A positivity property of the Satake isomorphism. (English) Zbl 0941.22006

Rapoport, Michael

Manuscr. Math. 101, No. 2, 153-166 (2000).

Let \(G\) be a connected reductive group over a local field \(F\) which is quasisplit and which is split over an unramified extension. In this paper the author considers the matrix of the Satake isomorphism in terms of the natural bases of the source and of the target, and proves that all coefficients of this matrix that are not zero are actually positive numbers. The theorem is proved by induction on the semi-simple rank of \(G\). In the case \(G=GL_n\) one also shows how this positivity result can be deduced from the theory of symmetric functions. The (positivity) result is then applied to an existence problem of \(F\)-crystals which is a partial converse to Mazur’s theorem relating the Hodge polygon and the Newton polygon.

Reviewer: Lucian Bădescu (Los Angeles)

MSC:

22E35	Analysis on \(p\)-adic Lie groups
14F30	\(p\)-adic cohomology, crystalline cohomology

Keywords:

reductive groups; local fields; Satake isomorphism; positivity result; symmetric functions; \(F\)-crystals; Hodge polygon; Newton polygon