Pseudo-reductive Galois descent in Bruhat-Tits buildings
Final Report Abstract
First funding period: The goal of the project was to determine the local structure of Bruhat-Tits buildings. This was a project that had been started by R. M. Weiss in his research monograph on affine buildings. Some cases were not treated in that reference because they were rather involved and required more sophisticated methods. We developed a theory of combinatorial descent in buildings which enabled to determine the local structure of all Bruhat-Tits buildings of rank at least three. We also applied our theory of descent to give an alternative existence proof for the exceptional quadrangles which turned out to be extremely useful in the second funding period of the project. In our theory of descent in buildings there is the notion of a descent group. Inspired from the theory of algebraic groups over fields we defined Galois groups of arbitrary spherical Moufang buildings and proved that each Galois group is a descent group. Second funding period: One major goal was to extend our result on the local structure to Bruhat-Tits trees. In order to get started we had to produce concrete models of the exceptional rank one groups that are characteristic free. The strategy was to apply our geometric version of Galois descent and this led us to the definition of a Tits polygon. These are geometric structures that are natural generalizations of Moufang polygons. Each exceptional rank one group can be realized as the fixed point set of a Galois involution acting on a Tits polygon. Eventually it became clear that a thorough understanding of the Tits polygons related to the exceptional groups is indispensable for producing the desired models of the exceptional rank one groups. We developed a general theory of Tits polygons and achieved classification results that are complete except for case of Tits quadrangles. We produced the desired models of the exceptional rank one groups. There are in fact two families of interest which are called of hexagonal and quadrangular type respectively. As a side product of our efforts in the hexagonal case we provided a characteristic-free approach to the famous 4-form on the 56-dimensional module for E7 . At some point we observed connections of our notion of a Tits polygon to concepts that already existed in the literature. In particular, our notion of a Tits triangle corresponds to the notion of a stable Moufang Veldkamp plane. We could show, under a mild assumption, that each such a plane is coordinatized by a stable alternative ring.
Publications
- Descent in buildings. Annals of Mathematics Studies, 190, Princeton University Press, Princeton, 2015, xvi+336 pp.
B. Mühlherr, H. P. Petersson, R.M. Weiss
(See online at https://doi.org/10.1365/s13291-016-0145-2) - Galois involutions and exceptional buildings. Enseign. Math. 62 (2016), 207-260
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.4171/LEM/62-1/2-1) - Rhizospheres in spherical buildings. Math. Ann. 369 (2017), 839-868
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.1007/s00208-016-1461-7) - Freudenthal triple systems in arbitrary characteristic. J. Algebra 520 (2019), 237-275
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.1016/j.jalgebra.2018.11.015) - Isotropic quadrangular algebras, J. Math. Soc. Japan 71 (2019), 1321-1380
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.2969/jmsj/80178017) - Root graded groups of rank 2, J. Comb. Algebra 3 (2019), 189-214
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.4171/JCA/30) - Tits triangles, Canad. Math. Bull. 62 (2019), 583-601
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.4153/S0008439518000140) - The exceptional Tits quadrangles, in: Transformation Groups, 2020, 56pp.
B. Mühlherr, R.M. Weiss
(See online at https://doi.org/10.1007/s00031-020-09573-5)
