Speaker: Vikraman Balaji
Date: December 04, 2024
Affiliation: Chennai Mathematical Institute

Abstract: Algebraic groups over local fields K has been extensively studied in the past 60 years. Bruhat-Tits theory develops the study of bounded subgroups of the group G(K ) where G is a connected reductive algebraic group under very general assumptions of being quasi-split over K . After the first big step in the paper by N.Iwahori and H.Matsumoto, this study was taken to Grothendieckian heights by F. Bruhat and J. Tits in their now well-known body of work, called Bruhat-Tits theory; their work extended over 20 years till the mid eighties. A main theme is to schematize these bounded groups and this is carried out in their papers in 1971 and 1984. In my paper with C.S. Seshadri (2015), a somewhat different geometrical approach was observed in the setting when K = k((t )). A novel yet foundational feature in my work with Seshadri is the theorem that any parahoric Bruhat-Tits group scheme over a complete discrete valuation ring O can be realized as an ?invariant direct image", i.e., obtained by taking Galois invariants of Weil restriction of scalars of a reductive group scheme, on a ramified cover of O . In my recent paper with Y. Pandey, we study the question of extending the schematic aspects of Bruhat-Tits theory to regular local rings. Such a study is already envisaged in the work of Bruhat and Tits. This paper provides a fairly complete higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. Application 1: I will briefly outline the solution to the problem of degeneration of moduli stack of G-torsors on smooth projective curves when the curve is allowed to degenerate to a nodal curve. In fact, the higher dimensional theory arose out of this work. Application 2: I will briefly indicate constructions of new modular desingularizations of moduli spaces of semistable principal bundles over curves. This comprehensively generalizes work of Seshadri, Narasimhan-Ramanan.