[Paper Review] Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and Connections

Daniel K. Nakano|arXiv (Cornell University)|Apr 13, 2014

Advanced Algebra and Geometry62 references3 citations

TL;DR

This paper establishes new connections between the cohomology theories of algebraic groups, finite groups, and Lie algebras—particularly focusing on $G(bF_q)$, $G_r$, and $G$—by leveraging geometric structures like the flag variety and nilpotent orbits. It provides explicit bounds and vanishing results for cohomology groups $\mathrm{H}^\bullet(G(\bbF_q),k)$, showing that for type $A_n$ and $C_n$, the first non-trivial cohomology appears at degrees $2p-3$ and $p-2$, respectively, under specified prime conditions.

ABSTRACT

This paper surveys results on the connections between the cohomology for algebraic groups, finite groups and Frobenius kernels that were presented at the Workshop and Summer School on Lie and Representation Theory at East China Normal University during July 2009.

Motivation & Objective

• To clarify the structural and cohomological relationships between the representation categories of algebraic groups $G$, finite groups $G(\bbF_q)$, and infinitesimal groups $G_r$.
• To address long-standing questions about the cohomology of finite Chevalley groups $G(\bbF_q)$, especially the vanishing and non-vanishing of $\mathrm{H}^i(G(\bbF_q),k)$.
• To develop new methods using geometric tools—such as the flag variety $G/B$ and nilpotent orbit theory—combined with root system combinatorics to compute extensions and cohomology.
• To establish precise upper bounds and explicit vanishing ranges for $\mathrm{H}^\bullet(G(\bbF_q),k)$ via Kostant’s partition functions and spectral sequences.
• To prove that for $G$ of type $C_n$ with $p > 2n$, the first non-trivial cohomology occurs at degree $p-2$, and for type $A_n$, it occurs at $2p-3$ under certain conditions.

Proposed method

• Utilizes the LHS spectral sequence to relate cohomology of $G(\bbF_q)$ to cohomology of $G$ and $G_1$, enabling transfer of information between categories.
• Applies Kostant’s partition function to bound the dimension of $\mathrm{H}^\bullet(G(\bbF_q),k)$ when $r=1$ and $p > h$, the Coxeter number.
• Employs root system combinatorics to derive lower bounds on cohomological degrees by analyzing weight multiplicities in symmetric powers of the dual of the unipotent radical $\mathfrak{u}^*$.
• Uses the Steinberg tensor product theorem and restriction functors to relate simple and projective modules across $G$, $G_r$, and $G(\bbF_q)$.
• Applies the support variety and complexity theory for finite-dimensional cocommutative Hopf algebras to analyze extension behavior in $\mathrm{mod}(G_r)$ and $\mathrm{mod}(G(\bbF_q))$.
• Constructs specific weights $\lambda = p\omega_1 + w\cdot 0$ such that $\mathrm{H}^{p-2}(G, H^0(\lambda) \otimes H^0(\lambda^*)^{(1)}) \neq 0$, proving non-vanishing at critical degrees.

Experimental results

Research questions

• RQ1 What is the precise vanishing range for $\mathrm{H}^i(G(\bbF_q),k)$ when $G$ is of type $C_n$ and $p > 2n$?
• RQ2 For type $A_n$, at what degree does the first non-trivial cohomology $\mathrm{H}^i(G(\bbF_p),k)$ first appear, and how does this depend on $p$ and $n$?
• RQ3 Can the cohomology of $G(\bbF_q)$ be bounded using Kostant’s partition function and root system data?
• RQ4 Is there a uniform bound on the dimension of $\mathrm{H}^\bullet(G(\bbF_q),k)$ for adjoint-type groups?
• RQ5 What is the relationship between the cohomology of $G(\bbF_q)$ and the cohomology of $G$ via the LHS spectral sequence?

Key findings

• For $G$ of type $C_n$ with $p > 2n$, $\mathrm{H}^i(G(\bbF_q),k) = 0$ for $0 < i < r(p-2)$, and $\mathrm{H}^{p-2}(G(\bbF_p),k) \cong k$.
• For type $A_n$ with $n \geq 2$ and $p > n+1$, the first non-trivial cohomology appears at degree $2p-3$ when $p > n+2$ and $n > 3$, with $\mathrm{H}^{2p-3}(G(\bbF_p),k) \cong k$.
• When $p = n+2$ and $n \geq 2$, $\mathrm{H}^{p-2}(G(\bbF_p),k) \cong k \oplus k$, indicating a non-trivial self-extension.
• For $n=2$ and $3 \mid (p-1)$, $\mathrm{H}^{2p-6}(G(\bbF_p),k) \cong k \oplus k$, while for $3 \nmid (p-1)$, the first non-trivial cohomology is at $2p-3$ with $\mathrm{H}^{2p-3}(G(\bbF_p),k) \cong k$.
• For $n=3$ and $p > 5$, $\mathrm{H}^{2p-6}(G(\bbF_p),k) \cong k$, showing a consistent pattern in the vanishing range.
• The weight $\lambda = (p-2n)\omega_1$ with $w$ a specific Weyl group element satisfies $\mathrm{H}^{p-2}(G, H^0(\lambda) \otimes H^0(\lambda^*)^{(1)}) \neq 0$, proving non-vanishing at the predicted degree.