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Tensor product structure of affine Demazure modules and limit constructions. (English) Zbl 1143.22010
Let $$\mathfrak g$$ be a simple Lie algebra over $$\mathbb C$$ and let $$\widehat{\mathfrak g}$$ be the associated affine Kac-Moody Lie algebra. Fix a Borel subalgebra $$\mathfrak b\subset\mathfrak g$$ and a Cartan subalgebra $$\mathfrak h\subset\mathfrak b$$. Any coweight $$\lambda^\vee\in\mathfrak h$$ of $$\mathfrak g$$ can be thought of as an element of the extended affine Weyl group $$\widetilde W^{\text{aff}}$$ and hence, for any dominant integral weight $$\Lambda$$ of $$\widehat{\mathfrak g}$$, we have the associated Demazure module $$V_{\lambda^\vee}(\Lambda)$$ (for the Lie algebra $$\widehat{\mathfrak g}$$). In the paper under review, the authors study the Demazure modules $$V_{-\lambda^\vee}(m\Lambda_0)$$ for the case when $$\lambda^\vee$$ is a dominant coweight and $$m$$ is any positive integer, where $$\Lambda_0$$ denotes the zeroeth fundamental weight of $$\widehat{\mathfrak g}$$. In this case, $$V_{-\lambda^\vee}(m\Lambda_0)$$ is a module for the standard maximal parabolic subalgebra of $$\widehat{\mathfrak g}$$. Viewed only as a $${\mathfrak g}$$-module, it is denoted by $$\overline V_{-\lambda^\vee}(m\Lambda_0)$$.
Let $$\lambda^\vee= \lambda^\vee_1+\cdots+ \lambda^\vee_\gamma$$ be any decomposition of the dominant $$\lambda^\vee$$ as a sum of dominant coweights $$\lambda^\vee_i$$. The authors’ first result asserts that
$\overline V_{-\lambda^\vee}(m\Lambda_0)\simeq\overline V_{-\lambda^\vee_1}(m\Lambda_0)\otimes\cdots\otimes\overline V_{-\lambda^\vee_r}(m\Lambda_0).$ Thus, the study of $$\overline V_{-\lambda^\vee}(m\Lambda_0)$$ for any dominant coweight $$\lambda^\vee$$ reduces to that of the fundamental coweights $$\omega^\vee_i$$. They determine the $${\mathfrak g}$$-module $$\overline V_{-\omega^\vee_i}(m\Lambda_0)$$ explicitly for any fundamental coweight $$\omega^\vee_i$$ in any classical $${\mathfrak g}$$ (i.e., $${\mathfrak g}$$ of type $$A_\ell$$, $$B_\ell$$, $$C_\ell$$, $$D_\ell$$). For the exceptional $${\mathfrak g}$$, the authors determine $$V_{-\omega^\vee_i}(m\Lambda_0)$$ for some of the $$\omega^\vee_i$$’s.
By comparing the known decomposition of the Kirillov-Reshetikhin modules due to Chari and the Kyoto school with the $${\mathfrak g}$$-module decomposition of $$V_{-\omega^\vee_i}(m\Lambda_0)$$ mentioned above, the authors of the paper under review show that in all the above cases the quantized version of the Demazure modules $$V_{-\omega^\vee_i}(m\Lambda_0)$$ are the Kirillov-Reshetikhin modules.
Finally, for any dominant integral weight $$\Lambda$$ of $$\widehat{\mathfrak g}$$, writing $$\Lambda= r\Lambda_0+\lambda$$ (for some dominant weight $$\lambda$$ of $${\mathfrak g}$$), they give a direct-limit construction of the integrable highest weight $$\widehat{\mathfrak g}$$-module $$V(\Lambda)$$ as a limit of the $${\mathfrak g}$$-modules
$\text{lt}_{m\to\infty}(\overline V_{-\theta^\vee}(r\Lambda_0))^{\otimes m}\otimes V(\lambda),$ where $$\theta$$ is the highest root of $${\mathfrak g}$$ and $$V(\lambda)$$ is the irreducible $${\mathfrak g}$$-module with highest weight $$\lambda$$.
The authors further extend these results to the twisted affine case.