Quasi-Classical Braverman--Kazhdan Intertwiners via Quiver Varieties

Abstract

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 $P^{\prime }$ with conjugate Levi subgroups, we construct $S{L}_n\times L^{\mathrm{ab}}$-equivariant isomorphisms $\Phi (P,P^{\prime }):{\overline{T^{\ast }(S{L}_n/[P,P])}}^{\mathrm{aff}}\to {\overline{T^{\ast }(S{L}_n/[P^{\prime },P^{\prime }])}}^{\mathrm{aff}}$ 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.

arXiv comment35 pages. Rewritten introduction and fixed typos. Comments welcome!