×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Graduate Studies in Mathematics 42 (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.