Isocrystals with additional structure. (English) Zbl 0597.20038

Kottwitz, Robert E.

Compos. Math. 56, 201-220 (1985).

From the author’s introduction: ”Let k be an algebraically closed field of characteristic \(p>0,\) and let K be the fraction field of the Witt ring W(k). The Frobenius automorphism of k induces an automorphism \(\sigma\) of K. An isocrystal is a finite dimensional vector space V over K together with a \(\sigma\)-semilinear bijection \(\Phi: V\to V.\) Let V be an n- dimensional vector space over \({\mathbb{Q}}_ p\) and let \(G=GL(V)\). For any element \(b\in G(K)\) we get an isocrystal \((V_ K,\Phi)\), where \(V_ K=V\otimes_{{\mathbb{Q}}_ p}K\) and \(\Phi =b\circ (id_ V\otimes \sigma)\). If b’ is \(\sigma\)-conjugate to b (in other words, if \(b'=gb\sigma (g)^{-1}\) for some \(g\in G(K))\), then the two isocrystals we get are isomorphic. This construction yields a bijection from the set of \(\sigma\)-conjugacy classes in G(K) to the set of isomorphism classes of n-dimensional isocrystals.”
The paper under review studies the set B(G) of \(\sigma\)-conjugacy classes in G(K) for arbitrary connected reductive groups G over \({\mathbb{Q}}_ p\). The results about B(G) are useful in studying the points mod p on Shimura varieties.

Reviewer: H.-J.Bartels

MSC:

20G05	Representation theory for linear algebraic groups
20G25	Linear algebraic groups over local fields and their integers
13K05	Witt vectors and related rings (MSC2000)
14L35	Classical groups (algebro-geometric aspects)

Keywords:

characteristic \(p\); Witt ring; Frobenius automorphism; isocrystals; \(\sigma\)-conjugacy classes; connected reductive groups; Shimura varieties

References:

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