Does the $K^1$-group of a complete flag variety vanish?

For $U(n)$ the Lie group of $n times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special case of $n=2$ this gives the sphere $S^2$. In this special case the topological $K$-theory group $K^1$ vanishes, i.e.
$$
K^1(S^2) = 0.
$$
Is this true for higher complete flag varieties? In explicit form, is it true that
$$
K^1(U(n)/T^n) = 0, ~~ forall n?
$$