A Gallery Model for Affine Flag Varieties via Chimney Retractions

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety

Regular sequences and random walks in affine buildings

We define and characterise regular sequences in affine buildings, thereby giving the "$p$-adic analogue" of the fundamental work of Kaimanovich. As applications we prove limit theorems for random walks on affine buildings and their automorphism groups

MV-Polytopes via affine buildings

We give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent and Littelmann. They used LS-gallery to define subsets in the Bott-Samelson variety that map to subsets of the affine Grassmannian, whose closure are MV-cycles. Since points in the Bott-Samelson variety correspond to galleries in the affine building one can look at the image of a point in such a special subset under all retractions at infinity. We prove that these images can be used to construct the corresponding MV-polytope in an explicit way, by using the GGMS strata. Furthermore we give a combinatorial construction for these images by using the crystal structure of LS-galleries and the action of the ordinary Weyl group on the coweight lattice.Comment: 34 page

On axiomatic definitions of non-discrete affine buildings

In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space. The sharpness of the axioms dealing with metric conditions is illustrated in an appendix. There it is shown that a space X defined over a model space with metric d is possibly a building only if the induced distance function on X satisfies the triangle inequality.Comment: Errors corrected, results extended. (This replaces the two earlier, separate preprints "Axioms of affine buidlings" arXiv:0909.2967v1 and "Affine $\Lambda$ buildings II" arXiv:0909.2059v1.

Retractions in Intersection Types

This paper deals with retraction - intended as isomorphic embedding - in intersection types building left and right inverses as terms of a lambda calculus with a bottom constant. The main result is a necessary and sufficient condition two strict intersection types must satisfy in order to assure the existence of two terms showing the first type to be a retract of the second one. Moreover, the characterisation of retraction in the standard intersection types is discussed.Comment: In Proceedings ITRS 2016, arXiv:1702.0187

Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields

In this paper, we prove some finiteness results about split Kac-Moody groups over local non-archimedean fields. Our results generalize those of "An affine Gindikin-Karpelevich formula" by Alexander Braverman, Howard Garland, David Kazhdan and Manish Patnaik. We do not require our groups to be affine. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group.Comment: International Mathematics Research Notices, Oxford University Press (OUP), 201