representations of affine Lie algebras which are unitarizable with respect to any form coming from a finite-dimensional Lie algebra. Let \(L(\mathfrak g) = \mathfrak g\otimes_{\mathbb C}\mathbb C[t,t^{-1}]\) be a loop algebra of a finite-dimensional simple Lie algebra \(\mathfrak g\) and \(\overline{L}(\mathfrak g)\) be a semi-direct product of \(L(\mathfrak g)\) and a derivation \(d\) defined by \([d, X\otimes t^ n]=n\,X\otimes t^ n\). Fix \(k\geq 1\) and let \(\mathfrak g_ k=\oplus_{k}\mathfrak g\) be a direct sum of \(k\)-copies of \(\mathfrak g\). Let \(\mathfrak a=(a_1,\ldots,a_k)\) be a sequence of distinct nonzero complex numbers. Define the Lie algebra homomorphism \(\Phi(\mathfrak a): L(\mathfrak g)\to\mathfrak g_ k\) by \(\Phi(\mathfrak a)(X\otimes t^n) = (a_1^nX,\ldots,a^n_kX)\), \(X\in\mathfrak g\), \(n\in\mathbb Z\). Let \((V,\tau)\) be a \(\mathfrak g_k\)-module and define an \(\overline{L}(\mathfrak g)\)-module \((L(V),\Pi(\mathfrak a))\) by \[ \Pi (\mathfrak a)(X\otimes t^n)(v\otimes t^m) = (\tau (\Phi (\mathfrak a)(X\otimes t^n))v)\otimes t^{n+m},\quad \Pi (\mathfrak a) dv\otimes t^m = mv\otimes t^ m \] for all \(X\in\mathfrak g\), \(v\in V\) and for all \(m,n\in\mathbb Z\). The author investigates the \(\overline{L}(\mathfrak g)\)-module \(L(V)\). Fix a Cartan subalgebra \(\mathfrak h\subset\mathfrak g\) and let \(\mathfrak h_k=\oplus_k \mathfrak h\). The author shows that if \((V,\tau)\) is an irreducible \(\mathfrak g_k\)-module which is also an \(\mathfrak h_k\)-weight module, then \((L(V),\Pi(\mathfrak a))\) is a completely reducible \(\overline{L}(\mathfrak g)\)-module, if further \(\mathfrak a=(a_1,\ldots,a_k)\) is chosen such that \((a_i/a_j)^{k!}\neq 1\) for \(i\neq j\), then \((L(V),\Pi(\mathfrak a))\) is an irreducible \(\overline{L}(\mathfrak g)\)-module and he gives a counter-example to show that it is not always irreducible. Further, he shows that \(L(V)\) is unitarizable with respect to a form of \(L(\mathfrak g)\) induced from any form of \(\mathfrak g\) iff the \(\mathfrak g_ k\)-module is unitarizable with respect to the corresponding form on \(\mathfrak g_k\) and \(| a_ i| =| a_ j|\) for all \(i,j\). In the special case where \(V\) is a finite-dimensional \(\mathfrak g_k\)-module, this was due to Chari and Pressley.
scientific article; zbMATH DE number 4081783