In this paper, we compute the $BP$-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and $BP$-operations to prove new results about equivariant maps between Stiefel manifolds.

Let $G$ be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of an anisotropic projective $G$-homogeneous variety in terms of motives of simpler $G$-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of $G$. We decompose the motive of a generalized Severi–Brauer variety $\mathrm{SB}_2(A)$ of a division algebra $A$ of degree 5 into a direct sum of twisted motives of the Severi–Brauer variety $\mathrm{SB}(B)$ of a division algebra $B$ Brauer-equivalent to the tensor square $A^{\otimes 2}$. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.

It is shown that, in characteristic zero, a finite subgroup of a general linear group is generated by pseudo-reflections if and only if its ring of coinvariants satisfies Poincaré duality.

For a parabolic subgroup H of the general linear group G = Gl(n, C), we characterize the Kähler classes of G/H and give a formula for the height of any two-dimensional cohomology class. As an application, we classify the automorphisms of the cohomology ring of G/H when this ring is generated by two-dimensional classes.