On the geometric realization of the inner product and canonical basis for quantum affine ${\mathfrak{s}\mathfrak{l}}_n$

We give a geometric interpretation of the inner product on the modified quantum group of ${\widehat{\mathfrak{s}\mathfrak{l}}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.