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On Ext1 for semisimple groups and infinitesimal subgroups

Published online by Cambridge University Press:  24 October 2008

Stephen Donkin
Stephen Donkin
Affiliation:
King's College, Cambridge
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Extract

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphism

is the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequence

is exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 2 , September 1982 , pp. 231 - 238
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

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