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$ {\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum
$\mathfrak {gl}_n$
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
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
of the loop algebra
$\widehat {\mathfrak {gl}}_{m|n}$
of
${\mathfrak {gl}}_{m|n}$
with those of affine symmetric groups
${\widehat {{\mathfrak S}}_{r}}$
. Then, we give a BLM type realization of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
via affine Schur superalgebras.
The first application of the realization of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
is to determine the action of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
on tensor spaces of the natural representation of
$\widehat {\mathfrak {gl}}_{m|n}$
. These results in epimorphisms from
$\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
to affine Schur superalgebras so that the bridging relation between representations of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
and
${\widehat {{\mathfrak S}}_{r}}$
is established. As a second application, we construct a Kostant type
$\mathbb Z$
-form for
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
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.
