关于$(C_1^\breve,C_1)$型nil-DAHA的有限维无重不可约模

中文摘要 AI 辅助

固定非零复数$r_0,r_1$，令$\breve{\boldsymbol{\textit{H}}}$表示由生成元$t_0,u_0,t_1,u_1$及关系$(t_i-r_i)(t_i-r_i^{-1})=0$（$i=0,1$）、$u_0^2=u_0$、$u_1^2=0$、$u_0t_0t_1u_1=0=t_1u_1u_0t_0$定义的$(C_1^\breve,C_1)$型nil-DAHA。设$A=u_0t_0$，$B=t_1u_1$，若有限维$\breve{\boldsymbol{\textit{H}}}$模中$A$和$B$可同时对角化且所有非零公共特征空间均为一维，则称该模为$(A,B)$无重，简称无重。本文研究具有特定有序基（称为适配块基）的有限维不可约无重模：对$D\boldsymbol{\text{≥}}1$，构造$2D$维$\breve{\boldsymbol{\textit{H}}}$模族$E_D$；对$D\boldsymbol{\text{≥}}0$，构造$(2D+1)$维$\breve{\boldsymbol{\textit{H}}}$模族$O_D$，确定这两类族中哪些成员是无重且不可约的，证明所有具有适配块基的有限维不可约无重$\breve{\boldsymbol{\textit{H}}}$模均同构于$E_D$或$O_D$型模，并确定同一家族中两个成员同构的条件。

英文摘要

Fix nonzero $r_0,r_1\in\mathbb{C}$. Let $\widetilde{\mathcal H}$ denote a nil-DAHA of type $(C_1^\vee,C_1)$ defined by generators $t_0,u_0,t_1,u_1$ and relations $(t_i-r_i)(t_i-r_i^{-1})=0$ for $i\in\{0,1\}$, $u_0^2=u_0$, $u_1^2=0$, and $u_0t_0t_1u_1=0=t_1u_1u_0t_0$. Set $A=u_0t_0$ and $B=t_1u_1$. A finite-dimensional $\widetilde{\mathcal H}$-module is called $(A,B)$-multiplicity-free, or simply multiplicity-free, if $A$ and $B$ are simultaneously diagonalizable and every nonzero common eigenspace is one-dimensional. We consider finite-dimensional irreducible multiplicity-free modules that have a certain ordered basis, which we call an adapted block basis. For $D\geq 1$, we construct a family of $2D$-dimensional $\widetilde{\mathcal H}$-modules $E_D$, and for $D\geq 0$, we construct a family of $(2D+1)$-dimensional $\widetilde{\mathcal H}$-modules $O_D$. We determine which members of these families are multiplicity-free and irreducible. We prove that every finite-dimensional irreducible $\widetilde{\mathcal H}$-module that is multiplicity-free and has an adapted block basis is isomorphic to a module of the form $E_D$ or $O_D$. We also determine when two members of the same family are isomorphic.

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19 pages