论文标题
dunkl总角动量代数的中心
The centre of the Dunkl total angular momentum algebra
论文作者
论文摘要
对于有限反射组的有限维表示$ v $，我们认为我们认为有理由Cherednik代数$ \ Mathsf {h} _ {t，c}（v，w）（v，w）$与$（v，w）$相关的$（v，w）$在parameters $ t \ neq 0 $ t \ neq 0 $和$ c $中。 dunkl总角动量代数$ o_ {t，c}（v，w）$作为lie superalgebra $ \ mathfrak {osp}（1 | 2）$包含dirac操作员的dunkl变形，内部$ \ mathsf { Clifford代数由$ V $产生。 我们表明，对于参数$ c $的每个值，$ o_ {t，c}（v，w）$的中心与单变量多项式环是同构。值得注意的是，中心的生成器会根据$（-1）_v $是组$ W $的一个元素而更改。使用此中心的描述，并使用伪标量表的投影从Clifford代数到$ o_ {t，c}（v，w，w）$，我们建立了类似于``vogan的猜想''的结果。
For a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum algebra $O_{t,c}(V,W)$ arises as the centraliser algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ containing a Dunkl deformation of the Dirac operator, inside the tensor product of $\mathsf{H}_{t,c}(V,W)$ and the Clifford algebra generated by $V$. We show that, for every value of the parameter $c$, the centre of $O_{t,c}(V,W)$ is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not $(-1)_V$ is an element of the group $W$. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into $O_{t,c}(V,W)$, we establish results analogous to ``Vogan's conjecture'' for a family of operators depending on suitable elements of the double cover $\tilde{W}$.
