EXOTIC SPRINGER FIBERS FOR ORBITS CORRESPONDING TO ONE-ROW BIPARTITIONS
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Springer Science+Business Media New York (2020)
EXOTIC SPRINGER FIBERS FOR ORBITS CORRESPONDING TO ONE-ROW BIPARTITIONS N. SAUNDERS
A. WILBERT
School of Computing and Mathematical Sciences University of Greenwich London SE10 9LS, United Kingdom
Department of Mathematics Boyd GSRC University of Georgia Athens, GA 30602, USA
[email protected]
[email protected]
Abstract. We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato’s original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley–Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov’s arc algebra to the exotic setting.
1. Introduction In [Kat09], Kato introduced the exotic nilpotent cone in a successful attempt to extend the Kazhdan–Lusztig–Ginzburg geometrization of affine Hecke algebras (see [KL87], [CG97]) from the one-parameter to the multi-parameter case by considering the equivariant algebraic K-theory of an exotic version of the Steinberg variety. Kato’s construction establishes a Deligne–Langlands type classification of irreducible modules of affine Hecke algebras of type C with only very mild restrictions on the parameters. Let N (gl2m ) ⊆ gl2m (C) be the ordinary nilpotent cone of type A and let S ⊆ gl2m (C) denote the Sp2m (C)-invariant complement of sp2m (C) in gl2m (C). The exotic nilpotent cone is the singular affine variety N = C2m × (S ∩ N (gl2m )). Many of its properties relating to, e.g., intersection cohomology of orbit closures (see [AH08] and [SS14]), theory of special pieces (see [AHS11]), and the Lusztig–Vogan bijection (see [Nan13]) have been explored in follow-up work to [Kat09]. In [Kat09], [Kat11], Kato used the exotic nilpotent cone to construct an exotic version of the Springer correspondence for the Weyl group of type C. This corresDOI: 10.1007/S00031-020-09613-0 Received November 5, 2019. Accepted April 7, 2020. Corresponding Author: A. Wilbert, e-mail: [email protected]
N. SAUNDERS, A. WILBERT
pondence is less intricate than the classical type C Springer correspondence (see [Spr76], [Spr78], [Sho79]) because it gives in fact a bijection between the orbits under the Sp2m (C)-action on N and the isomorphism classes of all complex, finitedimensional, irreducible representations of the type C Weyl group (i.e., it behaves more like the classical type A Springer correspondence for the symmetric group). In particular, since the irreducible representations of the Weyl group can be labe
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