arXiv:1209.0188v2 [math.RT] 12 Nov 2019 inter.T ealissaeet xacmlxrdciegroup reductive complex a fix statement, its recall To theory. tion nptn radical unipotent hi epcieLeagba,and algebras, Lie respective their h elno-enti oaiaintermi h unesnilre quintessential the is theorem localization Beilinson-Bernstein The ..TeDmzr ek Category Hecke Demazure References The groupoids for 4.2. Equivariance Descent Demazure 4.1. Appendix: 4. ..BiisnBrsenlclzto n h i-ek ler 14 Example: character infinitesimal 3.6. Specified sheaf Weyl 3.5. Universal algebra 3.4. nil-Hecke Symmetries the and localization 3.3. Beilinson-Bernstein 3.2. Quantization localization 3.1. Beilinson-Bernstein eigenvalues 3. Specified pattern 2.3. Descent 2.2. Recollections resolution 2.1. Grothendieck-Springer 2. Acknowledgements context 1.7. Categorical descent 1.6. Demazure limit 1.5. Classical 1.4. ..Cmaio with Comparison symmetry Hecke via 1.3. Beilinson-Bernstein Barr-Beck via 1.2. Beilinson-Bernstein 1.1. Introduction 1. ino netiigfntr arfie omo elgopsy group Weyl of form refined (a functors intertwining of tion n,w eosrt htall that demonstrate we ing, eut pl oabtaymdlswtottene ofi infi fix to need the without al modules derived arbitrary in to descent apply of results formalism the using theorem ization ntebscan space affine basic the on ecn aai ohcascladqatmvrin sacate a as versions quantum and classical alter both an in present we data descent appendix an In resolution. simultaneous Abstract. N epeetasml ro fasrnteigo h eie Be derived the of strengthening a of proof simple a present We SL ⊂ 2 VRTEHRS-HNR CENTER HARISH-CHANDRA THE OVER ELNO-ENTI LOCALIZATION BEILINSON-BERNSTEIN B n let and , K -theory G/N AI E-V N AI NADLER DAVID AND BEN-ZVI DAVID Ug hsi unu eso fdsetfrteGrothendieck-S the for descent of version quantum a is This . W H mdlsaeteivrat,o qiaetycoinvariants equivalently or invariants, the are -modules = h elgopof group Weyl the B/N 1. Introduction Contents eoeteuieslCra.Let Cartan. universal the denote 1 G . aieprpcie hc dnie the identifies which perspective, native eri emty h ruet and arguments The geometry. gebraic oia cino eaueoperators. Demazure of action gorical mty nmonodromic on mmetry) ieia hrce.Ruhyspeak- Roughly character. nitesimal G oe subgroup Borel , uti emti representa- geometric in sult lno-enti local- ilinson-Bernstein g , D b fteac- the of , , -modules n pringer and B ⊂ h G denote with 24 22 21 20 19 18 16 15 13 12 11 11 9 9 9 9 8 7 7 4 3 1 2 DAVID BEN-ZVI AND DAVID NADLER

∗ Fix λ ∈ h , and let Pλ = Dλ(G/B) denote the dg category of λ-twisted D-modules on the flag variety.1 By λ-twisted D-modules, one can take D-modules on the basic affine space G/N which are weakly H-equivariant and have monodromy λ along the H-orbits. (The notation Pλ reflects the analogy with principal series representations of real or p-adic groups.) Let Ug denote the universal enveloping algebra of g, and Zg ≃ O(h∗//W ) ⊂ Ug the Harish- Chandra center. Let Ug-mod[λ] denote the dg category of Ug-modules with infinitesimal character the W -orbit [λ] ∈ h∗//W ≃ Spec Zg. Theorem 1.1 ([BB]). For λ ∈ h∗ regular, localization and global sections provide inverse equiva- lences of derived categories o ∼ / ∆ : Ug-mod[λ] o / Pλ :Γ

Remark 1.2 (Refinements). There are several standard refinements of the above statement. 1) One can bound the cohomological amplitude of Γ by the length of the Weyl group element w ∈ W for which w · λ is dominant. In particular, when λ is dominant, the equivalence preserves the corresponding abelian categories. ∗ 2) For λ ∈ h singular, one can exhibit Ug-mod[λ] as the quotient of Pλ by the kernel of Γ. Alternatively, one can realize Ug-mod[λ] as the subcategory of Pλ left-orthogonal to the kernel of Γ. These discrepancies can be bridged via localization on partial flag varieties [K, BMR1, BK]. 3) The theorem respects the natural G-actions and so identifies equivariant objects on each side. Thus for a subgroup K ⊂ G, it provides an equivalence between Harish-Chandra (Ug-mod[λ],K)- modules and K-equivariant λ-twisted D-modules. 4) One can naturally extend the theorem to generalized monodromy and infinitesimal character and thus obtain an equivalence not for fixed regular λ ∈ h∗ but over its formal neighborhood λ∧ ∗ within h . Namely, let Pλ∧ = Dλ∧ (G/B) denote the dg category of weakly H-equivariant D-modules ∧ on G/N with monodromy in λ . Similarly, let Ug-mod[λ∧] denote the dg category of Ug-modules with infinitesimal character in the W -orbit [λ∧] or equivalently formal neighborhood [λ]∧. Then when λ is regular, localization and global sections provide inverse equivalences o ∼ / ∆ : Ug-mod[λ∧] o / Pλ∧ :Γ Alternatively, one can equip the original statement of Theorem 1.1 with the symmetries that control the extension to the formal neighborhood. Namely, one can pass to (co)modules for the (co)monad of the adjunction given by restriction and extension of modules along the inclusion of λ into λ∧. Despite its dramatic importance, which is hard to overestimate, there nevertheless remain several deficiencies of Theorem 1.1, even after taking into account the above refinements. First, and foremost of our motivations, the theorem applies to modules with fixed, or at most gen- eralized, infinitesimal character. This makes it difficult to apply to or even compare with questions of harmonic analysis, which typically involve the geometry or topology of families of representations (for example, the Plancherel formula and Baum-Connes conjecture). Second, it is unnatural and thus often confusing that the parameters on the two sides of the theorem do not match: to localize modules with infinitesimal character [λ] ∈ h∗//W , we must choose a representative lift λ ∈ h∗. It is worth noting the impact of this choice: the geometry of the corresponding D-modules – most basically, the dimension of their support – depends on this choice. Third, the theorem does not apply as stated to singular infinitesimal character where the category of D-modules is larger than the corresponding category of modules. As mentioned above, one can bridge this discrepancy via localization on partial flag varieties. But this seems to only increase the distance from a uniform statement over all infinitesimal characters.

1Since we are interested in performing algebraic operations on derived categories, it is important to work in a proper homotopical setting. The term “dg category” will stand throughout for a C-linear pre-triangulated (or stable) dg category, and all operations with dg categories will be taken in the derived sense, i.e., we work in the ∞-category of dg categories. See Section 1.6 below for a brief summary of our working context. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 3

In this paper, we present a natural refinement of the derived Beilinson-Bernstein localization the- orem which simultaneously corrects these three drawbacks by invoking descent. A naive paraphrase of the main result asserts: “the category of Ug-modules is equivalent to the Weyl group invariants in the category of all monodromic D-modules on G/N”. In what immediately follows, we first state a precise version of this result, then explain the paraphrase and a “classical limit” for quasicoherent sheaves on the Grothendieck-Springer resolution.

1.1. Beilinson-Bernstein via Barr-Beck. A natural setting for simultaneously localizing over all infinitesimal characters is the dg category P = Dmon(G/N) of all weakly H-equivariant D- modules on the basic affine space G/N (see for example [B]). The subscript “mon” stands for the term monodromic, which we will understand to be a synonym for weakly H-equivariant, and in particular not imply specified monodromy along the H-orbits. (The notation P reflects the analogy with the universal principal series representation of a real group or the universal unramified principal series representation of a p-adic group.) The natural right H-action on G/N induces an identification of Uh ≃ O(h∗) with central H- invariant differential operators on G/N. This equips P with a linear structure over the base h∗ with ∗ fiber at λ ∈ h the previously encountered dg category Pλ = Dλ(G/B) of λ-twisted D-modules. ∧ Similarly, restricting to the formal neighborhood λ , we recover the dg category Pλ∧ = Dλ∧ (G/B) of weakly H-equivariant D-modules with monodromy in λ∧. The natural left G-action on G/N induces an embedding of Ug in H-invariant differential oper- ators on G/N. This gives rise to an adjunction

∆ : Ug-mod o / P :Γ between the dg category of all Ug-modules and P. Note that here Γ denotes the H-invariants in the usual global sections functor. The functors intertwine the linear structure over h∗//W on the left and that over h∗ on the right. An important observation is that the adjunction is naturally ambidextrous: ∆ is canonically the right adjoint of Γ as well. This is a reflection of the Calabi-Yau structure of P, the key ingredient in the approach of [BMR1, BMR2] to establishing Beilinson-Bernstein equivalences (following an argument of [BKR] in the setting of the McKay correspondence, and extended to localization for quantum symplectic resolutions in [MN]). We will refer to the composition W = ∆ ◦ Γ ∈ End(P) as the Weyl functor. It naturally comes equipped with the structure of a monad and a comonad, or in other words, an algebra and coalgebra object in endofunctors of P. In fact, one could organize all of the structure in what could rightfully be called a Frobenius monad. We will use this term as evocative shorthand for the monad of an ambidextrous adjunction (and will not attempt to independently formalize it in the ∞-categorical setting; see Section 1.6 below for further discussion and references to the discrete setting). Another crucial feature of ∆ is its conservativity: no non-zero objects localize to zero (an easy consequence of the general localization formalism). We find ourselves in the setting of the Barr- Beck theorem, in its ∞-categorical form due to Lurie. Recall it states that given an adjunction F : C ←→ D : G between ∞-categories, if the right adjoint G is conservative, and any G-split simplicial object of D admits a colimit preserved by G, then G exhibits D as monadic over C. This allows us to describe Ug-mod in terms of P and the Weyl functor W with its natural structures. W Let us write PW and P for the respective dg categories of W-modules and W-comodules in P. Theorem 1.3 (Barr-Beck version of localization). There are canonical equivalences

W PW ≃ Ug-mod ≃P between the dg categories of W-modules and W-comodules in P and the category of Ug-modules. 4 DAVID BEN-ZVI AND DAVID NADLER

Remark 1.4. For fixed regular [λ] ∈ h∗//W , the arguments of [BMR1] utilize the ambidextrous property and the indecomposability of Dλ(G/B) to deduce the Beilinson-Bernstein equivalence. For singular parameters or families of infinitesimal characters, the indecomposability fails, but we recover Ug-mod as a summand of P in the form of an isotypic component Ug-mod = PW . Remark 1.5 (“Reverse” Beilinson-Bernstein and the nil-Hecke algebra). As we explain in Section 3.2, we can also run Barr-Beck in the opposite direction, accessing P through the global sections functor. This is no longer conservative, so the result is a description of globally generated monodromic D- modules on G/N: we have equivalences

glob Ug P ≃ f-mod Ug Ug Uh where f = ⊗Zg is the extended enveloping algebra, and (Pglob)H ≃ Ug-mod where H is the nil-Hecke algebra, the Uh-algebra generated by the Demazure divided-difference operators, which controls descent along the map z : h∗ → h∗//W from the dual Cartan to its coarse quotient by the Weyl group. Thus we see that the Weyl monad W in Theorem 1.3 combines the action of the nil-Hecke algebra with projection along the kernel of the global sections functor. See also 1.5 for a closely related appearance of the Demazure operators. 1.2. Beilinson-Bernstein via Hecke symmetry. In order to exploit the formal assertion of Theorem 1.3, we need to identify the Weyl functor W concretely in terms of the symmetries of P. By construction, the category P carries two fundamental commuting actions. On the one hand, the left G-action on G/N naturally equips P with the structure of de Rham G-category. By this, we mean an algebraic G-action that is infinitesimally trivialized, or in other words, the induced action of the formal group of G is trivialized. This can be formalized by saying that P is a module for the monoidal dg category D(G) of D-modules on G under convolution. On the other hand, the Hecke category H = Dbimon(N\G/N) of bimonodromic D-modules on N\G/N naturally acts on P on the right by intertwining functors. In the same way that P is linear over the base h∗, the Hecke category H is linear over the base h∗ × h∗, and its monoidal structure is compatible with convolution of bimodules over h∗ × h∗. One can view the Hecke category H as the monodromic generalization of the familiar finite Hecke category D(B\G/B), which in turn is a categorical form of the finite Hecke algebra or Artin braid group. More generally, one finds the Hecke category Hλ,µ = Dλ,µ(B\G/B) of λ, µ-twisted D-modules as the fiber of H at a point (λ, µ) ∈ h∗ × h∗. Similarly, restricting to the the formal ∧ ∧ neighborhood λ × µ , one finds the Hecke category Hλ∧,µ∧ = Dλ∧,µ∧ (B\G/B) of weakly H × H- equivariant D-modules with monodromy in λ∧ × µ∧. Remark 1.6. At first glance, it might look easier to work with strictly λ-twisted D-modules rather than generalized twisted D-modules with monodromy in λ∧. But since λ∧ is flat over the base h∗ unlike the point λ itself, tensor products and ultimately convolution patterns restrict to λ∧ in a less intricate way. Alternatively, one can equip strictly λ-twisted D-modules with the symmetries that control their extension to λ∧. Namely, one can pass to (co)modules for the (co)monad of the adjunction given by restriction and extension of modules along the inclusion of λ into λ∧. But keeping track of this extra Koszul dual structure can be less intuitive than simply working over λ∧. Remark 1.7. Here are some simple observations to help orient the reader. ′ The restriction Hλ∧,µ∧ vanishes unless µ = w · (λ + λ ), for some Weyl group element w ∈ W and ′ ∗ ∗ integral weight λ ∈ Λ ⊂ h . The restrictions Hλ∧,µ∧ and Hλ′∧,µ′∧ are non-canonically equivalent if ′ ′ ′ ′ there are Weyl group elements w, w ∈ W such that λ = w·λ, µ = w ·µ. The restrictions Hλ∧,µ∧ and ′ ′ ∗ ∗ Hλ′∧ ,µ′∧ are canonically equivalent if the difference of parameters is integral λ−λ ,µ−µ ∈ Λ ⊂ h . BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 5

The monoidal structure of H descends to compatible compositions / Hλ∧,µ∧ ⊗ Hµ∧,ν∧ / Hλ∧,ν∧

Moreover, the restriction maps H → Hλ∧,µ∧ are compatible with the above restricted compositions. In particular, the diagonal restriction map H → Hλ∧,λ∧ is a monoidal map of monoidal categories. Remark 1.8. A result of [BN] asserts that for any fixed λ ∈ h∗, the diagonally λ-twisted Hecke cat- egories Hλ,λ = Dλ,λ(B\G/B), Hλ∧,λ∧ = Dλ∧,λ∧ (B\G/B) are both categorified analogues of finite dimensional semisimple Frobenius algebras: they are the values on a point of extended oriented two- dimensional topological field theories. More precisely, they are two-dualizable Calabi-Yau algebra objects in dg categories. Along the way, as a simple application of results of [BN], we will establish the following basic relationship between the above commuting symmetries. Theorem 1.9. There is a monoidal equivalence

H≃ EndD(G)(P) between the Hecke category H and D(G)-linear endofunctors of P. Remark 1.10. Specializing over subsets of the base h∗ immediately provides analogous assertions for monodromic D-modules with prescribed monodromies. Observe that the adjoint G-action on g naturally equips Ug-mod with the structure of de Rham G-category. As was emphasized in [BD] (see also [FG]), the localization and global sections functors commute with the corresponding de Rham G-actions. Therefore, or as can be verified independently, the Weyl functor W must be represented by an object of the Hecke category H which we will also denote by W. It turns out to have a simple geometric description. Lemma 1.11. The universal Weyl sheaf W ∈ H, the Hecke kernel for the Weyl functor, is the sheaf of differential operators on N\G/N with its canonical weakly H-biequivariant structure. Remark 1.12. Differential operators on the stack N\G/N can be calculated by starting with differ- ential operators on G/N and then performing quantum Hamiltonian reduction for the left N-action. Thus one quotients differential operators on G/N by the vector fields generating the left N-action and then takes N-invariants. Theorem 1.13 (Hecke version of localization). The universal Weyl sheaf W is naturally an algebra and coalgebra in H. There are canonical equivalences W PW ≃ Ug-mod ≃P between the categories of W-modules and W-comodules in P and the category of Ug-modules. Altogether, one can view the universal Weyl sheaf W as a categorical idempotent, providing a lift of the idempotent in the groupoid algebra governing descent along the quotient map h∗ → h∗//W . It is clarifying to reverse our momentum and regard it as a family of idempotents as we vary the infinitesimal character. We include here a brief discussion of the two most extreme cases. Remark 1.14. It is worth pointing out first of all that while W ∈ H is the sheaf of differential operators on N\G/N so “large”, its fibers Wλ,µ ∈ Hλ,µ are regular holonomic twisted D-modules on B\G/B so “small”. This is a twisted instance of the general fact that on a quotient stack H\X such that X is smooth and H is affine algebraic acting with finitely many orbits in X, the sheaf of differential operators on H\X is regular holonomic. In fact, any coherent D-module on H\X is regular holonomic. At one extreme, when λ ∈ h∗ is regular, consider the direct sum of the potential targets for localization ∧ ∧ P[λ ] = Lw∈W Pw·λ 6 DAVID BEN-ZVI AND DAVID NADLER and the corresponding Hecke category

∧ ∧ ′ ∧ H[λ ] = Lw,w′∈W Hw·λ ,w ·λ The classical Beilinson-Bernstein theorem asserts that Γ is already a derived equivalence (and in fact t-exact up to a shift) when restricted to any summand. Moreover the classical action of principal series intertwining functors intertwine the different localizations [B]. Thus we can understand the statement of Theorem 1.13 as saying we perform all |W | localizations and keep track of the Weyl group action. Let us spell this out. First, for λ ∈ h∗ generic (with respect to the notion of integrality coming from coroot functionals), we have that ∧ ′ Hw·λ∧,w′·λ∧ ≃Q(λ ), for all w, w ∈ W , where Q(λ∧) denotes the dg category of quasicoherent sheaves on the formal neighborhood of λ ∈ h∗ or equivalently that of [λ] ∈ h∗//W . (Since λ is generic, only one Bruhat double-coset in G supports any bimonodromic modules with parameters in w · λ∧, w′ · λ∧, and all such modules are only constrained by the fact that the parameters must be related by w′w−1.) The monoidal structure of H[λ∧] is simply that of the groupoid algebra of the regular Weyl groupoid / W × W / W ∧ with scalars in the constant tensor category Q(λ ). Furthermore, the restriction W[λ∧] ∈ H[λ∧] of ∧ the universal Weyl sheaf corresponds to the direct sum of the structure sheaf Oλ∧ ∈Q(λ ) in each factor. Thus it provides the categorical analogue of the constant idempotent in the groupoid algebra of W . As expected, its modules and comodules in P[λ∧] are equivalent to a single copy of Pλ∧ . ∗ Leaving behind the generic case, for λ ∈ h regular integral, the Hecke category H[λ∧] now contains all of the combinatorics of Kazhdan-Lusztig theory. A common approach to capture this structure is to focus on the exceptional collections of standard or costandard objects associated to Schubert cells. Their convolutions provide dual realizations of the groupoid algebra of the natural Artin braid group action on the Weyl group / BW × W / W with scalars in the constant tensor category Q(λ∧). In fact, one can embed the regular Weyl groupoid inside the above braid groupoid by choosing appropriate lifts of minimal standard or costandard objects depending on whether multiplication increases or decreases length. With this observation in hand, the restriction W[λ∧] ∈ H[λ∧] corresponds to a direct sum of specified standard and costandard objects giving the constant idempotent of W . Thus as expected, its modules and comodules in P[λ∧] are equivalent to Ug-mod[λ∧] in the form of a single copy of Pλ∧ . Remark 1.15 (Borel-Weil-Bott). The above picture for λ ∈ h∗ regular integral gives a nice framework for understanding the Borel-Weil-Bott theorem. Let us focus on the objects of P[λ∧] given by the shifted line bundles O(w · λ)[ℓ(w)] → G/B, where ℓ(w) denotes the length of w ∈ W . Their global sections are the irreducible G-representation V[λ] placed in degree zero thanks to their original shifts by length. The summands of the idempotent W[λ∧] ∈ H[λ∧] intertwine the shifted line bundles, increasing or decreasing degree depending on what happens to length. Equivalently, we can apply the G-equivariant description of Theorem 1.13 to G-integrable repre- sentations (Ug-mod)G ≃ Rep(G), obtaining an equivalence with W-comodules in (P)G ≃ D(pt/N)H ≃ Rep(H). The comonad W is thereby identified with the Borel-Weil-Bott comonad on Rep(H), given by the standard parabolic induction/restriction adjunction. Finally, as λ ∈ h∗ specializes to become singular, the geometry becomes more interesting, ul- timately reflecting the nontrivial scheme structure of the quotient map h∗ → h∗//W along its ramification locus . In the most singular case λ = 0 ∈ h∗, recall that the target of localization P[0∧] ≃ P0∧ is dramatically different from the singular category Ug-mod[0∧]. The Hecke category H[0∧] = H0∧,0∧ controlling this difference can be viewed in dual standard and costandard ways as a categorical analogue of the Artin braid group algebra (rather than groupoid algebra as above). BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 7

To concretely discuss the restriction W[0∧] ∈ H[0∧], let us focus on its convolution against the object O(−ρ) ∈P[0∧] given by the canonical bundle of G/B. One can calculate that

T = W[0∧] ∗ O(−ρ) is the tilting sheaf given by the projective cover of the skyscraper at the closed Schubert cell. It is well known that T governs the singular category (as in the work of Soergel [S]); for example, the kernel of the global sections functor to Ug-mod[0∧] is its right-orthogonal.

1.3. Comparison with K-theory. We would like to interpret Theorem 1.13 as a refinement of results in K-theory which are closer in form to the naive paraphrase: “the category of Ug-modules is equivalent to the Weyl group invariants in the category of all monodromic D-modules on G/N”. First, let us recall K-theory versions of the Weyl character formula and Borel-Weil-Bott theorem from [BH] (where the context is equivariant KK-theory and the results are closely related to the Baum-Connes conjecture for Lie groups). To proceed in this setting, let Gc ⊂ G be a maximal com- pact subgroup, and Tc ⊂ Gc a maximal torus, so that we have G/B ≃ Gc/Tc. Natural morphisms relate the representation ring of Gc and the equivariant K-theory of the flag manifold ∗ o / ∗ ∆ : KGc (pt) KGc (Gc/Tc):Γ where Γ is the equivariant index (Borel-Weil-Bott construction) and ∆ is given by pullback followed · ∗ by multiplication by the virtual bundle Ω . Note that for a representation V ∈ KGc (pt), the virtual bundle ∆(V ) can be identified with the complex that fiberwise computes n-homology (where n is the unipotent radical of the stabilizer of a point in G/B). In other words, the virtual bundle ∆(V ) is the K-theory image of the Beilinson-Bernstein localization of V . The Weyl group W acts (non-holomorphically) from the right on Gc/Tc, and the main theorem of [BH] calculates (in the context of equivariant KK-theory) that Γ ◦ ∆ = |W | Id, and when restricted to the W -invariants, ∆ ◦ Γ= |W | Id . Identifying the virtual character of Ω· with the Weyl denominator, one recovers the Weyl character formula. In fact, the main theorem of [BH] calculates that ∆ ◦ Γ is given by the standard idempotent in the group algebra of W (the projector to the trivial representation), realized as a sum of standard intertwining operators. Note that the group algebra CW is a Frobenius algebra, and so W -invariants and coinvariants are both identified with the summand given by the image of the standard idempotent. In our present categorical setting, the action of the Weyl group on equivariant K-theory is replaced by the action of the Hecke category H = Dbimon(N\G/N) on the category P = Dmon(G/N). For ∗ fixed integral λ ∈ h , the corresponding Hecke category Hλ,λ = Dλ,λ(B\G/B) of Kazhdan-Lusztig theory has Grothendieck group the group algebra CW . But in fact standard bases of Hλ,λ provide actions on categories not of the Weyl group W but of the corresponding Artin braid group. The Frobenius monad W is the categorified analogue of the standard idempotent, with W-modules playing the role of W -invariants and W-comodules that of W -coinvariants.

1.4. Classical limit. The preceding description of Beilinson-Bernstein localization has a natural classical analogue for quasicoherent sheaves on the Springer resolution. Its interpretation as a form of proper descent becomes evident in this setting. We will proceed in the context of derived algebraic geometry, and in particular that of perfect stacks as introduced in [BFN]. In particular, let π : X → Y denote any morphism of perfect stacks, and X ×Y X the corresponding derived fiber product. First, consider the symmetric monoidal dg category Q(Y ) of quasicoherent sheaves equipped with tensor product, and its natural module dg category Q(X) under the pullback π∗. The dg category Q(X ×Y X) is monoidal with respect to convolution, acts on Q(X) by endofuntors with a natural Q(Y )-linear structure, thus leading to a monoidal equivalence ∼ / Q(X ×Y X) / EndQ(Y )(Q(X)). 8 DAVID BEN-ZVI AND DAVID NADLER

∗ The adjunction (π , π∗) on quasicoherent sheaves associated to π : X → Y defines a comonad ∨ ∗ ∨ T = π π∗ acting on Q(X), and it is easy to see that T is represented by the coalgebra object

A = OX×Y X ∈Q(X ×Y X). Observe that A is simply the groupoid coalgebra (functions on the groupoid with convolution coproduct) for the descent groupoid X ×Y X acting on X. When π is faithfully flat, then descent holds by [L4, 7], providing an equivalence Q(Y ) ≃Q(X)A. Such flatness will not hold in our setting, but we will know that π∗ is conservative and cocontinuous and thus descent holds by the Barr-Beck-Lurie theorem. Next, let us assume that π is proper and surjective on field points. Consider the symmetric monoidal dg category Q!(Y ) of ind-coherent sheaves equipped with !-tensor product, and its natural ! ! ! module dg category Q (X) under the pullback π . The dg category Q (X ×Y X) is monoidal with respect to convolution, acts on Q!(X) by endofuntors with a natural Q!(Y )-linear structure, thus leading to a monoidal functor which is typically not an equivalence

! / ! Q (X ×Y X) / EndQ!(Y )(Q (X)).

! The adjunction (π∗, π ) on ind-coherent sheaves associated to π : X → Y defines a monad ! ! T = π π∗ acting on Q (X), and it is easy to see that T is represented by an algebra object

! ! A = ωX×Y X/X ∈Q (X ×Y X) .

Observe that A! is the groupoid algebra (relative volume forms on the groupoid with convolu- tion product) for the descent groupoid X ×Y X acting on X. The proper descent theorem of [P, Proposition A.2.8] and [G2, 7.2.2] provides an equivalence

! ! Q (Y ) ≃Q (X)A! . Finally, let us now assume that X and Y are both smooth, so that we have canonical identifications Q(X) ≃ Q!(X) and Q(Y ) ≃ Q!(Y ). Moreover, let us assume that π is proper, surjective on field points, and crepant, or in other words, Calabi-Yau of dimension zero in that we are given a trivialization of its relative dualizing sheaf and thus an identification π∗ ≃ π!. In particular, this implies that π∗ ≃ π! are simultaneously continuous and cocontinuous, as well as conservative. Then ∨ T ≃ T is a Frobenius monad, the groupoid algebra ωX×Y X/X ≃ OX×Y X is a Frobenius algebra object, and there are equivalences

T Q(X) ≃Q(Y ) ≃Q(X)T . The prime example for these restrictive hypotheses is the Grothendieck-Springer simultaneous resolution π : g˜ → g with descent groupoid the Grothendieck-Steinberg variety. The resulting descent picture is precisely the classical limit of Beilinson-Bernstein localization. We develop the details of this in Section 2, in particular its various specializations over different regions in the adjoint quotient.

1.5. Demazure descent. In the Appendix, Section 4, we present a different picture of our results as a categorical form of the action of Demazure operators, inspired by [AK1, AK2]. Namely we show how the language of ind-coherent sheaves on inf-schemes [GR] allows a simple uniform description of the monads controlling both the classical Grothendieck-Springer descent and the quantum Beilinson- Bernstein localization. They both describe arise as images of the Demazure monad d, the algebra in the Demazure Hecke category D = (Q!(B\G/B), ∗) whose action on a D-module projects onto D- invariants. As a result we can interpret both the classical and quantum equivalences as a categorical form of taking invariants for the Demazure divided-difference operators. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 9

1.6. Categorical context. We will work throughout in the language of derived algebraic geometry following [L1, L2]; we refer the reader to [BFN, G2] for some gentle discussion of this context and its basic tools. We will work throughout over the complex numbers C. The words “category” and “dg category” will stand for either a C-linear pre-triangulated dg category or a C-linear stable ∞-category, and we refer to [G2] for a general homotopical treatment of dg categories and [Co] for an explicit comparison of the homotopy theories of dg categories and stable ∞-categories. Such categories fit into two related contexts: 1) DGC the symmetric monoidal ∞-category of stable presentable C-linear dg categories with morphisms colimit preserving functors, and 2) dgC the symmetric monoidal ∞-category of small stable idempotent-complete C-linear dg- categories with morphisms exact functors. We say that a functor between dg categories is continuous if it preserves coproducts, exact if it preserves zero objects and finite colimits, and compact if it preserves compact objects. Taking ind-objects defines a faithful symmetric monoidal functor Ind : dgC → DGC. It admits a left inverse on the subcategory of compact functors given by passing to compact objects. Any op category C ∈ dgC is dualizable with dual the opposite category C ∈ dgC. Thus any category Ind C ∈ DGC is dualizable with dual the restricted opposite category (Ind C)∨ = Ind(Cop) ∈ DGC. We will make heavy use of the theory of adjunctions, monads and comonads, and the Barr-Beck- Lurie theorem [L2]. Given a monad T or comonad T ∨ on a category C, we denote by

T ∨ CT = ModT (C) C = ComodT ∨ (C) the respective category of module objects or comodule objects. Working Definition 1.16 (Frobenius monads). The adjunctions appearing in this paper are am- bidextrous: the left adjoint is canonically the right adjoint of its right adjoint and vice versa. Thus their compositions provide endofunctors with compatible monadic and comonadic structures. We refer to an endofunctor arising in this way as a Frobenius monad, though we do not independently formalize this notion in the ∞-categorical setting (see [St, La] for the notion in the discrete setting). A natural context for considering Frobenius monads is the cobordism hypothesis with singulari- ties [L3], where the notion of ambidextrous adjunction captures an oriented domain wall between topological field theories (as explained pictorially by [La] in the discrete setting). 1.7. Acknowledgements. We would like to thank Jonathan Block and Nigel Higson for many inspiring conversations on representations of Lie groups, and in particular for asking what form Beilinson-Bernstein localization should take without specified infinitesimal character. Our collab- oration with Block and Higson was funded by the SQuaRE “The Baum-Connes Conjecture and Geometric Representation Theory” at the American Institute of Mathematics, and we are indebted to AIM for its support and hospitality. We would also like to thank Tom Nevins for helpful discus- sions of localization and the paper [MN]. DBZ is partially supported by NSF grant DMS-1103525. DN is partially supported by NSF grant DMS-0600909.

2. Grothendieck-Springer resolution 2.1. Recollections. We recall here the construction of the Grothendieck-Springer resolution of a reductive Lie algebra and the Steinberg variety. Let G be a complex reductive group. For a Borel subgroup B ⊂ G, let N ⊂ B denote its unipotent radical, and H = B/N the universal Cartan torus. Denote by g, b, n, and h the respective Lie algebras. Let W denote the Weyl group of g, and c = h//W the affine quotient. Fix a G-invariant inner product on g to obtain an identification g∗ ≃ g. Let B = G/B be the flag variety, and Be = G/N the base affine space. The natural projection B→Be is a G-equivariant torsor for the natural H-action on Be. Such torsors correspond to homo- morphisms B → H, and the base affine space B→Be corresponds to the tautological homomorphism B → B/N ≃ H. 10 DAVID BEN-ZVI AND DAVID NADLER

The cotangent bundle T ∗B→B classifies pairs of a Borel subalgebra b ⊂ g together with an element v ∈ (g/b)∗ ≃ n. The moment map for the natural G-action is given by the projection

∗ / ∗ µB : T B / g ≃ g µB(b, v)= v

∗ The cotangent bundle T Be → Be classifies pairs of an element xb ∈ Be over a Borel subalgebra b ⊂ g together with an element v ∈ (g/n)∗ ≃ b. The moment map for the natural G × H-action is given by the projection

µ e : T ∗B / g∗ × h∗ ≃ g × h µ e(x , v) = (v, [v]) B e B b where [v] ∈ h = b/n denotes the image of v ∈ b. The cotangent bundles are related by Hamiltonian reduction along the H-action

T ∗B = T ∗(B/H) ≃ (p ◦ µ e)−1(0)/H e h B where ph : g × h → h denotes projection. We will be interested in the quotient g = (T ∗B)/H classifying a Borel subalgebra b ⊂ g together e e g n ∗ b ∗ with an element v ∈ ( / ) ≃ . The moment map for the G-action on T Be descends to the Grothendieck-Springer resolution

µe : g / g µe(b, v)= v g e g

The Grothendieck-Springer resolution µe : g → g is projective, generically finite and G-equivariant. g e Moreover its relative dualizing sheaf is canonically trivial (and hence the same is true of any base change of µeg). To see this last claim, recall we have fixed a G-invariant inner product on g to obtain a G-equivariant identification g ≃ g∗. This provides an isomorphism g ≃ h ⊕ n ⊕ n∗, and thus in turn induces an isomorphism of lines ∧dim gg ≃ ∧dim hh. Thus a trivialization of ∧dim hh trivializes the canonical bundle of g. Furthermore, the partial moment map g → h is smooth with symplectic e fibers, hence a trivialization of ∧dim hh also trivializes the canonical bundle of g. e

2.1.1. The Grothendieck-Steinberg variety. The Grothendieck-Steinberg variety is the fiber product g×g g classifying triples of a pair of Borel subalgebras b1, b2 ⊂ g together with an element v ∈ b1 ∩b2. e(Notee here the derived fiber product coincides with the naive fiber product.) It has a microlocal interpretation involving the double coset spaces

Z = B\G/B ≃ G\B×B Ze = N\G/N ≃ G\B×e Be Namely, returning to the identification

g = (T ∗B)/H B = G/N e e e we have a similar identification

∗ G\(g ×g g) ≃ (T Z)/H × H e e e or after de-equivariantization

∗ g ×g g ≃ (pt × T Z)/H × H e e BG e

From this viewpoint, the fiber product in the construction of g ×g g arises as the moment map e ∗e ∗ equation for Hamiltonian reduction along the diagonal G-action for T B×e T Be. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 11

2.2. Descent pattern. Given a stack X, we will write Q(X) for the symmetric monoidal dg category of quasicoherent sheaves on X. All of the stacks X in play will be perfect in the sense of [BFN] and so the basic structure results for Q(X) will apply. The natural G-action on B = G/N and induced Hamiltonian G-action on g endows Q(g) with two e e e important compatible structures: an algebraic action of G as formalized by a Q(G)-module structure under convolution, and a Q(g)-module structure via pullback under µg˜ : g˜ → g. Altogether, these structures are captured by considering Q(g˜/G) as a Q(g/G)-module via the pullback under the induced map g˜/G → g/G. We will identify the symmetries of Q(g/G) preserving this structure. e First, the endomorphisms of Q(g) asa Q(g)-module are given (by [BFN]) by integral transforms e with kernels on the fiber product: we have a monoidal equivalence ∼ / ∗ Φ: Q(g ×g g) / End (Q(g)) Φ (−)= p (K⊗ p (−)) e e Q(g) e K 2∗ 1 where p ,p : g ×g g → g denote the projections. In particular, the identity functor corresponds to 1 2 e e e the integral kernel ∆e Oe ∈Q(g ×g g) obtained by pushforward along the diagonal map g∗ g e e / ∆e : g / g ×g g g e e e Similarly (again by [BFN]), the endomorphisms of Q(g) asa Q(G)-module are given by integral e transforms with G-equivariant kernels on the product End (Q(g)) ≃Q(G\(g × g)). Q(G) e e e Finally (again by [BFN]), the endomorphisms of Q(g) as a Hamiltonian G-category, or in other words, the endomorphisms of Q(g˜/G) as a Q(g/G)-module,e are given by integral transforms with equivariant kernels on the fiber product

End (Q(g/G)) ≃Q(G\(g ×g g)) Q(g/G) e e e or in other words, the monoidal dg category of equivariant quasicoherent sheaves on the Grothendieck- Steinberg variety. 2.2.1. Descent (co)monad. Consider the standard adjunction and Grothendieck duality adjunction on stable dg categories of quasi-coherent sheaves

∗ o o ! µe : Q(g) / Q(g): µeg∗ µeg∗ : Q(g) / Q(g): µe g e e g ! ∗ Since the relative dualizing sheaf of µeg is canonically trivial, we have a canonical equivalence µeg ≃ µeg, but we distinguish them to avoid confusion. By the projection formula, we can view these as adjunctions of Q(g)-module categories. The adjunctions are also evidently G-equivariant, and in total preserve the Hamiltonian G-structure. ! Let T = µegµeg∗ denote the resulting monad, or in other words, algebra object in the monoidal ∨ ∗ category of linear endomorphisms EndQ(g)(Q(g)). Likewise, let T = µegµeg∗ denote the resulting ∗ e ! ! comonad. Since µeg is conservative (µeg is surjective) and continuous, µeg is as well; and since µeg ∗ is cocontinuous, µeg is as well. Thus the Barr-Beck theorem provides canonical identifications of Hamiltonian G-categories ∨ Q(g) ≃Q(g) ≃Q(g)T e T e By base change and standard identities, the comonad T ∨ is given by tensoring with the sheaf ∗ of functions Oeg×geg = p1Og˜ with its canonical coalgebra structure. Likewise, the monad T is given ! by tensoring with the relative dualizing sheaf ωeg×geg/eg = p1Og˜ with its canonical algebra structure. ∨ ∗ ! Note that the identification of underlying functors T ≃ T reflects the equivalence p1 ≃ p1 which devolves by base change from the original ambidextrous adjunction of µg˜. 2.3. Specified eigenvalues. We describe here the above descent picture to distinguished loci within g. 12 DAVID BEN-ZVI AND DAVID NADLER

2.3.1. Regular locus. Over the open regular locus gr ⊂ g, we have a fiber square

µer r r g / r g = g ×g g / g e e

  / h π c = h//Wg Thus descent over gr ⊂ g is simply the base change of descent over the geometric invariant theory quotient. 2.3.2. Regular semisimple locus. Over the open regular semisimple locus grs ⊂ g, we have a fiber square µers rs rs g / rs g = g ×g g / g e e

  r / r r h / c = h /Wg πr In other words, we have a free W -action and quotient identification rs / rs rs rs Wg × g / g g ≃ g /W e e e Thus descent over grs ⊂ g is simply equivariance for the Weyl group W .

2.3.3. Nilpotent cone. Over the nilpotent cone N = g ×c {0}⊂ g, we have the base change / µe : g = g ×c {0} / N g0 e0 e where g is a non-reduced scheme with underlying reduced scheme the usual Springer resolution e0 ∗ b g g b ∗ n Ne ≃ T B classifying a Borel subalgebra ⊂ together with an element v ∈ ( / ) ≃ . By construction, descent along µeg0 is governed by the restricted algebra object

Oeg0×N eg0 ≃ Oeg×geg|N Remark 2.1. To work instead with the traditional reduced Springer resolution

µ e : N / N N e we must pass to ind-coherent sheaves. In applying the Barr-Beck theorem, we use that the adjunction

o ! µeg0∗ : Q(g0) / Q(N ): µe e g0 comprises a compact left adjoint and hence continuous right adjoint. But in contrast, this does not hold for the adjunction o ! µ e : Q(N ) / Q(N ): µ e N ∗ e / N

For example, Ne is smooth, hence all skyscraper sheaves on it are compact, but N is singular, hence many skyscraper sheaves on it are not compact. Rather we must pass to ind-coherent sheaves and work with the analogous adjunction

! o ! ! µ e : Q (N ) / Q (N ): µ e N ∗ e / N Here by construction, the adjunction comprises a compact left adjoint and hence continuous right ! ! ! adjoint, and thus µ e exhibits Q (N ) as monadic over Q (N ). N e 3. Beilinson-Bernstein localization Now we will repeat the constructions of the previous section after quantization of the natu- ral Poisson structures, that is, after turning on the noncommutative deformation from cotangent bundles to D-modules. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 13

3.1. Quantization. Let Ug be the universal enveloping algebra of g, and Zg ⊂ Ug the Harish- Chandra center. Let Ug-mod denote the dg category of Ug-modules. Informally speaking, Ug-mod consists of noncommutative modules on the Poisson manifold g ≃ g∗. Let Ug -perf ⊂ Ug-mod denote the small stable full dg subcategory of perfect modules so that Ug-mod ≃ Ind(Ug -perf). Lemma 3.1. There are canonical equivalences Ug -perf ≃ Ug -perfop Ug-mod ≃ Ug-mod∨ Proof. First, viewing Ug as a Ug-bimodule, we define the duality identification op ∼ / op / Ug -perf / Ug -perf M ✤ / HomUg(M, Ug[dim g]) Now let gop denote the vector space g with the opposite Lie bracket

[·, ·]gop = −[·, ·]g The negation map g → g, v 7→ −v provides a canonical isomorphism g ≃ gop and hence a canonical isomorphism Ug ≃ Ugop. This establishes the first assertion, and the second then follows from the standard identity Ug-mod∨ ≃ Ind(Ug -perfop) 

Remark 3.2. The equivalence Ug -perf ≃ Ug -perfop is a twisted form of the Serre duality equivalence Perf(g∗) ≃ Perf(g∗)op. Namely, the former invokes the negation on the vector space g∗ while the latter does not.

Let D e ∈ Q(B) denote the sheaf of differential operators on B. Let Dmon(B) denote the dg B e e e category of weakly H-equivariant D-modules on Be. Its objects are H-equivariant quasicoherent sheaves on B equipped with a compatible H-equivariant action of D e. Informally speaking, Dmon(B) e B e consists of noncommutative modules on the Poisson manifold g = (T ∗B)/H. e e Let DeB ∈ Q(B) denote the sheaf of H-invariant differential operators on Be. Then Dmon(Be) is equivalently the dg category of quasicoherent sheaves on B equipped with a compatible action of DB. e c Let Dmon(Be) ⊂ Dmon(Be) denote the full dg-subcategory of coherent modules so that Dmon(Be) ≃ c Ind(Dmon(Be)). Lemma 3.3. Verdier duality provides canonical equivalences

c c op ∨ Dmon(Be) ≃ Dmon(Be) Dmon(Be) ≃ Dmon(Be) Proof. The first assertion is Verdier duality, and the second follows from the standard identity ∨ c op  Dmon(Be) ≃ Ind(Dmon(Be) ). Remark 3.4. When keeping track of additional structures, it is useful to keep in mind the opposite op −1 base affine space Be = N\G. The inverse map G → G, g 7→ g provides a canonical isomorphism op Be ≃ Be . Consider the localization adjunction / ∗ Ug o / mon γ : -mod o D (Be): γ∗

∗ e e γ (M)= DB ⊗Ug M γ∗(M) = Hom(DB, M) Informally speaking, this is a quantization of the standard adjunction for the Grothendieck-Springer resolution µe : g → g. g e 14 DAVID BEN-ZVI AND DAVID NADLER

Proposition 3.5. The right adjoint γ∗ is continuous and compact, and hence itself admits a con- tinuous right adjoint γ!. Furthermore, there is a canonical identification γ! ≃ γ∗. ∗ Proof. γ∗ is continuous since its left adjoint γ preserves compact objects. Moreover γ∗ itself preserves compact objects, indeed it’s clear that coherent D-modules get taken to finitely presented Ug-modules. We also have a compatibility with duality of the form

c c op Dmon(Be) ≃ Dmon(Be)

op γ∗ γ∗   Ug -perf ≃ Ug -perfop

! ∗ To check the equivalence γ ≃ γ , we apply the Rees construction. Namely, we have a Gm- 1 ∗ ! equivariant A -family of adjunctions (γ ,γ∗,γ ) degenerating to the classical version of the local- ization functors, associated to the Gm-equivariant Poisson map which is the Grothendieck-Springer ∗ ! resolution µeg : g → g. We need to show the canonical map γ → γ is an equivalence, i.e., its cone vanishes. It’s enoughe to check this when applied to the Rees construction of Ug itself. At ~ = 0, the cone vanishes by the Calabi-Yau property of µeg. Hence it vanishes for all ~ by semicontinuity.  3.1.1. Linearity. Let Zg ⊂ Ug be the Harish-Chandra center, and Uh the universal enveloping algebra of h. We also have the canonical embedding Zg ⊂ Uh as the ρ-shifted Weyl invariants. Ug Zg mon Uh Zg Observe that -mod is naturally -linear, and D (Be) is naturally -linear and hence - linear. Lemma 3.6. The adjunctions

∗ / / ! Ug o mon mon o Ug γ : -mod o D (Be): γ∗ γ∗ : D (Be) o -mod : γ are naturally Zg-linear. Uh Proof. This follows from the compatibility between the action → Γ(DfB) and the Harish-Chandra  .[homomorphism Zg ֒→ Uh, see [BMR1, Lemma 3.1.5 3.2. Beilinson-Bernstein localization and the nil-Hecke algebra. In this section we describe how one can easily apply Barr-Beck-Lurie to identify representations of g with modules over the nil-Hecke algebra inside globally generated monodromic D-modules on Be. This realization won’t be used in what follows, where we apply Barr-Beck-Lurie in the opposite direction. Let Ug Ug Uh f = ⊗Zg denote the extended enveloping algebra. We will use the following proposition of Miliˇci´cdescribing monodromic differential operators on the basic affine space: Proposition 3.7. [Mi1, Lemma 3.1], [Mi2, Theorem C.6.5] There is an equivalence

γ (D e) ≃ Ug ∗ B f i.e. we have an isomorphism of global sections H Γ(G/N, D e) = Γ(G/B, D) ≃ Ug B e f and vanishing of higher cohomologies i R Γ(G/B, De)=0 (i> 0). Proposition 3.8. Localization and global sections induce an equivalence glob mon Ug D (Be) ≃ f-mod, where the left hand side is the category of globally generated monodromic D-modules, i.e., the category generated by the sheaf of differential operators. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 15

Proof. The functor γ∗ has both left and right adjoints. Thus to apply the Barr-Beck-Lurie theorem (in the monadic form) we just need to ensure conservativity. This can be always be achieved formally in the setting of stable categories by killing the objects that are sent to zero by the given functor, i.e., passing to the left orthogonal of the subcategory of γ∗-null objects. Equivalently, we pass to the category generated by the image of the left adjoint γ∗ – in other words the category of globally ∗ generated objects. Finally the monad γ∗γ itself is identified by Proposition with the extended enveloping algebra, as desired.  The nil-Hecke algebra of Kostant and Kumar H is the subalgebra of endomorphisms of C[h∗] generated by Sym(h) = C[h∗] and the Demazure, or divided-difference, operators associated to simple reflections σi ∈ W with associated simple roots αi

Ai = (1 − σi)/αi. It has the interpretation as the algebra which controls descent along the map z : h∗ → h∗//W from the dual Cartan to its coarse quotient by the Weyl group: namely, there is an equivalence of categories Q(h∗//W ) ≃ Q(h∗)H (see e.g. [?], which proves the corresponding statement for the coarse quotient of the reflection representation by any Coxeter group). Hence applying the descent Ug H Ug to -modules we can identify them with -modules in f-modules, and hence from Proposition 3.8 we find the following: Corollary 3.9. There is an equivalence of categories glob H mon Ug (D (Be) ) ≃ -mod where the nil-Hecke algebra acts on the left hand side through its Uh-linearity. 3.3. Symmetries. We next introduce quantum analogues of the previously encountered Hamil- tonian G-actions. Following [BD, FG], by a de Rham G-category, we mean a dg category with an algebraic G-action that is infinitesimally trivialized, or in other words, the induced action of the formal group of G is trivialized. This can be formalized by saying that a dg category is a module for the monoidal dg category D(G) of D-modules on G under convolution. The primary examples are the dg category D(X) of D-modules on a G-variety X, and the dg category Ug-mod with its conjugation G-action. These can be unified by considering more ′ generally D(G)-modules of the form DG′-mon(X) where X is a G × G -variety, and we take weakly G′-equivariant D-modules. In particular, for G = X = G′ with G acting on the left and G′ on the right, we have the D(G)-linear equivalence Ug-mod ≃ DG′-mon(G). (Since the action is free and transitive, we can trivialize the underlying quasi-coherent sheaf of a weakly equivariant D-module, and then realize invariant sections as a module over invariant vector fields.) Informally speaking, g∗ ∗ mon this is a quantum analogue of the identification ≃ (T G)/G. Similarly, D (Be) comes equipped with a natural D(G)-module structure. Remark 3.10. Within the stable setting, there are two equivalent dual formulations of a de Rham G-category. By definition, we have taken de Rham G-category to mean a module for the monoidal dg category D(G) of D-modules on G under convolution. But one can observe that for X a smooth variety, D(X) is dualizable as a plain dg category. Furthermore, it is self-dual so that for maps f : X → Y of smooth varieties, pullback is dual to pushforward (by the projection formula). Thus a de Rham G-category could equivalently be taken to mean a comodule for D(G) equipped with its coconvolution coalgebra structure. The following is evident from the constructions. Lemma 3.11. The adjunctions

∗ / / ! Ug o mon mon o Ug γ : -mod o D (Be): γ∗ γ∗ : D (Be) o -mod : γ are naturally D(G)-linear. 16 DAVID BEN-ZVI AND DAVID NADLER

Consider the stack Ze = N\G/N. Let Dbimon(Ze) denote the dg category of H × H-weakly equivariant D-modules on Ze. Informally speaking, Dbimon(Ze) consists of noncommutative modules on the Grothendieck-Steinberg stack G\(g ×g g). e e Convolution equips Dbimon(Ze) with a natural monoidal structure, and Dmon(Be) with a natural right Dbimon(Ze)-module structure commuting with its natural left D(G)-module structure. This is a quantum analogue of the convolution pattern for sheaves on g ×g g acting by integral transforms on sheaves on g respecting the Hamiltonian G-structure. Thee followinge is the quantum analogue e of the result quoted from [BFN] earlier that such integral transforms are precisely the symmetries respecting the Hamiltonian G-structure. Theorem 3.12. Convolution provides a monoidal equivalence ∼ / Φ: Dbimon(Ze) EndD(G)(Dmon(Be))

Proof. Let us begin by forgetting the D(G)-module structure of Dmon(Be). Then by a monodromic version of [BN, Theorem 1.14], we have a monoidal equivalence ′ ∼ / ′ ∗ Φ : Dbimon(B×e Be) End(Dmon(Be)) Φ (K)(−)= p2∗(K⊗ p1(−)) where p1,p2 : B×e Be → Be denote the projections. Returning D(G)-module structures to the picture, Φ′ is evidently D(G)-linear by standard iden- tities. Moreover, D(G)-linear endomorphisms of Dmon(Be) are simply the invariants

EndD(G)(Dmon(Be)) = HomD(G)(D(pt), End(Dmon(Be)) By descent along pt → BG, the invariants can be calculated as comodules

OG HomD(G)(D(pt), End(Dmon(Be)) ≃ End(Dmon(Be)) for the canonical coalgebra OG ∈ D(G) given by the structure sheaf. On the other hand, by another application of descent, OG-comodules in Dbimon(Be × Be) are precisely G-equivariant bimonodromic D-modules on Be × Be. Now the theorem follows from the  identification Ze ≃ G\(B×e Be). 3.4. Universal Weyl sheaf. Recall the adunctions

∗ / / ! Ug o mon mon o Ug γ : -mod o D (Be): γ∗ γ∗ : D (Be) o -mod : γ with ambidextrous identification γ! ≃ γ∗. ! Let T = γ γ∗ denote the resulting monad, or in other words, algebra object in the monoidal ∨ ∗ category of endomorphisms EndD(G)(Dmon(Be)). Likewise, let T = γ γ∗ denote the resulting comonad. Lemma 3.13. γ! is conservative, and hence γ∗ is as well. Proof. From Proposition 3.2 we find ! e Hom(DB,γ (M)) ≃ Hom(Ug ⊗Zg Uh,M) Since Uh is free (of rank |W |) over Zg, we conclude that this Hom space vanishes if and only if M ≃ 0, hence M 6≃ 0 implies γ!(M) 6≃ 0.  Since γ! is continuous and conservative and γ∗ is cocontinuous and conservative, the Barr-Beck theorem provides canonical identifications

mon Ug ∨ mon ModT (D (Be)) ≃ -mod ≃ ComodT (D (Be)) We would like to explicitly describe the integral kernel giving rise to T ≃ T ∨ under the equivalence ∼ / Φ: Dbimon(Ze) EndD(G)(Dmon(Be)) BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 17

Definition 3.14. bimon The universal Weyl sheaf W ∈ D (Ze) is the sheaf of differential operators on Ze with its canonical H × H-weakly equivariant structure. Remark 3.15. Let us spell out this definition. By quantum Hamiltonian reduction (and under the identification Ze ≃ G\(G/N × G/N)), the pullback of W along the natural quotient map r : G/N × G/N / N\G/N is the G-strongly equivariant D-module ∗ r W = DG/N×G/N /(g) where (g) ⊂ DG/N×G/N is the left ideal generated by vector fields arising from the diagonal G-action on G/N × G/N. Thus W is the quantum analogue of the structure sheaf of G\(g ×g g). e e Remark 3.16. One can write down an explicit Frobenius algebra structure on W but its construction is an explicit unwinding of that given by the adjunctions of Theorem 3.17 below. In parallel with the commutative case, the identification of W as the integral kernel giving rise to T ≃ T ∨ can be viewed as a microlocal version of G-equivariant base change along the diagram

D (B× B) H×H e g e ♣♣ ◆◆◆ ♣♣♣ ◆◆◆ ♣p1∗ ∗ ◆ ♣♣ p2 ◆◆ ♣w ♣♣ ◆◆ Dmon(B) Dmon(B) eg e ❖❖ ∗ ♦ ❖❖ γ γ∗ ♦♦ ❖❖❖ ♦♦♦ ❖❖❖ ♦♦♦ ❖❖ ♦w ♦♦ Ug-mod This is formalized in the argument for the following assertion. Theorem 3.17. The monoidal equivalence ∼ / Φ: Dbimon(Ze) EndD(G)(Dmon(Be)) takes the universal Weyl sheaf W to the endofunctor T ≃ T ∨. Thus W inherits the structures of algebra and coalgebra in Dbimon(Ze), and we have equivalences W mon Ug mon D (Be)W ≃ -mod ≃ D (Be) . Proof. On the one hand, given an object M ∈ Dmon(Be), we find ∗ T (M) ≃ γ γ M≃ D ⊗Ug Hom e (D , M) ∗ eB DB eB Thus it is the Ug-coinvariants of the intermediate functor

T ′(M)= D ⊗ Hom e (D , M) eB DB eB Moreover, under the identification ′ ∼ / Φ : Dbimon(B×e Be) End(Dmon(Be)) ′ ⊠ D ⊠ observe that T corresponds to the integral kernel DeB (DeB). (Indeed the integral kernel M N represents the functor N ⊗ Hom(DM, −)). Moreover, the Calabi-Yau structure on G/N allows us to identify DeB with its dual (with suitable H-monodromic structure). On the other hand, as discussed Ug ⊠ above, W is simply the -coinvariants of DeB DeB. Thus since all functors are continuous, taking Ug-coinvariants can be equivalently performed on the integral kernel or on the result of the integral transform. All of the above arguments are manifestly G-equivariant and so descend to give the assertion.  18 DAVID BEN-ZVI AND DAVID NADLER

3.5. Specified infinitesimal character. Recall that the commutative algebra Uh ⊗ Uh = O(h∗ × h∗ bimon ) acts by central endomorphisms on D (Ze). The action factors through the closed subscheme Γ ⊂ h∗ × h∗ given by the union of the graphs of Weyl group elements

∗ ∗ Γ= `w∈W Γw Γw = {(λ, wλ) ∈ h × h }

To better understand the universal Weyl sheaf W ∈ Dbimon(Ze), let us restrict one of its mon- odromies and calculate its resulting fiber. The composite projection to either factor  Γ / h∗ × h∗ / h∗ is a finite flat map. All of what follows is symmetric in the two projections, so let us focus on the projection to the second factor. First, let us identify the fibers of Dbimon(Ze) along the projection to the second factor. For simplicity, let us also forget the H-weak equivariance along the first factor, i.e. consider D-modules on N\G/N which are H-weakly equivariant on the right. The fiber of this category over λ ∈ h∗ is canonically equivalent to the dg category Dλ(N\B) of N-strongly equivariant λ-twisted D-modules on the flag variety B. This in turn is the full subcategory of those λ-twisted D-modules on B that are locally constant along Schubert cells. To recover the corresponding fiber of Dbimon(Ze) we should then reimpose weak H-equivariance on the right. Now let us identify the corresponding fiber

Wλ ∈ Dλ(N\B) of the universal Weyl sheaf. This is a regular holonomic λ-twisted D-module on the flag variety B locally constant along Schubert cells. For concreteness, we will consider several specific cases: (1) λ generic (regular and not at all integral), (2) λ regular and integral, and (3) λ = 0 trivial. (1) When λ is generic, we have a direct sum decomposition

Wλ ≃⊕w∈W Wλ,wλ Each summand admits the description as a standard or equivalently costandard extension off of a Schubert cell

Wλ,wλ ≃ jw!Oλ,w ≃ jw∗Oλ,w where jw : Bw → B is the inclusion of the w-Schubert cell for w ∈ W , and Oλ,w is the λ-twisted structure sheaf of Bw. (2) Suppose λ is regular and integral, and let w0 ∈ W be the Weyl group element such that λ = w0λ0 for dominant λ0. We can tensor by the line bundle O(λ) to obtain an identification ∼ / / D(B) / Dλ(B) M ✤ / M ⊗ O(λ)

This is convenient since we will describe Wλ in terms of the natural monoidal structure on the dg category D(B\B) of B-equivariant D-modules on the flag variety B. Namely, under the above identification, we have a direct sum decomposition

Wλ ≃⊕w∈W Wλ,wλ

Each summand admits the description as the convolution of standard extensions Tw∗ = jw∗Ow and costandard extensions Tw! = jw!Ow off of Schubert cells

1 1 W ≃ T 0 ∗ T − ≃ T − λ,wλ w ! ww0 ∗ w0ww0 ∗

(3) When λ = 0 is trivial, the fiber W0 is the maximal tilting sheaf on B. Namely, within D(N\B), it is the projective cover of the skyscraper sheaf at the closed Schubert cell. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 19

3.6. Example: SL2. We offer here a brief discussion of the structure of the universal Weyl sheaf W in the case when G = SL2. Already here one can see how intricate topology is packaged in the simple algebra of W. More specifically, the simple notion of differential operators on N\SL2/N interpolates between standard, costandard and tilting sheaves as we specialize parameters. Let us identify h∗ ≃ A1 so that Λ∗ ≃ Z with the reflection action of the Weyl group W ≃ Z/2 centered at −ρ = −1 ∈ Z ⊂ A1. Thus −1 ∈ Z ⊂ A1 is the unique singular parameter, the rest of the integers Z \{−1} are regular integral, and the rest of the parameters h∗ \ Z are generic (not at all integral). Let us identify 2 SL2/N ≃ A \{(0, 0)} = Spec C[x, y] \{(0, 0)} so that the left action of B has orbits  / i : V = Gm ×{0} / SL2/N and the complement 1  / j : U = A × Gm / SL2/N and the right action of H ≃ Gm is the usual scaling dilation. Thus the left action of the diagonal torus T ⊂ B coincides with the right action of H on the closed orbit V and is its inverse on the open orbit U. Note that the left action of N ≃ A1 consists of the individual points of the closed orbit V and the slices A1 ×{y}⊂ U of the open orbit. The N-equivariant ring of differential operators on G/N is given by Hamiltonian reduction

W = DG/N /(n) where (n) ⊂ DG/N is the left ideal generated by the vector field y∂x arising from the left N-action. Thus the coisotropic (but not Lagrangian) singular support of W is the union of conormals to N-orbits and explicitly cut out by the single equation yξx = 0. Let us specialize W to prescribed monodromies for the right H-action. For λ ∈ h∗ ≃ A1, the corresponding fiber is the quotient Wλ = DG/N /(n, hλ) where (n, hλ) ⊂ DG/N is the left ideal generated by the vector field y∂x arising from the left N- action, and the differential operator x∂x + y∂y − λ coming from prescribing the monodromy of the right H-action. Thus Wλ is regular holonomic with singular support the union of conormals to B-orbits and explicitly cut out by the equations yξx = xξx + yξy = 0. From these equations, we see its characteristic cycle is the weighted sum of conormals to B-orbits ∗ ∗ cc(Wλ)=2 · TV + TU ∗ Now when λ ∈ h \ Z is generic, Wλ splits as a direct sum

Wλ = i∗LV,λ ⊕ j∗LU,λ of the standard extensions of the local systems on B-orbits

LV,λ = DV /(x∂x − λ) LU,λ = DU /(∂x, y∂y − λ) Observe that because λ is generic, the standard extension off of the open orbit U is equivalent to the costandard extension j!LU,λ ≃ j∗LU,λ (Of course, there is no difference in the standard and costandard extensions off of the closed orbit V .) The characteristic cycles of the summands are the sums of conormals to B-orbits

∗ ∗ ∗ cc(i∗LV,λ)= TV cc(j∗LU,λ)= TV + TU

When λ ∈ Z \ {−1} becomes regular integral, Wλ continues to split as a direct sum but now in a more delicate form. Namely, when λ ∈{0, 1, 2,...} is “positive”, we have

Wλ = i∗LV,λ ⊕ j∗LU,λ 20 DAVID BEN-ZVI AND DAVID NADLER and when λ ∈ {−2, −3, −4,...} is “negative”, we have

Wλ = i∗LV,λ ⊕ j!LU,λ From the perspective of the monadic symmetries governing localization, this asymmetry reflects the choice of global sections functor. For example, going back to Borel-Weil-Bott, we see that the global sections of line bundles leads to the asymmetry of cohomological shifts. Finally, when λ = −1 is singular, W0 no longer splits as a direct sum but becomes the indecom- posable tilting extension of the structure sheaf of the open orbit tilt W0 = OU = DG/N /(y∂x, x∂x + y∂y) Thus it is self-dual and admits an increasing filtration 0 1 W0 ⊂W0 ⊂W0 with associated graded

0 1 0 1 W0 ≃ i∗OV W0 /W0 ≃ OG/N W0/W0 ≃ i∗OV

To see any of the preceding identifications of Wλ explicitly, one can restrict to the transverse line A1 ≃{x =1} to find 2 Wλ|{x=1} ≃ DA1 /(y ∂y − λy) ≃ DA1 /(y∂yy − (λ + 1)y)

In particular, when λ = −1 is singular, we find the traditional algebraic presentation DA1 /(y∂yy) 1 of the tilting extension of the structure sheaf of Gm to all of A .

4. Appendix: Demazure Descent In this section we present a simple uniform description of the Grothendieck-Springer monad on Q(g) expressing descent to g∗ and the Weyl monad on D(G/N) expressing localization of e H Ug-mod. Namely we show they both come from the Demazure monad d, an algebra in the coherent or Demazure Hecke category D = (Q!(B\G/B), ∗), via lax actions of D on the corresponding categories. We are grateful to Arkhipov and Kanstrup [AK1] for suggesting the relevance of Demazure algebras to the results of this paper. In order to see the Grothendieck-Springer resolution and Beilinson-Bernstein localization on equal footing, it is convenient to pass from sheaves on cotangents and D-modules to the dual perspective of sheaves on Dolbeault and de Rham spaces, respectively. The de Rham functor of a smooth scheme X can be described as the quotient of X by the formal neighborhood of the diagonal, while the Dolbeault functor is the relative classifying stack of the formal group of the tangent bundle of X. The general formalism of inf-schemes [GR] handles such objects (more generally, quotients of schemes by formal groupoids) on an equal footing with ordinary schemes. The extension of the functor Q! of ind-coherent sheaves to (correspondences of) inf-schemes developed in [GR] allows one to treat D-modules (together with strong or weak equivariance) and coherent sheaves on an equal footing. Notation 4.1. In this appendix, for a smooth scheme X we denote by X either ! (1) the de Rham space XdR, in which case Q (X)= D(X), (2) the Dolbeault space XDol, i.e., the classifying space of the formal group of the tangent bundle of X, in which case Q!(X)= Q!(T ∗X).

1 (We could likewise work with the Hodge inf-scheme XHod → A /Gm interpolating XdR and XDol [Si], with sheaves given by modules for the Rees algebra of D.) Example 4.2. (1) For a G-space X, the categories Q!(X/G) and Q!(X/G) recover the cate- gories of weakly and strongly G-equivariant D-modules on X, respectively. The pullback functor p! along p : X/G → X/G is identified with the standard functor forgetting from strong to weak equivariance. BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 21

(2) The category Q!(G/G) is identified with Q(T ∗G)G ≃ Q(g∗) in the Dolbeault setting and D(G)G ≃ Ug-mod in the de Rham setting. (3) The category Q!(G/NH) is identified with Q((T ∗G/N)/H) = QC(g) (Dolbeault) and e D(G/N)H (de Rham). The two spaces above are related by a correspondence G/B ✉ ❋❋ p ✉✉ ❋ q ✉✉ ❋❋ ✉✉ ❋❋ ✉z ✉ ❋# G/NH G/G

! ! ! Lemma 4.3. The functor q∗p : Q (G/NH) → Q (G/G) is identified with pushforward π∗ along the Grothendieck-Springer resolution π : g → g∗ in the Dolbeault setting and with global sections e γ∗ : D(G/N)H → Ug-mod in the de Rham setting. It is now easy to express the resulting monads geometrically by composing this correspondence with its opposite:

(4.1) G/B ×pt/B B\G/B P ♥♥♥ PPP ♥♥♥ PPP ♥♥♥ PPP ♥w ♥♥ PP' G/B G/B ✉ PPP ♥♥ ■■ ✉✉ PP ♥♥ ■■ ✉✉ PPP ♥♥♥ ■■ ✉✉ PPP ♥♥♥ ■■ ✉z ✉ PP' ♥w ♥♥ ■$ G/NH G/G G/NH We see that the Grothendieck-Springer and Beilinson-Bernstein adjunctions are controlled by the groupoid B\G/B through its action on G/B. We will now make this relation precise. 4.1. Equivariance for groupoids. In order to gain some perspective on the Demazure monad, it is useful to consider it in the setting of groupoid actions. Thus for an ind-proper morphism p : X → Y let G = X ×Y X denote the descent groupoid, with source and target the two projections π1, π2 : G → X. (We will apply this to p : X = pt/B → Y = pt/G, with G = B\G/B.) Let H = Q!(G), the Hecke category of G, denote the monoidal category of ind-coherent sheaves under convolution. It is inherited on applying Q! to the structure on G of algebra object in corre- spondences. (See [GR, Sections II.2.5.1, III.3.6.3] for a general treatment of ind-proper groupoids and more generally monoid or Segal stacks and the corresponding convolution categories.) The diagonal embedding (unit map) i : X → G induces a monoidal functor Q!(X) → H. Let H, the Hecke algebra of G, denote the descent monad for p : X → Y , an algebra object ! ! ! structure on the functor π2,∗π1 ≃ p p∗ ∈ End(Q (X)). Thus by ind-proper descent we have an identification ! G ! H-modQ!(X) ≃Q (X) ≃Q (Y ) of H-modules, G-equivariant sheaves on X and sheaves on Y . Definition 4.4. For a H = Q!(G)-module M, we define the category of G-equivariant objects to be G ! M := HomH(Q (X), M). The following proposition (while not strictly needed) gives a handy picture of equivariance as modules for a categorical form of the averaging idempotent in a finite group algebra:

Proposition 4.5. (1) The object e = ωG ∈ H has a canonical structure of algebra object, which is taken to the Hecke algebra H under the monoidal action map H → End(Q!(X)). Thus ! ! G ! we have an equivalence Q (Y ) = Q (X) ≃ Q (X)e of equivariant sheaves with e-modules in Q!(X). H (2) More generally, for any H-module M we have an identification M ≃ Me of equivariant objects with e-modules in M. 22 DAVID BEN-ZVI AND DAVID NADLER

Proof. The first assertion follows from base-change and ind-proper descent. It follows from the ind-properness of p that H is rigid, while the QCA property of X implies the self-duality of Q!(X) over k. Together these properties imply that Q!(X) is naturally self-dual over H. It follows that for any H-module M we have G ! M = HomH(Q (X), M) ! ≃ Q (X) ⊗H M

≃ He ⊗H M

≃ Me where the last step is [BFN][Proposition 4.1], or rather its straightforward extension from symmetric monoidal to monoidal categories from [BFN2][Proposition 3.1]. (In particular we have Q!(X)G = Q!(Y ) so the notation is consistent.)  4.2. The Demazure Hecke Category. Definition 4.6. Let D = (Q!(B\G/B), ∗) denote the Demazure Hecke category. The Demazure monad d ∈ Alg(D) is the dualizing complex ωB\G/B with its natural algebra structure. Example 4.7. [AK2] For any G-space Z, we have the D-module category Q!(Z/B). The equiv- ariants are given by ! D ! ! Q (Z/B) ≃Q (Z/B)d ≃Q (Z/G), where the last equivalence follows from proper descent: since Z/B → Z/G is a base change of pt/B → pt/G, the descent monad in End(Q!(Z/B)) is given by the image of d ∈ Alg(D) → Alg(End(Q!(Z/B))). Remark 4.8. A corresponding result on the level of K-groups was proved in [HLS], extending De- mazure’s description of the Weyl character formula (the case Z = pt). Example 4.9. Specializing to Z = G, we find an action of D by G-endomorphisms of Q!(G/B)– i.e. (in the de Rham setting) a monoidal functor ! D = Q (B\G/B) → EndD(G)D(G)B ≃ DB(G)B to bi-weakly B-equivariant D-modules on G. This in turn comes (by passing to [weak] B-equivariants) from the induction functor Q!(G) →Q!(G)= D(G) associated to the group homomorphism G → G, which is monoidal for convolution. The Demazure monad d is taken to [the natural B bi-equivariant r structure on] the induced D-module DG on G. We will be interested in transporting (lax) D-actions along adjunctions. Given an adjunction φ : N o / M : ψ there is a natural lax monoidal structure on the functor Ψ: End(M) → End(N ), Ψ(T )= ψ ◦ T ◦ φ – e.g., on the level of individual endomorphisms we have a map Ψ(T ) ◦ Ψ(U)= ψ ◦ T ◦ φ ◦ ψ ◦ U ◦ φ −→ ψ ◦ T ◦ U ◦ φ. = Ψ(T ◦ U). Thus if M is a module category for D, i.e., we are given a monoidal functor D → End(M), then N becomes a lax module category via the lax monoidal functor D → End(M) → End(N ). In particular, we obtain a functor (4.2) Alg(D) −→ Alg(End(N )) from algebras in D to monads on N . We apply this construction to the adjunction

p! : N = Q!(G/NH) o / M = Q!(G/B): p!,R BEILINSON-BERNSTEIN LOCALIZATION OVER THE HARISH-CHANDRA CENTER 23 where p! forgets strong to weak N-equivariance (which preserves compact objects) and p!,R is its continuous right adjoint (informally, taking invariants for Un, rather than coinvariants as in the left adjoint). Remark 4.10. Geometrically, such lax actions come from considering groupoid objects in the cat- egory of correspondences. For example, if G  X is a groupoid and X → Y a morphism, then G forms a groupoid object in the category of correspondences over Y – in particular the composition is given by a correspondence o / G×Y Go G×X G / G .

In the setting of Diagram (4.1) we see that G = G/B ×pt/B B\G/B  X = G/B forms a groupoid in correspondences over Y = G/NH. ! ! Proposition 4.11. (1) The categories N = Q (G/NH), i.e., Q (g) (Dolbeault) and D(G/N)H (de Rham), carry natural lax D-module structures. These actionse come from the natural lax monoidal functor (induction) ! ! D = Q (B\G/B) →Q (HN\G/NH)= DH (N\G/N)H (where the last identification with the Hecke category is in the de Rham version). (2) Under the resulting functor (4.2) the Demazure monad d is taken to the Grothendieck- Springer and Weyl monads, respectively. Thus we find an identification of the Weyl monad with the universal Weyl sheaf W of Lemma 1.11. (3) The resulting categories of modules for the Demazure monad are identified as Q!(g)d ≃Q(g∗) d e and D(G/N)H ≃ Ug-mod. Proof. The lax action is the result of the (p!,p!,R) adjunction above, or geometrically of the corre- spondence action from Remark 4.10. By construction it commutes with the strong G-action, hence factors through a lax monoidal functor to the Hecke category, which is given by pushforward along the composition B\G/B → B\G/B → HN\G/NH – the first pushforward (induction) encodes the action of D on Q!(G/B) as in Example 4.9, while the second (imposing strong N bi-equivariance, i.e., taking invariants for the n-actions) is only lax monoidal. The second assertion follows from Lemma 4.3, so ultimately from basechange for the diagram (4.1). In particular, the description of the action via induction provides an identification of the Weyl monad with the induction of the Demazure monad, so that the underlying sheaf is the sheaf of differential operators on N\G/N: imposing strong N-biequivariance on the (weakly biequivariant) induced module Dr on G produces the induced module Dr on N\G/N, which is identified with D by the Calabi-Yau structure on G/N. Since we’ve already established monadicity in both Grothendieck- Springer and Beilinson-Bernstein situations, the third assertion follows.  Remark 4.12. There is a parallel comonadic story where we use (as in [AK2]) the quasicoherent ver- ∗ ∗ sion D = (Q(B\G/B), ∗) and the natural coalgebra object d = OB\G/B. The category Q(G/NH) is endowed with an op-lax D∗-module structure and d∗ is taken to the Grothendieck-Springer and Beilinson-Bernstein comonads. Remark 4.13 (Borel-Weil-Bott). Passing to G-equivariant sheaves, we find the Borel-Weil-Bott ad- junction (parabolic induction/restriction) Q!(G\G/NH)= Rep(H) o / Q!(G\G/G)= Rep(G) in both Dolbeault and de Rham settings, recovering the original appearance of Demazure operators in the Weyl character formula. Remark 4.14 (Whittaker objects). Let us briefly indicate the case of Whittaker modules. Fix a non- degenerate character ψ of n. We may then pass to Whittaker objects in any strong G-category, i.e., objects strongly equivariant for N against the character ψ (see e.g. [Be]). When applied to Ug-mod, 24 DAVID BEN-ZVI AND DAVID NADLER we obtain the category of Whittaker Ug-modules, which by Kostant and Skryabin is identified with modules for Z(Ug). The same category arises in the classical limit, as sheaves on the Kostant slice ∗ g //ψN ≃ c. On the other hand, Whittaker D-modules on G/N are identified with D-modules on the big cell D(N\Nw0HN/N ≃ D(H). Thus we find the following adjunction:

(N,ψ) ! ∗ o / (N,ψ) ! ∗ D(G/N)H ≃ D(H)H ≃Q (h ) o / Ug-mod ≃ Zg-mod ≃Q (h //W ) . In the classical limit we find the same adjunction, realized as the restriction of the Grothendieck- Springer resolution to the Kostant slice. We thus find that in the Whittaker setting the Beilinson- Bernstein or Demazure monads are taken to the nil-Hecke algebra H, which controls descent from h∗ to the coarse quotient h∗//W .

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Department of Mathematics, University of Texas, Austin, TX 78712-0257 E-mail address: [email protected]

Department of Mathematics, University of California, Berkeley, CA 94720-3840 E-mail address: [email protected]