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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.
This work extends earlier results by several mathematicians including Sanderson, Kuniba-Misra-Okado-Takagi-Uchiyama, Magyar, Hatayama-Kuniba-Okado-Takagi-Tsuboi, Hatayama-Kirillov-Kuniba-Okado-Takagi-Yamada, Kashiwara, Yamane, Kang-Kashiwara-Kuniba-Misra.

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
22E46 Semisimple Lie groups and their representations
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI
[1] Graduate Studies in Mathematics 42 (2002)
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