[Paper Review] A Survey on Springer Theory

Julia Sauter|arXiv (Cornell University)|Jul 3, 2013

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper presents a unified geometric framework for Springer theory over ℂ, defining a Springer triple and Steinberg variety to construct a convolution algebra on equivariant Borel-Moore homology and K-theory—called the Steinberg algebra—whose indecomposable projective graded modules arise from the decomposition theorem. The key contribution is a generalization of classical Springer theory and quiver-graded Springer theory as special cases of this algebraic-geometric construction.

ABSTRACT

This is not standard in the sense that we understand a Springer map to be a collapsing of homogeneous bundles. Apart from that we use mostly techniques from Chriss and Ginzbergs book but we work in the equivariant derived category of Bernstein and Lunts. We define Steinberg algebras as (equivariant) Borel-Moore homology algebras of the associated Steinberg varieties. The data of the BBD-decomposition theorem applied to the Springer map give a parametrization of projective graded and simple graded modules over the Steinberg algebra. Also, the projective graded modules are equivalent to a category of shifts of perverse sheaves. This has as a consequence for example the Springer correspondence. We call classical Springer Theory what is usually considered as Springer Theory. The main results are parametrizations of simple modules of different types of Hecke algebras. Our second main example is quiver-graded Springer theory (due to Lusztig), here the Steinberg algebras are the quiver Hecke algebras. We also explain Lusztig's and Khovanov-Lauda's monoidal categorification of the negative half of the quantum group using the categories of shifts of perverse sheaves and projective graded modules over the quiver Hecke algebra respectively.

Motivation & Objective

• To generalize classical Springer theory into a uniform geometric construction applicable to a wide class of non-commutative algebras and their modules.
• To establish a systematic framework for studying Steinberg algebras via convolution products on equivariant Borel-Moore homology and K-theory.
• To clarify the role of Springer fibres as subvarieties of flag varieties and their homological structure via the decomposition theorem.
• To connect the theory to known algebras such as Weyl group group algebras, affine Hecke algebras, KLR algebras, and quiver Schur algebras.
• To identify open problems and partial answers in the representation theory of Steinberg algebras, including semi-simplicity, global dimension, and canonical bases.

Proposed method

• Define a Springer triple via a reductive group G, parabolic subgroups P_i, and P_i-stable subrepresentations F_i of a G-representation V.
• Construct the Springer map π: E → V as a union of associated homogeneous vector bundles E_i = G ×^{P_i} F_i over G/P_i.
• Form the Steinberg variety Z = E ×_V E as the fiber product, equipped with projections p and m to V and (⊔G/P_i) × (⊔G/P_i).
• Equip the equivariant Borel-Moore homology H_*^A(Z) with a convolution product * defined via pullback, intersection, and pushforward: c1,2 * c2,3 = (q1,3)_*(p1,2^*c1,2 ∩ p2,3^*c2,3).
• Use the decomposition theorem for perverse sheaves to decompose the Steinberg variety into indecomposable projective graded modules over the Steinberg algebra.
• Study Springer fibre modules as homology groups of fibers of π, equipped with module structures via convolution.

Experimental results

Research questions

• RQ1 Are Springer fibre modules always semi-simple as modules over the Steinberg algebra?
• RQ2 Which Steinberg algebras are affine cellular algebras, and which have finite global dimension?
• RQ3 Do Steinberg algebras admit Kazhdan-Lusztig polynomials and a canonical basis?
• RQ4 Can noncommutative resolutions of singularities be constructed for Springer maps using constructible sheaves?
• RQ5 Is there a Schur-Weyl duality relating classical and quiver-graded Springer theories via Morita equivalence of Steinberg algebras?

Key findings

• The Steinberg algebra H_*^A(Z) is an associative algebra under the convolution product, with components supported on Z_{i,j} ×_V Z_{k,l} only when j = k.
• The decomposition theorem implies that the equivariant Borel-Moore homology of the Steinberg variety decomposes into indecomposable projective graded modules over the Steinberg algebra.
• For classical Springer theory, the Steinberg algebra H_*^G(Z) is isomorphic to the group algebra ℂ[W] of the Weyl group W.
• In quiver-graded Springer theory for the oriented cycle quiver, H_*^G(Z) is isomorphic to the quiver Schur algebra, as shown by Stroppel and Webster.
• Kato's exotic Springer theory realizes the Steinberg algebra as the Hecke algebra with unequal parameters of type C_n^{(1)}, with an associated exotic Deligne-Langlands correspondence.
• KLR algebras for Dynkin quivers are affine cellular and have finite global dimension, as shown by Brundan, Kleshchev, and McNamara, and are conjectured to be the only such cases.