Koszul duality for Kac–Moody groups and characters of tilting modules

Abstract

We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic ℓ in terms of ℓ-Kazhdan-Lusztig polynomials, for ℓ > h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if ℓ ≥ 2h − 2. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.1.3. The Kac-Moody case and quantum groups. These ideas were later generalized by Bezrukavnikov-Yun [BY] to the case where G is replaced by a general Kac-Moody group G . Let B ⊂ G be a Borel subgroup, and let U ⊂ B be its unipotent radical. An important new idea in [BY] (also suggested in [BG]) is that a richer version of Koszul duality can be obtained if one "deforms" the categories of semisimple complexes on G /B and tilting perverse sheaves on G ∨ /B ∨ along a polynomial ring. The B-constructible semisimple complexes are thus replaced by the B-equivariant semisimple complexes, and the tilting perverse sheaves are replaced by the so-called "free-monodromic" objects constructed (via a very technical procedure) by Yun using certain pro-objects in the derived category of G ∨ /U ∨ , see [BY, Appendix A]. These deformed categories each have a monoidal structure, given by an appropriate kind of convolution product. The main result of [BY] is an equivalence of monoidal categoriesrelating B-equivariant semisimple complexes on G /B and free-monodromic tilting perverse sheaves attached to G ∨ . From this, Bezrukavnikov-Yun then deduce a Kac-Moody analogue of (1.1). As in §1.2, this result has a combinatorial motivation in terms of Kazhdan-Lusztig polynomials [Yu], and a representation-theoretic motivation in terms of analogues of the category O for Kac-Moody Lie algebras.But a third motivation for the work in [BY], specifically in the case of affine Kac-Moody groups, came from the hope of uniting two geometric approaches to the study of representations of Lusztig's quantum groups at a root of unity (see e.g. [Be, §1.2]), which we review below. Let Rep 0 (U ζ ) denote the principal block of the category of finite-dimensional representations of Lusztig's quantum group U ζ associated with an adjoint semisimple complex algebraic group G, specialized at a root of unity ζ.The first approach comes from [ABG]. The main result of [ABG, Part I] relates 3 Rep 0 (U ζ ) to the derived category of equivariant coherent sheaves on the Springer resolution N of G, denoted by D b Coh G×Gm ( N ). Then the main result of [ABG, Part II] states that D b Coh G×Gm ( N ) is equivalent to the derived category of Iwahoriconstructible perverse sheaves on the affine Grassmannian Gr of the Langlands dual semisimple group G ∨ . Together, these results give a new proof of Lusztig's character formula for simple modules in Rep 0 (U ζ ). (This character formula was already known when [ABG] appeared, by combining work of Kazhdan-Lusztig [KL2], Lusztig [Lu2] and Kashiwara-Tanisaki [KT].)For each s ∈ J, choose, once and for all, a...

An Iwahori-Whittaker model for the Satake category

“…In[JMW1] the authors consider the setting of "ordinary" constructible complexes. However, as observed already in[RW, §11.1] or[AMRW, §6.2], their considerations apply verbatim in our Iwahori-Whittaker setting.…”

“…Let k be an algebraic closure of k, and assume that the characteristic of k is strictly bigger than the Coxeter number of G. Then the formula of Proposition 4.18 is related to Donkin's tensor product theorem for tilting modules of the Langlands dual k-group G ∨ k as follows. In [RW,AR2,AMRW] the authors construct a "degrading functor" η : Parity IW (Fl • , k) → Tilt prin (G ∨ k ), where Fl • is the connected component of the base point in Fl, Parity IW (Fl • , k) is the category of (I + u , χ * I + (L k ψ ))-equivariant parity complexes on Fl • , and Tilt prin (G ∨ k ) denotes the category of tilting objects in the (non-extended) principal block of the category of finite-dimensional G ∨ k -modules. We expect that Donkin's tensor product theorem (see [Ja,§E.9]) can be explained geometrically by an isomorphism of complexes involving the functor Z .…”

p-Jones-Wenzl idempotents

“…The central role that the canonical basis of the Hecke algebra (and its associated Kazhdan-Lusztig polynomials) was believed to play is now known to be played by the p-canonical basis (and its associated p-Kazhdan-Lusztig polynomials). The most groundbreaking papers in this direction are (in our opinion) the paper by Williamson [Wil17] commonly known as "Torsion explosion" (that broke down the old paradigm), the paper by Riche and Williamson [RW18] known as the "Tilting manifesto" (that crystallized the emerging philosophy) and the recent paper by Achar, Makisumi, Riche, and Williamson [AMRW19] (that proved the combinatorial part of the conjecture in the tilting manifesto).…”

Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

“…This paper draws inspiration from [AR1], which (in the setting of parity complexes on flag varieties) introduced the notions of "mixed derived category" and "mixed perverse sheaves." These notions have since found important applications in modular representation theory; see in particular [AR1,ARd2,MaR,AR2,AMRW2].…”

“…One of the main results of [AMRW2] is that when the realization h comes from a Kac-Moody root datum, there is an equivalence of triangulated categories between RE(h, W ) and RE(h * , W ), known as Koszul duality. The main reason for the restriction to the Kac-Moody setting is that some of the arguments make use of the perverse t-structure from [AR1].…”