基于零锥几何的量子群的上同调
Cohomology for quantum groups via the geometry of the nullcone
摘要：
令$ζ$是奇数$\ell>1$的复数$\ell$的单位根.对于任何复单李代数$\mathfrak g$，令$u_ζ=u_ζ({\mathfrak g})$是相关的“小”量子包络代数。一般来说，对量子群的表示理论知之甚少（分别是，代数群）当$l$（分别，$p$）小于底层根系的Coxeter数$h$。例如，Lusztig关于有理不可约$G$ -模的特征标的猜想规定：$p \geq h$。本文的主要结果为小量子群的上同调代数$\opH^\bullet(u_ζ,{\mathbb C})$提供了一个令人惊讶的统一答案。当$\ell>h$，这个上同调代数已计算金斯堡和库马尔\cite{GK}。我们的结果需要强大的工具，从复杂的几何和详细的知识的零锥的$\mathfrak g$的几何。通过这种方式，该方法指出了在小特征的限制包络代数$u$中获得类似结果的困难，尽管它们也提供了一些已知结果的澄清。最后，我们证明了若$M$是有限维$u_ζ$ -模，则$\opH^\bullet(u_ζ,M)$是有限维生成的$\opH^\bullet(u_ζ,\mathbb C)$ -模，并在$u_ζ$的支撑簇理论上得到了新的结果.
英文摘要：
Let $ζ$ be a complex $\ell$th root of unity for an odd integer $\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let $u_ζ=u_ζ({\mathfrak g})$ be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\opH^\bullet(u_ζ,{\mathbb C})$ of the small quantum group. When $\ell>h$, this cohomology algebra has been calculated by Ginzburg and Kumar \cite{GK}. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of $\mathfrak g$. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra $u$ in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if $M$ is a finite dimensional $u_ζ$-module, then $\opH^\bullet(u_ζ,M)$ is a finitely generated $\opH^\bullet(u_ζ,\mathbb C)$-module, and we obtain new results on the theory of support varieties for $u_ζ$.
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