85,264 research outputs found
Local-global principles for embedding of fields with involution into simple algebras with involution
In this paper we prove local-global principles for embedding of fields with
involution into central simple algebras with involution over a global field.
These should be of interest in study of classical groups over global fields. We
deduce from our results that in a group of type D_n, n>4 even, two weakly
commensurable Zariski-dense S-arithmetic subgroups are actually commensurable.
A consequence of this result is that given an absolutely simple algebraic
K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having
the same set of isomorphism classes of maximal K-tori as G, is necessarily
K-isomorphic to G. These results lead to results about isolength and
isospectral compact hyperbolic spaces of dimension 2n-1 with n even
A geometric approach to (semi)-groups defined by automata via dual transducers
We give a geometric approach to groups defined by automata via the notion of
enriched dual of an inverse transducer. Using this geometric correspondence we
first provide some finiteness results, then we consider groups generated by the
dual of Cayley type of machines. Lastly, we address the problem of the study of
the action of these groups in the boundary. We show that examples of groups
having essentially free actions without critical points lie in the class of
groups defined by the transducers whose enriched dual generate a torsion-free
semigroup. Finally, we provide necessary and sufficient conditions to have
finite Schreier graphs on the boundary yielding to the decidability of the
algorithmic problem of checking the existence of Schreier graphs on the
boundary whose cardinalities are upper bounded by some fixed integer
Hecke Modules from Metaplectic Ice
We present a new framework for a broad class of affine Hecke algebra modules,
and show that such modules arise in a number of settings involving
representations of -adic groups and -matrices for quantum groups.
Instances of such modules arise from (possibly non-unique) functionals on
-adic groups and their metaplectic covers, such as the Whittaker
functionals. As a byproduct, we obtain new, algebraic proofs of a number of
results concerning metaplectic Whittaker functions. These are thus expressed in
terms of metaplectic versions of Demazure operators, which are built out of
-matrices of quantum groups depending on the cover degree and associated
root system
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