论文标题
抛物线式Hecke subselgebras的Freeness和Trace猜想
The freeness and trace conjectures for parabolic Hecke subalgebras
论文作者
论文摘要
The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace.第一个猜想最近已成为一个定理，而第二个猜想（已知是真正的反思组的猜想）仅被证明是针对某些非现实的非现实复合体反射组（所有排名$ 2 $，但一个）。关于抛物线寄生虫hecke submalgebra $ \ Mathcal {h}（h}（w'）$的两个最基本的猜想，与抛物线子组$ w $ of $ w $相关联$ \ MATHCAL {H}（w'）$的规范对称跟踪是$ \ MATHCAL {H}（w）$ to $ \ MATHCAL {h h}（h}（w'）$的规范对称跟踪的限制。到目前为止，这两个猜想仅在实际反射组中是正确的。我们证明了所有复杂反思组的排名$ 2 $，而BMM对称痕量猜想的持有。
The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic Hecke subalgebra $\mathcal{H}(W')$ associated with a parabolic subgroup $W'$ of $W$ state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W')$-module of rank $|W|/|W'|$, and that the canonical symmetrising trace of $\mathcal{H}(W')$ is the restriction of the canonical symmetrising trace of $\mathcal{H}(W)$ to $\mathcal{H}(W')$. Until now, these two conjectures have only be known to be true for real reflection groups. We prove them for all complex reflection groups of rank $2$ for which the BMM symmetrising trace conjecture is known to hold.
