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Affine quiver Schur algebras and \(p\)-adic \(\mathrm{GL}_n\). (English) Zbl 1475.20010

The (affine) Schur algebra is defined as the endomorphism algebra of certain permutation modules for the (affine) Iwahori-Matsumoto Hecke algebra, while the quiver Schur algebra attached to the cyclic quiver is defined as the \(\mathrm{GL}_\mathbf{d}\)-equivariant Borel-Moore homology of a “Steinberg type” variety equipped with the convolution product.
In the paper under review, an isomorphism between suitable completions of the above two algebras is established. This isomorphism endows the completed (affine) Schur algebra a grading, which is illustrated in the explicit example of \(\mathrm{GL}_2(\mathbb{Q}_5)\) in characteristic \(3\). The authors also provide some fundamental constructions for the (affine) Schur algebra, such as the generating sets, explicit faithful representations and geometrically adapted bases.
Reviewer: Li Luo (Shanghai)

MSC:

20C08 Hecke algebras and their representations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
20G43 Schur and \(q\)-Schur algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
22E57 Geometric Langlands program: representation-theoretic aspects
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References:

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