标题：Simple $\sl_{d+1}$-modules from Witt algebra modules
报告人：郭向前（广州大学）
时间：2024年11月8日 15:20-16:20
地点：36-507
摘要：Let $d\ge1$ be an integer, $\W_d$ be the Witt algebra. For any admissible $\W_d$-module $P$ and any $\gl_d$-module $V$, one can form a $\W_d$-module $\F(P,V)$, which as a vector space is $P\ot V$.
Since $\W_d$ has a natural subalgebra isomorphic to $\sl_{d+1}$, we can view $\F(P,V)$ as an $\sl_{d+1}$-module. Taking $P=\Omega(\bl)$, the rank-$1$ $U(\mh)$-free $\W_d$-module and $V=V(\ba,b)$, the irreducible cuspidal module over $\gl_d$, we get the special $\sl_{d+1}$-module $\F(\bl;\ba,b)=\F(\Omega(\bl),V(\ba,b))$, which are $U(\fh)$-free modules of infinite rank. We determine the necessary and sufficient condition for the $\sl_{d+1}$-module $\F(\bl;\ba,b)$ to be irreducible and for the reducible case, we construct their proper submodules explicitly. At last, using the above results, we deduce an explicit irreducibility criterion for the generalized Verma modules induced from $V(\ba,b)$ and obtain a family of irreducible affine modules from $\F(\bl;\ba,b)$, which can be viewed as the nonweight version of loop modules.
报告人介绍：郭向前，2007年毕业于中科院数学所，获理学博士学位，研究方向为李理论及其应用。现为广州大学教授，博士生导师，美国《数学评论》评论员，德国《数学文摘》评论员。在Trans. Amer. Math. Soc.，J. Lond. Math. Soc.， Proc. Edinb. Math. Soc.， Doc. Math.， Commun. Contemp. Math., Israel J. Math., J. Algebra等杂志发表高水平论文四十余篇SCI刊物发表学术论文50余篇，主持完成国家自然科学基金面上项目2项，在研1项。
