Quasi-Classical Braverman--Kazhdan Intertwiners via Quiver Varieties

We show that Braverman–Kazhdan normalized intertwiners for $S{L}_n(\mathbf{C})$ have a quasi-classical incarnation governed by type $A$ quiver varieties. More precisely, for standard parabolic subgroups $P$ and $P^{\mathrm{\prime }}$ with conjugate Levi subgroups, we construct $S{L}_n\times L^{\mathrm{a}\mathrm{b}}$-equivariant isomorphisms $\mathrm{\Phi }(P,P^{\mathrm{\prime }}):{\overline{T^{\ast }(S{L}_n\mathrm{/}[P,P])}}^{\mathrm{a}\mathrm{f}\mathrm{f}}\to {\overline{T^{\ast }(S{L}_n\mathrm{/}[P^{\mathrm{\prime }},P^{\mathrm{\prime }}])}}^{\mathrm{a}\mathrm{f}\mathrm{f}}$ between the affinizations of the cotangent bundles of the corresponding Braverman–Kazhdan spaces, and we prove that these isomorphisms satisfy Coxeter relations. The construction uses $SL$-gauge analogues of Lusztig–Maffei–Nakajima reflection functors, thereby extending Wang’s quiver-variety realization of the quasi-classical Gelfand–Graev action from the Borel case to arbitrary parabolic subgroups. In this way, we complete the quasi-classical Braverman–Kazhdan intertwiner story for $S{L}_n(\mathbf{C})$ and obtain a systematic source of non-isomorphic varieties whose affinized cotangent bundles are isomorphic.