Torus-enriched Motivic Bruhat Complexes and Maximal Compact Groups

Bruhat decompositions give cellular models for split algebraic groups, flag varieties, and maximal compact groups, but motivic boundaries retain orientation and torus-translation data lost in the flag quotient. Over a perfect field of characteristic zero, let the group be connected, split, semisimple, and simply connected. Fixing a Borel subgroup with split maximal torus and unipotent radical, we construct a torus-enriched motivic cellular complex for the basic affine space and compute its boundary in every degree. Each cover in Bruhat order contributes a two-face operator determined by a transported coroot, a tail determinant weight, and an explicit Milnor--Witt frame degree. Bott--Samelson purity proves the formula, while the unipotent torsor identifies the complex with that of the group. Over the real numbers, realization identifies it at chain level with the extended-Weyl complex of a maximal compact subgroup, while torus augmentation gives the flag complex. A single motivic complex therefore interpolates between the two incidence theories. A finite torus-support filtration makes this explicit; after inversion of two it splits by the characters of the component group of the real split torus, and the support spectral sequence degenerates. Calculations in the rank-three special linear and exceptional rank-two cases exhibit the first higher differentials beyond the previously known range.