A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to my wife

We give a new approach to unramified descent in Bruhat-Tits theory of reductive groups over a discretely valued field k, with Henselian local ring and perfect residue field, which appears to be conceptually simpler, and more geometric, than the original approach of Bruhat and Tits. We are able to derive the main results of the theory over k from the theory over the maximal unramified extension K of k. Even in the most interesting case for number theory and representation theory, where k is a locally compact nonarchimedean field, the geometric approach described in this paper appears to be considerably simpler than the original approach. Introduction. Let k be a discretely valued field whose local ring is Henselian. Throughout this paper, we will assume that the residue field of k is perfect. The Bruhat-Tits theory of reductive groups over k has two parts. The first part is the theory over the maximal unramified extension K of k; because of our assumption on k, any reductive k-group G is quasi-split over K (see 1.7) and hence G(K) has a rather simple structure. This part of the theory is due to Iwahori-Matsumoto, Hijikata and Bruhat and Tits. The second part, called the “unramified descent” (or “étale descent”), is due to Bruhat and Tits. This part gives us the Bruhat-Tits theory over k, and also the Bruhat-Tits building of G(k), from the Bruhat-Tits theory over K and the Bruhat-Tits building B(G/K) of G(K), using descent of valuation of root datum from K to k. This second part is somewhat technical; see [BrT1, §9] and [BrT2, §5]. The purpose of this paper is to present an alternative approach to unramified descent which appears to be conceptually simpler, and more geometric, than the approach in [BrT1], [BrT2] in that it does not use descent of valuation of root datum from K to k to show that B(G/K)Γ, where Γ is the Galois group of K/k, is an affine building. In this approach, we will use the Bruhat-Tits theory, and the buildings, only over the maximal unramified extension K of k and derive the main results of the theory for reductive groups over k. In §4, we will describe a natural filtration of the root groups Ua(k) and also a valuation of the root datum of G/k relative to a maximal k-split torus S, using the geometric results of §§2, 3 that provide the Bruhat-Tits building of G(k). The approach described here appears to be considerably simpler than the original approach even for reductive groups over locally compact nonarchimedean fields (i.e., discretely valued complete fields with finite residue field). In §5, we prove results over discretely valued fields with