A REALIZATION OF THE ENVELOPING SUPERALGEBRA UQ((gl)over-capm|n)

In [2], Beilinson-Lusztig-MacPherson (BLM) gave a beautiful realization for quantum gl(n )via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra U-Q((gl) over cap (m vertical bar n)) of the loop algebra (gl) over cap (m vertical bar n) of gl(m vertical bar n) with those of affine symmetric groups (S) over cap (r). Then, we give a BLM type realization of U-Q ((gl) over cap (m vertical bar n)) via affine Schur superalgebras. The first application of the realization of U-Q((gl) over cap (m vertical bar n)) is to determine the action of U-Q((gl) over cap (m vertical bar n)) on tensor spaces of the natural representation of <(gl)(over cap>m vertical bar)(n). These results in epimorphisms from U-Q((gl) over cap (m vertical bar n)) to affine Schur superalgebras so that the bridging relation between representations of U-Q((gl) over cap (m vertical bar n) and (S) over cap (r) is established. As a second application, we construct a Kostant type Z-form for U-Q((gl) over cap (m vertical bar n)) whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur-Weyl duality in arbitrary characteristics.