Quantum groups, the loop Grassmannian, and the Springer resolution

Abstract

We establish equivalences of the following three triangulated categories: \[ D quantum ( g ) ⟷ D coherent G ( N ~ ) ⟷ D perverse ( G r ) . D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}). \] Here, D quantum ( g ) D_\text {quantum}(\mathfrak {g}) is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g \mathfrak {g} ; the category D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for G G ( = = semisimple group with Lie algebra g \mathfrak {g} ), and the category D perverse ( G r ) D_\text {perverse}({\mathsf {Gr}}) is the derived category of perverse sheaves on the Grassmannian G r {\mathsf {Gr}} associated with the loop group L G ∨ LG^\vee , where G ∨ G^\vee is the Langlands dual group, smooth along the Schubert stratification. The equivalence between D quantum ( g ) D_\text {quantum}(\mathfrak {g}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between D perverse ( G r ) D_\text {perverse}(\mathsf {Gr}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p p -adic reductive group, while the other is in terms of equivariant K K -theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting U q ( g ) U_q(\mathfrak {g}) -modules, as will be explained in a separate paper.

On support varieties for Lie superalgebras and finite supergroup schemes

“…Let Γ(g 0 [2]) (1) be the divided power algebra on g 0 , its vector space structure twisted by the Frobenius map (λ → λ p ) on k, and considered as a homologically graded superspace with g 0 concentrated in Z-degree 2. The algebra structure on W (g 0 ) together with the natural coalgebra structure on Γ(g 0 [2]) (1) induces on…”

“…In [31,38] (see also [18,Lemma 3.3.1]) it is shown that a twisting cochain t can always be constructed such that the resulting chain complex (X(g), d t ) is a V (g)-free resolution of the trivial module. The proof of this fact depends, however, on the choice of a fixed basis for g 0 , so the resolution (X(g), d t ) need not be natural in g. In the construction, the action of t on Γ i (g 0 [2]) (1) is defined by induction on i so that the following properties are satisfied: i = 0: If ε : W (g) → k denotes the natural augmentation map on W (g), then ε • t = 0. i = 1: If x is one of the fixed basis vectors for g 0 , then t(γ 1 (x)) = x p−1 x − x [p] . Here γ 1 (x) is one of the divided power generators for Γ(g 0 [x]) ( , if g 0 is abelian, then t can be constructed to be trivial in homological degrees greater than 2, i.e., such that t(Γ i (g 0 [2]) (1) ) = 0 for i > 1.…”

“…Then identifying Γ(s 0 [2]) (1) with its image under Γ(ι), t ′ can be extended to a twisting cochain t : (1) to agree with t ′ and defining t arbitrarily on any complementary subspace of Γ i (g 0 [2]) (1) such that (in the notation of [18]) d • t 2i = r i for i > 2. It is possible to define t on Γ i (s 0 [2]) (1) to agree with t ′ because the image of Y (s) under Y (ι) is an exact subcomplex of Y (g). Now since t extends t ′ , the inclusion ι extends to a monomorphism of projective resolutions X(ι) : (X(s), d t ′ ) ֒→ (X(g), d t ).…”

“…Then H • (G, k) identifies with H • (V (g), k), and as discussed in [18, §3.5] the May spectral sequence for G (5.1.7) can be constructed from a filtration on the resolution X(g). In terms of this construction, the subalgebra S(g * 0 [2]) (1) of Hom V (g) (X(g), k) identifies with the row j = 0 of (5.1.7). In particular, S(g * 0 [2]) (1) consists of cocycles in Hom V (g) (X(g), k).…”

Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

“…For a subcategory C p (s) in the category C p (considered as either the W(p)-or U q s (2)-representation category), the derived category of C p (s) is equivalent to the derived category of coherent sheaves on a noncommutative extension of CP 1 [24]. The algebra Ext • is then the coordinate ring of this noncommutative extension.…”