4,760 research outputs found
Local models of Shimura varieties and a conjecture of Kottwitz
We give a group theoretic definition of "local models" as sought after in the
theory of Shimura varieties. These are projective schemes over the integers of
a -adic local field that are expected to model the singularities of integral
models of Shimura varieties with parahoric level structure. Our local models
are certain mixed characteristic degenerations of Grassmannian varieties; they
are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and
the second-named author to mixed characteristics and to the case of general
(tamely ramified) reductive groups. We study the singularities of local models
and hence also of the corresponding integral models of Shimura varieties. In
particular, we study the monodromy (inertia) action and show a commutativity
property for the sheaves of nearby cycles. As a result, we prove a conjecture
of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby
cycles gives a function which is central in the parahoric Hecke algebra.Comment: 88 pages, several corrections and change
Twisted loop groups and their affine flag varieties
We develop a theory of affine flag varieties and of their Schubert varieties
for reductive groups over a Laurent power series local field k((t)) with k a
perfect field. This can be viewed as a generalization of the theory of affine
flag varieties for loop groups to a "twisted case"; a consequence of our
results is that our construction also includes the flag varieties for Kac-Moody
Lie algebras of affine type. We also give a coherence conjecture on the
dimensions of the spaces of global sections of the natural ample line bundles
on the partial flag varieties attached to a fixed group over k((t)) and some
applications to local models of Shimura varieties.Comment: LaTex, 73 page
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