European Congress of Mathematics Kraków, 2 – 7 July, 2012

doi:10.4171/120

2013

DOI: 10.4171/120

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European Congress of Mathematics Kraków, 2 – 7 July, 2012

Abstract: A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (Qp, +) but our main interest here will be on non-locally compact groups.In recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, finite comb… Show more

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Cited by 3 publications

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“…4 Langlands work is in slightly less generality, and perhaps for this reason, the formula for the spherical function is nowadays just referred to as Macdonald's formula.…”

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confidence: 99%

Iwahori–Hecke algebras for p-adic loop groups

2015

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This paper is a continuation of Braverman and Kazhdan (Ann Math (2) 174 (3): 2011) in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case.

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“…4 Langlands work is in slightly less generality, and perhaps for this reason, the formula for the spherical function is nowadays just referred to as Macdonald's formula.…”

mentioning

confidence: 99%

Iwahori–Hecke algebras for p-adic loop groups

2015

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Small covers of graph-associahedra and realization of cycles

Gaifullin¹

2016

Sb. Math.

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Abstract. An oriented connected closed manifold M n is called a URC-manifold if for any oriented connected closed manifold N n of the same dimension there exists a non-zero degree mapping of a finite-fold covering M n of M n onto N n . This condition is equivalent to the following: For any n-dimensional integral homology class of any topological space X, a multiple of it can be realized as the image of the fundamental class of a finite-fold covering M n of M n under a continuous mapping f : M n → X. In 2007 the author gave a constructive proof of the classical result by Thom that a multiple of any integral homology class can be realized as an image of the fundamental class of an oriented smooth manifold. This construction yields the existence of URC-manifolds of all dimensions. For an important class of manifolds, the so-called small covers of graph-associahedra corresponding to connected graphs, we prove that either they or their two-fold orientation coverings are URC-manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a URC-manifold. In dimensions 4 and higher, this manifold is simpler than all previously known URC-manifolds.

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Small covers of graph-associahedra and realization of cycles

Gaifullin¹

2016

Математический сборник

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Ориентированное связное замкнутое многообразие $M^n$ называется $\mathrm{URC}$-многообразием, если для любого другого ориентированного связного замкнутого многообразия $N^n$ той же размерности найдется отображение ненулевой степени некоторого конечнолистного накрытия $\widehat{M}^n$ многообразия $M^n$ на многообразие $N^n$. Это условие эквивалентно следующему: любой $n$-мерный класс целочисленных гомологий любого топологического пространства $X$ может быть с некоторой кратностью реализован как образ фундаментального класса многообразия $\widehat{M}^n$, являющегося конечнолистным накрытием многообразия $M^n$, при непрерывном отображении $f\colon \widehat{M}^n\to X$. В 2007 г. автором было получено конструктивное доказательство классического результата Р. Тома о том, что любой целочисленный класс гомологий с некоторой кратностью реализуется образом фундаментального класса ориентированного гладкого многообразия. Из этой конструкции следует существование $\mathrm{URC}$-многообразий всех размерностей. Для важного класса многообразий - так называемых малых накрытий над граф-ассоциэдрами, отвечающими связным графам, - мы доказываем, что все они (или их ориентируемые двулистные накрытия) являются $\mathrm{URC}$-многообразиями. В частности, мы получаем, что двулистное накрытие малого накрытия над обычным ассоциэдром Сташефа является $\mathrm{URC}$-многообразием. В размерностях 4 и выше это многообразие гораздо проще, чем все примеры $\mathrm{URC}$-многообразий, которые были известны ранее. Библиография: 39 названий.

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European Congress of Mathematics Kraków, 2 – 7 July, 2012

Cited by 3 publications

References 42 publications

Iwahori–Hecke algebras for p-adic loop groups

Iwahori–Hecke algebras for p-adic loop groups

Small covers of graph-associahedra and realization of cycles

Small covers of graph-associahedra and realization of cycles

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