Cohomological growth rates and Kazhdan-Lusztig polynomials

In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_\zeta$ (at roots of unity $\zeta$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $\zeta$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $\Phi$. For quantum groups $U_\zeta$ with a fixed $\Phi$, we show the sequence $\{\max_{L\,{\text{irred}}}\dim \opH^n(U_\zeta,L)\}_n$ has polynomial growth independent of $\zeta$. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for $\Ext^n_{U_\zeta}$ are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that $\{\log\max_{L\,\text{\rm irred}}\dim\opH^n(G,L)\}$ has polynomial growth $\leq 3$ for any fixed prime $p$ (and $\leq 4$ if $p$ is allowed to vary with $n$). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov, and Lubotzky \cite{GKKL}.