We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A ${\hat{\mathfrak{gl}}}_n$-variant of the embedding is also established.

We extend the $\imath$Hall algebra realization of $\imath$quantum groups arising from quantum symmetric pairs, which establishes an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type to the $\imath$Hall algebra associated to an arbitrary $\imath$quiver (not necessarily virtually acyclic). This generalizes Lu-Wang's result.

Let W be a real symplectic space and (G, G') an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let $\widetilde {\text{G}}$ be the preimage of G in the metaplectic group $\widetilde {\text{Sp}}$(W). Given an irreducible unitary representation $\Pi$ of $\widetilde {\text{G}}$ that occurs in the restriction of the Weil representation to $\widetilde {\text{G}}$, let $\Theta_\Pi$ denote its character. We prove that, for a suitable embedding $T$ of $\widetilde {\text{Sp}}$(W) in the space of tempered distributions on W, the distribution $T(\check\Theta_\Pi)$ admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of $\Pi'$, the representation of $\widetilde {\text{G'}}$ dual to $\Pi$, by elementary means.