Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the Hecke associated to Weyl group of type C. establish multiplication formulas on and algebra. Then we construct monomial bases canonical for The formula allows us a stabilization property family algebras that leads modified version ${\mathbf K}^{\mathfrak c}_n$. show c}_n$ is coideal subalgebra quantum ${\bf U}(\hat{\mathfrak{gl}}_n)$, $\big({\mathbf U}(\hat{ \mathfrak{gl}}_n), {\mathbf c}_n)$ forms symmetric pair. shown admit stably bases. also formulate several variants (modified) above, as well their This work provides new algebraic approach which complements sheds light our previous geometric subject. In appendix by four authors, length groups classical types are obtained in symmetrized fashion.