Presenting affine infinitesimal Schur algebras

Let U(glˆn) be the universal enveloping algebra of glˆn. A certain Z-form UZ(glˆn) of U(glˆn) was introduced in [15] such that the affine Schur algebra S▵(n,r)Z is a homomorphic image of UZ(glˆn). Let k be a field and let Uk(glˆn)=UZ(glˆn)⊗k. Let S▵(n,r)k=S▵(n,r)Z⊗k. By tensoring with k, we have a surjective algebra homomorphism ξr,k:=ξr⊗id:Uk(glˆn)↠S▵(n,r)k. For h⩾1 and r∈N let s▵(n,r)h=ξr,k(s▵(n)h), where s▵(n)h is a certain subalgebra of Uk(glˆn). The algebra s▵(n,r)h is the affine analogue of infinitesimal Schur algebras introduced in [7]. In this paper we will give a presentation of s▵(n,r)1 by generators and defining relations.