Motivic Springer Theory

Catharina Stroppel; Jens Niklas Eberhardt

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Motivic Springer Theory

Jens Niklas Eberhardt, Catharina Stroppel

Representation Theory · Submitted 1 Sep 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives.

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We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives.	result 0.72	0.94	0.87	asserted 1.00
s1	To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures.	method 0.66	0.95	0.10	asserted 1.00
s2	As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.	result 0.97	0.93	0.12	asserted 1.00

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Original abstract

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.

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Denote the automorphism group of $M$ by $(M)⊂(V).$ Restriction yields a natural morphism $:(M)→P.$ The morphism $:(M)→P$ has a section $θ:P→(M).$ We follow similar arguments to [ref] .

Cell decompositions of quiver flag varieties for nilpotent representations of the oriented cycle · Affine pavings for quiver flag varieties in type $A$ · 0.95
We will use the following description of weight exact functors due to Sosnilo, see [ref] .

Theorem of the heart in negative K-theory for weight structures · Weight complex functor · 0.93

Compares with 1 paper

A similar result was shown in [ref] using slightly different arguments.

Equivariant motives and geometric representation theory. (with an appendix by F. H\"ormann and M. Wendt) · Equivariant mixed Tate motives · 0.83

Cites as support 1 paper

The works of Cerulli-Irelli–Esposito–Franzen–Reineke [ref] and Maksimau [ref] show that type $A$ partial quiver flag varieties admit an affine pavings.

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology · Conditions (PT) and (FO) · 1.00

Cites as background 7 papers

Soergel–Virk–Wendt [ref] introduced an equivariant version of the above formalism.

Equivariant motives and geometric representation theory. (with an appendix by F. H\"ormann and M. Wendt) · Equivariant motivic sheaves · 0.87
Denote by $R_𝐝$ the quiver Hecke (KLR) algebra associated to $Q$ and $𝐝$ as defined by Khovanov–Lauda [ref] and Rouquier [ref] .

2-Kac-Moody algebras · Quiver Hecke and quiver Schur algebras · 0.84
We refer to [ref] and [ref] for more details.

Quiver Schur algebras and q-Fock space · Quiver flag varieties · 1.00

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