In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in terms of elements of canonical basis being fixed by the involution (coming from Grothendieck-Serre duality) and the pairing (given by graded Exts) of two canonical basis elements $C_a$, $C_b$ being in $\delta_{ab} + v {\mathbb Z}[v]$. Does there exist an analogous characterization of canonical basis in the ${\rm GL}_n \times {\mathbb C}^*$-equivariant K-theory of the Springer resolution $\tilde N$ or of the preimage $\tilde U$ of the union of nilpotents with at most two Jordan blocks in $\tilde N$?
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