}}) \ar[r]^<<<<<{\sim} & K^{V_{\mathbf{GL_0}}}(V_{\mathbf{GL_0}}/V_{\mathbf{\whB}}), }$$ that we call *the characteristic isomorphism in the equivariant $K$-theory of the flag variety $V_{\mathbf{GL_0}}/V_{\mathbf{\whB}}$*.  We have a commutative diagram *specialization at $\bfq=0$* $$\xymatrix{ \bbZ[X,Y,z_0^{\pm0}] \ar[rr]^{c_{V_{\mathbf{GL_0}}}}_{\sim} 