Affine Laumon space and contragredient dual Verma module of $U_q(\widehat{\mathfrak{gl}_n})$

Abstract

We study the action of the quantum group $U_q(\widehat{\mathfrak{gl}_n})$ on the equivariant K-theory of affine Laumon spaces. We show that, at any highest weight away from the critical level, this can be identified with the contragredient dual Verma module of $U_q(\widehat{\mathfrak{gl}_n})$, improving earlier results of Braverman-Finkelberg and Negu{\c{t}}. The proof uses a variant of stable envelopes first introduced by Maulik-Okounkov in the study of Nakajima quiver varieties.