Filtrations of the discrete series of \(SL_ 2(q)\) via crystalline cohomology. (English) Zbl 0724.20030
Given a power \(q\) of \(p\), let \(k\) be a sufficiently large field of characteristic \(p\). One defines a certain plane projective curve \(C\) over \({\mathbb{F}}_ p\), left invariant by the action of \(SL_ 2(q)\). Then \(H^ 1_{cris}(C/k)\) is an \(SL_ 2(q)\)-module. The inverse images of the powers of the Frobenius morphism define a filtration on \(H^ 1_{cris}(C)\) by \(SL_ 2(q)\)-submodules. One compares this filtration mod \(p\) with a similar filtration of the Weyl modules \(V_ j=S^ j((k^ 2)^*)^*\) [cf. J. E. Humphreys, Proc. Symp., Univ. Durham 1978, 259-288 (1980; Zbl 0472.20015) and J. C. Jantzen, ibid., 291-300 (1980; Zbl 0472.20016)].
Reviewer: J.H.de Boer (Nijmegen)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 20G05 | Representation theory for linear algebraic groups |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 20G40 | Linear algebraic groups over finite fields |
| 20C33 | Representations of finite groups of Lie type |
Keywords:
crystalline cohomology; representation theory; algebraic groups of Lie type; plane projective curves; Frobenius morphism; filtrations; Weyl modulesReferences:
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