GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED

Abstract. in the 1960's noboru iwahori and hideya matsumoto, eiichi abe and kazuo suzuki, and michael stein discovered that chevalley groups g = g(ф;r) over a semilocal ring admit remarkable gauss decomposition g = tuu-u, where t = t(ф;r) is a split maximal torus, whereas u = u(ф;r) and u- = u-(ф;r) are unipotent radicals of two opposite borel subgroups b = b(ф;r) and b- = b-(ф;r) containing t. it follows from the classical work of hyman bass and michael stein that for classical groups gauss decomposition holds under weaker assumptions such as sr(r) = 1 or as (r) = 1. later the third author noticed that condition sr(r) = 1 is necessary for gauss decomposition. here, we show that a slight variation of tavgen's rank reduction theorem implies that for the elementary group e = e(ф;r) condition sr(r) = 1 is also suffcient for gauss decomposition. in other words, e = huu-u, where h = h(ф;r) = t e. this surprising result shows that stronger conditions on the ground ring, such as being semi-local, asr(r) = 1, sr(r; ф) = 1, etc., were only needed to guarantee that for simply connected groups g = e, rather than to verify the gauss decomposition itself.