Document Zbl 1234.20056 - zbMATH Open

From the authors’ summary: “Given a finite root system \(\Phi\), we show that there is an integer \(c=c(\Phi)\) such that \(\dim\text{Ext}^1_G(L,L')<c\), for any reductive algebraic group \(G\) with root system \(\Phi\) and any irreducible rational \(G\)-modules \(L,L'\). There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for \(\text{Ext}^n\), for any integer \(n\geqslant 0\), using a constant depending only on \(n\) and the root system. When \(L\) is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of \(L=L(\lambda)\), even allowing \(\lambda\) to vary, provided the \(p\)-adic expansion of \(\lambda\) is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan-Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin’s tilting module conjecture.”
That is, the appendix shows that a projective principal indecomposable module of a Frobenius kernel is the restriction of a tilting module often enough for the purpose of this paper. The bound \(c=c(\Phi)\) seems in principle effective. It is a combination of two bounds. One bound depends on \(p\), the other depends on the Lusztig character formula and thus applies only for \(p\) large enough.

MSC:

20G10	Cohomology theory for linear algebraic groups
20G40	Linear algebraic groups over finite fields
20C33	Representations of finite groups of Lie type
20G15	Linear algebraic groups over arbitrary fields
20G42	Quantum groups (quantized function algebras) and their representations

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