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Projective and Whittaker functors on category 𝒪\mathcal{O}.

Juan Camilo Arias and Erik Backelin Erik Backelin, Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bogotá, COLOMBIA erbackel@uniandes.edu.co Juan Camilo Arias, Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bogotá, COLOMBIA. Tel: +571 3394999 ext. 3705 Fax: +571 3324427 jc.arias147@uniandes.edu.co
Abstract.

We show that the Whittaker functor on a regular block of the BGG-category 𝒪\mathcal{O} of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Miličić’s equivalence between the category of Whittaker modules and a singular block of 𝒪\mathcal{O}. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.

2000 Mathematics Subject Classification:
Primary 17B10, 18G15, 17B20

1. Introduction

Let 𝔤\mathfrak{g} be a complex semisimple Lie algebra, 𝔟\mathfrak{b} a Borel subalgebra, 𝔥\mathfrak{h} a Cartan subalgebra and 𝔫=[𝔟,𝔟]\mathfrak{n}=[\mathfrak{b},\mathfrak{b}]. Let 𝒪\mathcal{O} be the Bernstein-Gelfand-Gelfand (BGG) category of representations of 𝔤\mathfrak{g}. Let U=U​(𝔤)U=U(\mathfrak{g}) be the universal enveloping algebra of 𝔤\mathfrak{g} and ZZ its center. Let f:U​(𝔫)→ℂf:U(\mathfrak{n})\to\mathbb{C} be an algebra homomorphism and 𝒩f\mathcal{N}_{f} the corresponding category of Whittaker modules (finitely generated UU-modules that are locally finite over ZZ and locally annihilated by Ker⁡f\operatorname{Ker}f). For a UU-module VV let Γf​(V)\Gamma_{f}(V) denote the submodule of vectors annihilated by some power of Ker⁡f\operatorname{Ker}f. In [1] the Whittaker functor

Γ¯f:𝒪→𝒩f,M↦Γf​(M¯)\overline{\Gamma}_{f}:\mathcal{O}\to\mathcal{N}_{f},\ M\mapsto\Gamma_{f}(\overline{M})

was introduced. Here M¯\overline{M} denote the completion of MM, i.e. the product of its weight spaces. It is exact and maps Verma modules to standard Whittaker modules. In this paper we shall establish a few basic properties of this functor that we haven’t found in the literature.

When ff is a regular character (i.e. f​(Eα)≠0f(E_{\alpha})\neq 0 for all simple root vectors EαE_{\alpha}) Kostant [10] showed that 𝒩f\mathcal{N}_{f} is equivalent to the category of finite dimensional ZZ-modules. The functor is M↦W​hf​(M)={m∈M|Ker⁡f⋅m=0}M\mapsto Wh_{f}(M)=\{m\in M|\operatorname{Ker}f\cdot m=0\}.

On the other hand, if we fix a dominant weight λ∈𝔥∗\lambda\in\mathfrak{h}^{*} and the corresponding block 𝒪λ^⊂𝒪\mathcal{O}_{\widehat{\lambda}}\subset\mathcal{O} there is Soergel’s functor 𝕍:𝒪λ^→Z−mod\mathbb{V}:\mathcal{O}_{\widehat{\lambda}}\to Z-\operatorname{mod}, 𝕍​(M)=Hom⁡(Pw0⋅λ,M)\mathbb{V}(M)=\operatorname{Hom}(P_{w_{0}\cdot\lambda},M), where Pw0⋅λP_{w_{0}\cdot\lambda} is the anti-dominant projective. It was shown in [1] that if we restrict Γ¯f\overline{\Gamma}_{f} to 𝒪λ^\mathcal{O}_{\widehat{\lambda}} and compose it with Kostant’s equivalence W​hfWh_{f} the resulting functor is naturally equivalent to 𝕍\mathbb{V} (see Proposition 2 for a short proof). Thus one may think of Γ¯f\overline{\Gamma}_{f} as a ”partial” Soergel functor, for a general character ff.

Let λ\lambda be dominant, integral and regular and μ\mu be dominant and integral and assume that ff is a character such that Wμ=WfW_{\mu}=W_{f}. Then using the machinery of Harish-Chandra bimodules Soergel and Miličić, [13] established an equivalences between (an enhancement of) 𝒪μ^\mathcal{O}_{\widehat{\mu}} and the block 𝒩λ^f\mathcal{N}^{f}_{\widehat{\lambda}}. Harish-Chandra bimodule theory provides an equivalence κ:𝒪λ^​⟶∼​𝒪λ′\kappa:\mathcal{O}_{\widehat{\lambda}}\overset{\sim}{\longrightarrow}\mathcal{O}^{\prime}_{\lambda}. If we compose κ\kappa with the translation to the wall functor Θλμ:𝒪λ′→𝒪μ^′\Theta^{\mu}_{\lambda}:\mathcal{O}^{\prime}_{\lambda}\to\mathcal{O}^{\prime}_{\widehat{\mu}} and then compose the result with the equivalence σ:𝒪μ^​⟶∼​𝒩λ^f\sigma:\mathcal{O}_{\widehat{\mu}}\overset{\sim}{\longrightarrow}\mathcal{N}^{f}_{\widehat{\lambda}} we obtain a functor τ=σ​Θλμ​κ:𝒪λ^→𝒩λ^f\tau=\sigma\Theta^{\mu}_{\lambda}\kappa:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{N}^{f}_{\widehat{\lambda}}. We show in Theorem 1 that τ\tau is naturally equivalent to Γ¯f\overline{\Gamma}_{f}.

As a consequence we establish that Γ¯f\overline{\Gamma}_{f} has a left and a right adjoint. (It has eluded us to directly prove the existence of these adjoints without using τ\tau.) We also prove that Γ¯f\overline{\Gamma}_{f} is a quotient functor, Corollary 2.

The idea behind the proofs of these facts is that the Whittaker functor Γ¯f\overline{\Gamma}_{f} commutes not just with projective functors but actually with morphisms between projective functors in the following sense: Let 𝒫\mathcal{P} and 𝒫′\mathcal{P}^{\prime} be projective endofunctors of the category of all ZZ-finite UU-modules; in particular such functors act on 𝒪\mathcal{O} and on Whittaker modules. Let ϕ:𝒫→𝒫′\phi:\mathcal{P}\to\mathcal{P}^{\prime} be a natural transformation. Then we are constructing equivalences Θ𝒫:𝒫∘Γ¯f→Γ¯f∘𝒫\Theta_{\mathcal{P}}:\mathcal{P}\circ\overline{\Gamma}_{f}\to\overline{\Gamma}_{f}\circ\mathcal{P} such that Γ¯f∘ϕ\overline{\Gamma}_{f}\circ\phi (composing a natural transformation with a functor) and ϕ∘Γ¯f\phi\circ\overline{\Gamma}_{f} (precomposing a natural transformation with a functor) becomes equivalent after conjugation with Θ\Theta: Θ𝒫′​(ϕ∘Γ¯f)=(Γ¯f∘ϕ)​Θ𝒫\Theta_{\mathcal{P}^{\prime}}(\phi\circ\overline{\Gamma}_{f})=(\overline{\Gamma}_{f}\circ\phi)\Theta_{\mathcal{P}}. This is showed in Proposition 3. We also show that the functor τ\tau commutes with projective functors and their morphisms in this sense. With this in hand, in order to establish Theorem 1 it is enough to observe that Γ¯f​(Δλ)≅τ​(Δλ)\overline{\Gamma}_{f}(\Delta_{\lambda})\cong\tau(\Delta_{\lambda}).

1.0.1.

We wonder if the commutativity between Γ¯f\overline{\Gamma}_{f} and projective functors may be of interest in the study of the so called Rouqier complexes, see [7], [11]. Rouquier complexes occur naturally when one constructs projective resolutions of Verma modules by iterated use of wall-crossing functors. They are mostly interpreted as complexes of Soergel bimodules and they have been of fundamental importance in establishing Hodge theoretical properties of the category of Soergel bimodules. In their original form however Rouquier complexes are complexes of projective functors (and as such carry more information) and the results of this paper may perhaps help to find symmetries of them. One may construct Rouqier complexes of Whittaker modules and it follows from the results here that these have good exactness properties, see Corollary 1. One may also use the composition of the Whittaker functor and its adjoint to provide endofunctors of 𝒪λ^\mathcal{O}_{\widehat{\lambda}} that commute with all projective endofunctors of 𝒪λ^\mathcal{O}_{\widehat{\lambda}}. In section 3.2 we added some computations of the adjoints of Γ¯f\overline{\Gamma}_{f} in order to facilitate this.

1.0.2. Acknowledgements

We like to thank Paul Bressler for many useful conversations.

2. Preliminaries.

In this section we collect the known facts that we will need about category 𝒪\mathcal{O}, Harish-Chandra bimodules, Whittaker modules and the Whittaker functor.

2.1. Root data

Let 𝔤⊃𝔟⊃𝔥\mathfrak{g}\supset\mathfrak{b}\supset\mathfrak{h} be a complex semisimple Lie algebra containing a Borel and a Cartan subalgebra and put 𝔫:=[𝔟,𝔟]\mathfrak{n}:=[\mathfrak{b},\mathfrak{b}]. Let WW be the Weyl group and 𝔹⊂𝔥∗\mathbb{B}\subset\mathfrak{h}^{*} the simple roots and ρ\rho the half sum of the positive roots. We consider the d​o​tdot-action of WW on 𝔥∗\mathfrak{h}^{*} defined by w⋅λ=w​(λ+ρ)−ρw\cdot\lambda=w(\lambda+\rho)-\rho; we let S​(𝔥)WS(\mathfrak{h})^{W} be the invariants with respect to the dot-action. Let w0∈Ww_{0}\in W be the longest element.

Let U=U​(𝔤)U=U(\mathfrak{g}) be the universal enveloping algebra of 𝔤\mathfrak{g} and Z⊂UZ\subset U its center. For λ∈𝔥∗\lambda\in\mathfrak{h}^{*} let Jλ⊂ZJ_{\lambda}\subset Z be the maximal ideal that annihilates a Verma module with highest weight λ\lambda. Let Wλ={w∈W|w⋅λ=λ}W_{\lambda}=\{w\in W|w\cdot\lambda=\lambda\}. Let WλW^{\lambda} be the set of longest representatives of the left cosets W/WλW/W_{\lambda} and Wλ{}^{\lambda}W the set of longest representatives of the right cosets Wλ∖WW_{\lambda}\setminus W.

Let f:U​(𝔫)→ℂf:U(\mathfrak{n})\to\mathbb{C} be an algebra homomorphism. Let 𝔹f={α∈𝔹|f​(Eα)≠0}\mathbb{B}_{f}=\{\alpha\in\mathbb{B}|f(E_{\alpha})\neq 0\}, where Eα∈𝔫E_{\alpha}\in\mathfrak{n} is the Chevalley generator corresponding to α∈𝔹\alpha\in\mathbb{B}, and let WfW_{f} be the subgroup of WW generated by simple reflections sαs_{\alpha}, α∈𝔹f\alpha\in\mathbb{B}_{f}.

We say that ff is non-degenerate if 𝔹f=𝔹\mathbb{B}_{f}=\mathbb{B} and that ff is trivial if 𝔹f=∅\mathbb{B}_{f}=\emptyset.

2.1.1.

Let 𝔤\mathfrak{g}-mod\operatorname{mod} denote the category of all (left) UU-modules and let ℱ{\mathcal{F}} be the category of finite dimensional UU-modules. Let ℳ\mathcal{M} denote the category of finitely generated UU-modules such that the action of the subalgebra Z⊂UZ\subset U is locally finite. For λ∈𝔥∗\lambda\in\mathfrak{h}^{*} define full subcategories of ℳ\mathcal{M}:

ℳλ={V∈ℳ|Jλ​V=0},ℳλ^={V∈ℳ|∃n>0:Jλn​V=0}.\mathcal{M}_{\lambda}=\{V\in\mathcal{M}|J_{\lambda}V=0\},\ \mathcal{M}_{\widehat{\lambda}}=\{V\in\mathcal{M}|\,\exists n>0:\,J^{n}_{\lambda}V=0\}.

We let iλ:ℳλ^→ℳi_{\lambda}:\mathcal{M}_{{\widehat{\lambda}}}\to\mathcal{M} denote the inclusion functor and define

p​rλ:ℳ→ℳλ^,M↦{m∈M|∃n>0:Jλn⋅m=0}.pr_{\lambda}:\mathcal{M}\to\mathcal{M}_{{\widehat{\lambda}}},\ M\mapsto\{m\in M|\,\exists n>0:\,J^{n}_{\lambda}\cdot m=0\}.

Then we have the block decomposition ℳ=⊕λ∈𝔥∗ℳλ^\mathcal{M}=\oplus_{\lambda\in\mathfrak{h}^{*}}\mathcal{M}_{\widehat{\lambda}}. For any full subcategory 𝒞\mathcal{C} of ℳ\mathcal{M} we define full subcategories 𝒞λ=𝒞∩ℳλ\mathcal{C}_{\lambda}=\mathcal{C}\cap\mathcal{M}_{\lambda} and 𝒞λ^=𝒞∩ℳλ^\mathcal{C}_{\widehat{\lambda}}=\mathcal{C}\cap\mathcal{M}_{\widehat{\lambda}}.

2.2. Category 𝒪\mathcal{O}.

See [8]. Let 𝒪\mathcal{O} be the BGG-category of finitely generated UU-modules which are locally finite over 𝔫\mathfrak{n} and semisimple over 𝔥\mathfrak{h}. Any M∈𝒪M\in\mathcal{O} thus has the weight space decomposition M=⊕ν∈𝔥∗MνM=\oplus_{\nu\in\mathfrak{h}^{*}}M^{\nu} and each MνM^{\nu} is a finite dimensional vector space. 𝒪\mathcal{O} is a full subcategory of ℳ\mathcal{M}. Also, let 𝒪′⊃𝒪\mathcal{O}^{\prime}\supset\mathcal{O} be the category of finitely generated UU-modules which are locally finite over 𝔟\mathfrak{b} and locally finite over ZZ.

For λ∈𝔥∗\lambda\in\mathfrak{h}^{*} we get full subcategories 𝒪λ⊂Oλ^⊂𝒪\mathcal{O}_{\lambda}\subset O_{\widehat{\lambda}}\subset\mathcal{O} and 𝒪λ′⊂Oλ^′⊂𝒪′\mathcal{O}^{\prime}_{\lambda}\subset O^{\prime}_{\widehat{\lambda}}\subset\mathcal{O}^{\prime} and block decompositions 𝒪=⊕λ∈𝔥∗𝒪λ^\mathcal{O}=\oplus_{\lambda\in\mathfrak{h}^{*}}\mathcal{O}_{\widehat{\lambda}} and 𝒪′=⊕λ∈𝔥∗𝒪λ^′\mathcal{O}^{\prime}=\oplus_{\lambda\in\mathfrak{h}^{*}}\mathcal{O}^{\prime}_{\widehat{\lambda}}.

For M=⊕ν∈𝔥∗Mν∈𝒪M=\oplus_{\nu\in\mathfrak{h}^{*}}M^{\nu}\in\mathcal{O} we let M∗=⊕ν∈𝔥∗Homℂ⁡(Mν,ℂ)M^{*}=\oplus_{\nu\in\mathfrak{h}^{*}}\operatorname{Hom}_{\mathbb{C}}(M^{\nu},\mathbb{C}) with the 𝔤\mathfrak{g}-module structure given by the Chevalley involution: (x​f)​(m)=f​(xt​r​m)(xf)(m)=f(x^{tr}m) for x∈𝔤x\in\mathfrak{g}, f∈M∗f\in M^{*} and m∈Mm\in M. Then M∗∈𝒪M^{*}\in\mathcal{O} and M​⟶∼​M∗∗M\overset{\sim}{\longrightarrow}M^{**}.

For λ∈𝔥∗\lambda\in\mathfrak{h}^{*} denote by ℂλ\mathbb{C}_{\lambda} the corresponding one dimensional representation of 𝔟\mathfrak{b} (by means of the projection 𝔟→𝔥\mathfrak{b}\to\mathfrak{h}). Let Δλ=U​(𝔤)⊗U​(𝔟)ℂλ\Delta_{\lambda}=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_{\lambda} be the Verma module with highest weight λ\lambda, ∇λ\nabla_{\lambda} its dual, LλL_{\lambda} its simple quotient and PλP_{\lambda} a projective cover of LλL_{\lambda} in 𝒪λ^\mathcal{O}_{\widehat{\lambda}} and Iλ=Pλ∗I_{\lambda}=P^{*}_{\lambda} an injective hull of LλL_{\lambda}.

Assume that a module M∈𝒪M\in\mathcal{O} has a filtration 0=F0⊂F1⊂…⊂Fn=M0=F_{0}\subset F_{1}\subset\ldots\subset F_{n}=M such that Fi/Fi−1≅ΔλiF_{i}/F_{i-1}\cong\Delta_{\lambda_{i}} for each ii. Then we say that MM has a Verma flag and we write (M:Δλ)=#{i|λi=λ}(M:\Delta_{\lambda})=\#\{i|\,\lambda_{i}=\lambda\} for the corresponding Verma flag multiplicity. For any V∈𝒪V\in\mathcal{O} we write [V:L][V:L] for the Jordan Hölder-multiplicity of LL in VV, for LL a simple module. Any projective module admits a Verma flag and BGG-reciprocity states (Pλ:Δμ)=[Δμ:Lλ](P_{\lambda}:\Delta_{\mu})=[\Delta_{\mu}:L_{\lambda}].

2.3. Whittaker modules and functor.

Let 𝒩\mathcal{N} be the category of finitely generated UU-modules which are locally finite over U​(𝔫)U(\mathfrak{n}) and locally finite over ZZ. This is a finite length category. Let f:U​(𝔫)→ℂf:U(\mathfrak{n})\to\mathbb{C} be an algebra homomorphism and denote by ℂf\mathbb{C}_{f} the corresponding one dimensional representation of U​(𝔫)U(\mathfrak{n}).

Let 𝒩f\mathcal{N}^{f} be the full subcategory of 𝒩\mathcal{N} whose objects are locally annihilated by some power of Ker⁡f\operatorname{Ker}f. Objects of 𝒩f\mathcal{N}^{f} are called Whittaker modules. (Thus, if ff is the trivial homomorphism then 𝒩f=𝒪′\mathcal{N}^{f}=\mathcal{O}^{\prime}.) Since 𝔫\mathfrak{n} is nilpotent we have by [5]

𝒩=⊕f:U​(𝔫)→ℂ𝒩f.\mathcal{N}=\oplus_{f:U(\mathfrak{n})\to\mathbb{C}}\,\mathcal{N}^{f}.

The categories 𝒩\mathcal{N} and 𝒩f\mathcal{N}^{f} also decomposes over the center ZZ:

𝒩=⊕λ∈𝔥∗𝒩λ^,𝒩f=⊕λ∈𝔥∗𝒩λ^f.\mathcal{N}=\oplus_{\lambda\in\mathfrak{h}^{*}}\mathcal{N}_{{\widehat{\lambda}}},\;\;\mathcal{N}^{f}=\oplus_{\lambda\in\mathfrak{h}^{*}}\mathcal{N}^{f}_{\widehat{\lambda}}.
Lemma 1.

a) Let MM be a 𝔤\mathfrak{g}-module and assume that each m∈Mm\in M is annihilated by some power of Eα−f​(Eα)E_{\alpha}-f(E_{\alpha}) for α∈𝔹\alpha\in\mathbb{B}. Then MM is locally finite over 𝔫\mathfrak{n}. In particular, if MM is finitely generated over UU then M∈𝒩fM\in\mathcal{N}^{f}.

b) For E∈ℱE\in{\mathcal{F}} and M∈𝒩fM\in\mathcal{N}^{f} we have E⊗M∈𝒩fE\otimes M\in\mathcal{N}^{f}.

Proof.

a) Define a new 𝔫\mathfrak{n}-action on MM by x∗m=(x−f​(x))​mx*m=(x-f(x))m. Then the generators EαE_{\alpha} of 𝔫\mathfrak{n} acts nilpotently on mm. By [3], Lemma 7.3.7. this implies that U​(𝔫)∗mU(\mathfrak{n})*m is finite dimensional for each m∈Mm\in M.

b) Let e⊗m∈E⊗Me\otimes m\in E\otimes M. Then (Eα−f​(Eα))​(e⊗m)=Eα​e⊗(Eα−f​(Eα))​m(E_{\alpha}-f(E_{\alpha}))(e\otimes m)=E_{\alpha}e\otimes(E_{\alpha}-f(E_{\alpha}))m and therefore by induction

(Eα−f​(Eα))n​(e⊗m)=∑j=0n(nj)​Eαj​e⊗(Eα−f​(Eα))n−j​m.(E_{\alpha}-f(E_{\alpha}))^{n}(e\otimes m)=\sum^{n}_{j=0}\binom{n}{j}E^{j}_{\alpha}e\otimes(E_{\alpha}-f(E_{\alpha}))^{n-j}m.

Since EE is finite dimensional Eαj​e=0E^{j}_{\alpha}e=0 for j>>0j>>0 and since also (Eα−f​(Eα))n−j​m=0(E_{\alpha}-f(E_{\alpha}))^{n-j}m=0 for n−j>>0n-j>>0 we get that the above sum vanishes for n>>0n>>0. Thus b) follows from a). ∎

2.3.1. Standard Whittaker modules

See [12], [13]. Let ff be fixed and let 𝔭\mathfrak{p} denote the parabolic subalgebra of 𝔤\mathfrak{g} generated by 𝔟\mathfrak{b} and EαE_{\alpha} for α∈𝔹f\alpha\in\mathbb{B}_{f}. Let 𝔩\mathfrak{l} be the reductive Levi factor of 𝔭\mathfrak{p} and put Jλ(𝔩)=AnnZ​(𝔩)⁡U​(𝔩)⊗U​(𝔟∩𝔩)ℂλJ^{(\mathfrak{l})}_{\lambda}=\operatorname{Ann}_{Z(\mathfrak{l})}U(\mathfrak{l})\otimes_{U(\mathfrak{b}\cap\mathfrak{l})}\mathbb{C}_{\lambda}, for λ∈𝔥∗\lambda\in\mathfrak{h}^{*}.

Consider the U​(𝔩)U(\mathfrak{l})-module U​(𝔩)/Jλ(𝔩)⊗U​(𝔫∩𝔩)ℂfU(\mathfrak{l})/J^{(\mathfrak{l})}_{\lambda}\otimes_{U(\mathfrak{n}\cap\mathfrak{l})}\mathbb{C}_{f} as a U​(𝔭)U(\mathfrak{p})-module by means of the projection 𝔭→𝔭/r​a​d​𝔭≅𝔩\mathfrak{p}\to\mathfrak{p}/rad\,\mathfrak{p}\cong\mathfrak{l} and define the standard Whittaker module

Δλ​(f)=U⊗U​(𝔭)(U​(𝔩)/Jλ(𝔩)⊗U​(𝔫∩𝔩)ℂf),λ∈𝔥∗.\Delta_{\lambda}(f)=U\otimes_{U(\mathfrak{p})}(U(\mathfrak{l})/J^{(\mathfrak{l})}_{\lambda}\otimes_{U(\mathfrak{n}\cap\mathfrak{l})}\mathbb{C}_{f}),\;\lambda\in\mathfrak{h}^{*}.

Note that when ff is the trivial homomorphism then Δλ​(f)=Δλ\Delta_{\lambda}(f)=\Delta_{\lambda}. Δλ​(f)\Delta_{\lambda}(f) has a unique irreducible quotient Lλ​(f)L_{\lambda}(f).

Also, define

Δλn​(f)=U⊗U​(𝔭)(U​(𝔩)/(Jλ(𝔩))n⊗U​(𝔫∩𝔩)ℂf)\Delta^{n}_{\lambda}(f)=U\otimes_{U(\mathfrak{p})}(U(\mathfrak{l})/(J^{(\mathfrak{l})}_{\lambda})^{n}\otimes_{U(\mathfrak{n}\cap\mathfrak{l})}\mathbb{C}_{f})

for n≥1n\geq 1.

2.4. The Whittaker functor

For a 𝔤\mathfrak{g}-module VV we define

Γf​(V)={v∈V|(Ker⁡f)n​v=0,for⁡n>>0}\Gamma_{f}(V)=\{v\in V|(\operatorname{Ker}f)^{n}v=0,\operatorname{for}n>>0\}

and

W​hf​(V)={v∈V|(Ker⁡f)​v=0}.Wh_{f}(V)=\{v\in V|(\operatorname{Ker}f)v=0\}.

Then Γf​(V)\Gamma_{f}(V) is a 𝔤\mathfrak{g}-submodule of VV while W​hf​(V)Wh_{f}(V) is merely a ZZ-submodule. It is clear that Γf\Gamma_{f} and W​hfWh_{f} yield functors. W​hfWh_{f} was introduced by Kostant, [10].

For M=⊕λ∈𝔥∗Mλ∈𝒪M=\oplus_{\lambda\in\mathfrak{h}^{*}}M^{\lambda}\in\mathcal{O} we define the completion M¯=∏λ∈𝔥∗Mλ\overline{M}=\prod_{\lambda\in\mathfrak{h}^{*}}M^{\lambda}. This has a natural 𝔤\mathfrak{g}-module structure making M⊆M¯M\subseteq\overline{M} a submodule. Let Z−modf​dZ-\operatorname{mod}_{fd} denote the category of finite dimensional ZZ-modules. In [1] the second author introduced the functors

Γ¯f:𝒪→𝒩f,M↦Γf​(M¯)​ and \overline{\Gamma}_{f}:\mathcal{O}\to\mathcal{N}^{f},\ M\mapsto\Gamma_{f}(\overline{M})\hbox{ and }
W​h¯f:𝒪→Z−modf​d,M↦W​hf​(M¯).\overline{Wh}_{f}:\mathcal{O}\to Z-\operatorname{mod}_{fd},\ M\mapsto Wh_{f}(\overline{M}).

Γ¯f\overline{\Gamma}_{f} is called the Whittaker functor; it is exact for any ff. The functor W​h¯f\overline{Wh}_{f} is exact if and only if ff is non-degenerate. These functors commute with the action of ZZ. Assume that ff is non-degenerate: [10] (see also [14] for a geometric proof) showed that the functor

W​hf:𝒩f→Z−modf​dWh_{f}:\mathcal{N}^{f}\to Z-\operatorname{mod}_{fd}

is an equivalence of categories; its quasi-inverse is M↦U⊗U​(𝔫)⊗ZMM\mapsto U\otimes_{U(\mathfrak{n})\otimes Z}M. Here the left ZZ-module structure on MM is the given one and the U​(𝔫)U(\mathfrak{n})-module structure is the unique one such that Ker⁡f⋅M=0\operatorname{Ker}f\cdot M=0.

In [1] it was proved that

Proposition 1.

For any λ∈𝔥∗\lambda\in\mathfrak{h}^{*} we have Γ¯f​(Δλ)≅Δλ​(f)\overline{\Gamma}_{f}(\Delta_{\lambda})\cong\Delta_{\lambda}(f). If, moreover, λ\lambda is dominant and x∈Wμx\in{}^{\mu}W then Γ¯f​(Lx⋅λ)≅Lx⋅λ​(f)\overline{\Gamma}_{f}(L_{x\cdot\lambda})\cong L_{x\cdot\lambda}(f) and for x∉Wμx\notin{}^{\mu}W we have Γ¯f​(Lx⋅λ)=0\overline{\Gamma}_{f}(L_{x\cdot\lambda})=0.

2.4.1.

Let λ\lambda be integral and dominant. Let J=AnnZ⁡(Pw0⋅λ)J=\operatorname{Ann}_{Z}(P_{w_{0}\cdot\lambda}) and put C=Z​(g)/JC=Z(g)/J. Let 𝕍:𝒪λ^→Z−mod\mathbb{V}:\mathcal{O}_{{\widehat{\lambda}}}\to Z-\operatorname{mod} be Soergel’s functor 𝕍​(M)=Hom𝒪⁡(Pw0⋅λ,M)\mathbb{V}(M)=\operatorname{Hom}_{\mathcal{O}}(P_{w_{0}\cdot\lambda},M) [16]. Thus, C=𝕍​(Pw0⋅λ)C=\mathbb{V}(P_{w_{0}\cdot\lambda}). Since 𝕍\mathbb{V} is fully faithful on projective objects we conclude that J​𝒪λ^=0J\mathcal{O}_{{\widehat{\lambda}}}=0. Thus, also J​Γ¯f​(M)=J​W​h¯f​(M)=0J\overline{\Gamma}_{f}(M)=J\overline{Wh}_{f}(M)=0 for M∈𝒪λ^M\in\mathcal{O}_{{\widehat{\lambda}}}. The following result was proved in [1]. We include here a short proof that Soergel once explained to us.

Proposition 2.

Assume that ff is non-degenerate. Then W​h¯f|𝒪λ^\overline{Wh}_{f}|_{\mathcal{O}_{\widehat{\lambda}}} is equivalent to 𝕍\mathbb{V}.

Proof.

The assumption on ff implies that W​h¯f\overline{Wh}_{f} is exact. Let v∈W​h¯f​(Pw0⋅λ)v\in\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}) be such that v¯≠0\overline{v}\neq 0 in W​h¯f​(Lw0⋅λ)\overline{Wh}_{f}(L_{w_{0}\cdot\lambda}) under the surjection W​h¯f​(Pw0⋅λ)↠W​h¯f​(Lw0⋅λ)\overline{Wh}_{f}(P_{w_{0}\cdot\lambda})\twoheadrightarrow\overline{Wh}_{f}(L_{w_{0}\cdot\lambda}) induced by the surjection Pw0⋅λ↠Lw0⋅λP_{w_{0}\cdot\lambda}\twoheadrightarrow L_{w_{0}\cdot\lambda}. By Yoneda Lemma

Homf​u​n​c​t​o​r​s⁡(𝕍,W​h¯f)=HomZ⁡(𝕍​(Pw0⋅λ),W​h¯f​(Pw0⋅λ))=\operatorname{Hom}_{functors}(\mathbb{V},\overline{Wh}_{f})=\operatorname{Hom}_{Z}(\mathbb{V}(P_{w_{0}\cdot\lambda}),\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}))=
=HomZ⁡(C,W​h¯f​(Pw0⋅λ))=HomC⁡(C,W​h¯f​(Pw0⋅λ))=W​h¯f​(Pw0⋅λ).=\operatorname{Hom}_{Z}(C,\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}))=\operatorname{Hom}_{C}(C,\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}))=\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}).

Let h:𝕍→W​h¯fh:\mathbb{V}\to\overline{Wh}_{f} be the natural transformation that corresponds to v∈W​h¯f​(Pw0⋅λ)v\in\overline{Wh}_{f}(P_{w_{0}\cdot\lambda}). Then we see that hLw0⋅λ:𝕍​(Lw0⋅λ)→W​h¯f​(Lw0⋅λ)h_{L_{w_{0}\cdot\lambda}}:\mathbb{V}(L_{w_{0}\cdot\lambda})\to\overline{Wh}_{f}(L_{w_{0}\cdot\lambda}) is an isomorphism since it is non-zero and both sides are one dimensional vector spaces. Also, since 𝕍​(Lx⋅λ)=W​h¯f​(Lx⋅λ)=0\mathbb{V}(L_{x\cdot\lambda})=\overline{Wh}_{f}(L_{x\cdot\lambda})=0 for x≠w0x\neq w_{0} we get that hLx⋅λh_{L_{x\cdot\lambda}} is an isomorphism as well. Hence, by exactness of both functors and the five lemma hMh_{M} is an isomorphism for all M∈𝒪λ^M\in\mathcal{O}_{{\widehat{\lambda}}}. ∎

2.5. Projective functors

Let E∈ℱE\in{\mathcal{F}} and M∈ℳM\in\mathcal{M}; then there is the functor

TE:ℳ→ℳ,TE​(M)=E⊗M.T_{E}:\mathcal{M}\to\mathcal{M},\ T_{E}(M)=E\otimes M.

A projective functor from ℳλ^\mathcal{M}_{\widehat{\lambda}} to ℳμ^\mathcal{M}_{\widehat{\mu}} is a direct summand in a functor p​rμ​TE​iλpr_{\mu}T_{E}i_{\lambda}. Projective functors are exact and they have left and right adjoints which coincide.

By [4] we have the following important: Assume that λ\lambda is dominant and let 𝒫,𝒫′:ℳλ^→ℳμ^\mathcal{P},\mathcal{P}^{\prime}:\mathcal{M}_{\widehat{\lambda}}\to\mathcal{M}_{\widehat{\mu}} be projective functors. Let g∈Hom⁡(𝒫​(Δλ),𝒫′​(Δλ))g\in\operatorname{Hom}(\mathcal{P}(\Delta_{\lambda}),\mathcal{P}^{\prime}(\Delta_{\lambda})). Then there is a natural transformation ϕ:𝒫→𝒫′\phi:\mathcal{P}\to\mathcal{P}^{\prime} such that ϕ​(Δλ)=g\phi(\Delta_{\lambda})=g. Moreover, ϕ\phi is uniquely determined by gg if gg is an isomorphism and if gg is an idempotent then we can chose ϕ\phi to be an idempotent as well. Thus decomposing 𝒫​(Δλ)\mathcal{P}(\Delta_{\lambda}) into a direct sum of indecomposables and decomposing 𝒫\mathcal{P} is the same thing.

Assume that λ,μ∈𝔥∗\lambda,\mu\in\mathfrak{h}^{*} such that μ−λ\mu-\lambda is integral and Wλ⊆WμW_{\lambda}\subseteq W_{\mu}. Let VV be a finite dimensional irreducible 𝔤\mathfrak{g}-module with extremal weight μ−λ\mu-\lambda. Then there is the translation to the wall functor

(2.1) Θλμ:ℳλ^→ℳμ^,M↦p​rμ​(TV​(M)).\Theta^{\mu}_{\lambda}:\mathcal{M}_{{\widehat{\lambda}}}\to\mathcal{M}_{{\widehat{\mu}}},\;M\mapsto pr_{\mu}(T_{V}(M)).

Its left and right adjoint is translation out of the wall;

(2.2) Θμλ:ℳμ^→ℳλ^,M↦p​rλ​(TV′​(M)).\Theta^{\lambda}_{\mu}:\mathcal{M}_{{\widehat{\mu}}}\to\mathcal{M}_{{\widehat{\lambda}}},\;M\mapsto pr_{\lambda}(T_{V^{\prime}}(M)).

Here V′V^{\prime} is a finite dimensional irreducible 𝔤\mathfrak{g}-module with extremal weight λ−μ\lambda-\mu. We have Θλμ​Θμλ≅I​dℳμ^|Wμ|\Theta^{\mu}_{\lambda}\Theta^{\lambda}_{\mu}\cong Id^{|W_{\mu}|}_{\mathcal{M}_{\widehat{\mu}}}.

2.5.1. Projective functors on 𝒪\mathcal{O}

Any projective functor 𝒫:ℳλ^→ℳμ^\mathcal{P}:\mathcal{M}_{\widehat{\lambda}}\to\mathcal{M}_{\widehat{\mu}} descends to functors 𝒫:𝒪λ^→𝒪μ^\mathcal{P}:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{O}_{\widehat{\mu}} and 𝒫:𝒪λ^′→𝒪μ^′\mathcal{P}:\mathcal{O}^{\prime}_{\widehat{\lambda}}\to\mathcal{O}^{\prime}_{\widehat{\mu}}. Those are the projective functors on 𝒪\mathcal{O} and 𝒪′\mathcal{O}^{\prime}; they are exact, maps projectives to projectives and commute with duality: 𝒫​(M∗)≅𝒫​(M)∗\mathcal{P}(M^{*})\cong\mathcal{P}(M)^{*}.

Let λ\lambda and μ\mu be as in Section 2.5. Thus we get translation to and out of the wall: Θλμ:𝒪λ^→𝒪μ^\Theta^{\mu}_{\lambda}:\mathcal{O}_{{\widehat{\lambda}}}\to\mathcal{O}_{{\widehat{\mu}}} and Θμλ:𝒪μ^→𝒪λ^\Theta^{\lambda}_{\mu}:\mathcal{O}_{{\widehat{\mu}}}\to\mathcal{O}_{{\widehat{\lambda}}}.

We have

Θλμ​Δx⋅λ≅Δx⋅μ,x∈W,\Theta^{\mu}_{\lambda}\Delta_{x\cdot\lambda}\cong\Delta_{x\cdot\mu},\,x\in W,
Θλμ​Lx⋅λ=Lx⋅μ,x∈Wμ,\Theta^{\mu}_{\lambda}L_{x\cdot\lambda}=L_{x\cdot\mu},\,x\in W^{\mu},

and Θλμ​L​(x⋅λ)=0\Theta^{\mu}_{\lambda}L(x\cdot\lambda)=0, for x∉Wμx\notin W^{\mu}.

For any projective module P∈𝒪λ^P\in\mathcal{O}_{\widehat{\lambda}} there is a unique (upto isomorphism) projective functor 𝒫:𝒪λ^→𝒪λ^\mathcal{P}:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{O}_{\widehat{\lambda}} such that 𝒫​(Δλ)=P\mathcal{P}(\Delta_{\lambda})=P.

2.5.2. Projective functors on 𝒩f\mathcal{N}^{f}

By Lemma 1 any projective functor 𝒫:ℳλ^→ℳμ^\mathcal{P}:\mathcal{M}_{\widehat{\lambda}}\to\mathcal{M}_{\widehat{\mu}} descends to a functor 𝒫:𝒩λ^f→𝒩μ^f\mathcal{P}:\mathcal{N}^{f}_{\widehat{\lambda}}\to\mathcal{N}^{f}_{\widehat{\mu}}. Those are our projective functors on 𝒩f\mathcal{N}^{f}. 111Contrary to the case of 𝒪\mathcal{O} it is a priori not clear whether an indecomposable projective functor on ℳ\mathcal{M} when restricted to 𝒩f\mathcal{N}^{f} remains indecomposable. However we only need those projective functors on 𝒩f\mathcal{N}^{f} that are restrictions of projective endofunctors of ℳ\mathcal{M}.

2.6. Harish-Chandra bimodules and Soergel-Miličić’s equivalence.

Throughout this section we fix dominant integral weights λ\lambda and μ\mu with λ\lambda regular. 222One could weaken the integrality condition to λ−μ\lambda-\mu is integral here. But for the sake of simplicity we have assumed both λ\lambda and μ\mu are integral.

2.6.1.

Let XX be a U−UU-U-bimodule. Then we have the adjoint action a​dad of 𝔤\mathfrak{g} on XX given by a​d​(g)​x=g​x−x​gad(g)x=gx-xg and the sub-bimodule Xa​d​f⊆XX_{adf}\subseteq X consisting of ad-finite vectors. If XX is ad-finite (i.e. if X=Xa​d​fX=X_{adf}) then XX is finitely generated as a bimodule iff X is finitely generated as a left module iff XX is finitely generated as a right module.

The category of Harish Chandra bimodules ℋ\mathcal{H} is the category of finitely generated U−UU-U-bimodules which are locally finite with respect to the adjoint action of 𝔤\mathfrak{g} and to the left (or equivalently the right) action of the center ZZ. The category of Harish-Chandra bimodules decomposes into blocks ℋμ^λ^:=ℋλ^∩ℋμ^{}_{{\widehat{\lambda}}}\mathcal{H}_{{\widehat{\mu}}}:={}_{{\widehat{\lambda}}}\mathcal{H}\cap\mathcal{H}_{{\widehat{\mu}}}, for λ,μ∈𝔥∗\lambda,\mu\in\mathfrak{h}^{*}, where

ℋλ^={X∈ℋ|∃n>0,Jλn​X=0},{}_{{\widehat{\lambda}}}\mathcal{H}=\{X\in\mathcal{H}|\;\exists n>0,J^{n}_{\lambda}X=0\},
ℋμ^={X∈ℋ|∃n>0,X​Jμn=0}.\mathcal{H}_{{\widehat{\mu}}}=\{X\in\mathcal{H}|\;\exists n>0,XJ^{n}_{\mu}=0\}.

Similarly we define ℋλ,ℋμ,ℋμλ{}_{\lambda}\mathcal{H},\mathcal{H}_{\mu},{}_{\lambda}\mathcal{H}_{\mu} and ℋμλ^{}_{{\widehat{\lambda}}}\mathcal{H}_{\mu}. There is an autoequivalence V↦s(V)=:VsV\mapsto s(V)=:V^{s} of ℋ\mathcal{H} where Vs=VV^{s}=V as a set and the U−UU-U-bimodule action is given by u∗v∗u′:=(u′)t​v​utu*v*u^{\prime}:=(u^{\prime})^{t}vu^{t} for v∈Vsv\in V^{s}. Since the Chevalley involution fixes ZZ we have (ℋμ^λ^)s=ℋλ^μ^({}_{{\widehat{\lambda}}}\mathcal{H}_{{\widehat{\mu}}})^{s}={}_{{\widehat{\mu}}}\mathcal{H}_{{\widehat{\lambda}}}.

For E∈ℱE\in{\mathcal{F}} we consider the U−UU-U-bimodule El=EE^{l}=E as a set and with action u∗e∗u′=u​eu*e*u^{\prime}=ue and the U−UU-U-bimodule Er=EE^{r}=E as a set and action u∗e∗u′=(u′)t​eu*e*u^{\prime}=(u^{\prime})^{t}e. Note that (El)s=Er(E^{l})^{s}=E^{r}.

Consider the canonical projections (p​rμ)l:ℋ→ℋμ^(pr_{\mu})^{l}:\mathcal{H}\to{}_{{\widehat{\mu}}}\mathcal{H} and (p​rμ)r:ℋ→ℋμ^(pr_{\mu})^{r}:\mathcal{H}\to\mathcal{H}_{{\widehat{\mu}}}. Since the left and right UU-action commute we see that (p​rμ)l​(ℋλ)=ℋλμ^(pr_{\mu})^{l}(\mathcal{H}_{\lambda})={}_{{\widehat{\mu}}}\mathcal{H}_{\lambda} (p​rμ)l​(ℋλ^)=ℋλ^μ^(pr_{\mu})^{l}(\mathcal{H}_{{\widehat{\lambda}}})={}_{{\widehat{\mu}}}\mathcal{H}_{{\widehat{\lambda}}} and similarly for (p​rμ)r(pr_{\mu})^{r}. If VV is any finite dimensional bimodule and X∈ℋX\in\mathcal{H} then V⊗X∈ℋV\otimes X\in\mathcal{H} with the canonical bimodule structure. Note that (V⊗X)s=Vs⊗Xs(V\otimes X)^{s}=V^{s}\otimes X^{s}.

Similarly, if 𝒫\mathcal{P} is a projective functor on ℳ\mathcal{M} we get projective functors 𝒫l,𝒫r:ℋ→ℋ\mathcal{P}^{l},\mathcal{P}^{r}:\mathcal{H}\to\mathcal{H}.

2.6.2. Equivalences with category 𝒪\mathcal{O}

By results of Bernstein and Gelfand [4], Soergel [15] and Soergel and Miličić [13] we have mutually inverse equivalences

Fμ:𝒪μ^′⇆ℋλ^μ^:GμF_{\mu}:\mathcal{O}^{\prime}_{{\widehat{\mu}}}\leftrightarrows{}_{{\widehat{\mu}}}\mathcal{H}_{{\widehat{\lambda}}}:G_{\mu}

where Fμ(X)=lim→nHomℂ(Δλn,X)a​d​fF_{\mu}(X)={\varinjlim}_{n}\operatorname{Hom}_{\mathbb{C}}(\Delta^{n}_{\lambda},X)_{adf} and Gμ​(Y)=lim←n⁡Y⊗UΔλnG_{\mu}(Y)={\varprojlim}_{n}Y\otimes_{U}\Delta^{n}_{\lambda}. (Here the UU-bimodule structure on Homℂ⁡(Δλn,X)\operatorname{Hom}_{\mathbb{C}}(\Delta^{n}_{\lambda},X) is given by (u​f​u′)​(m)=u⋅f​(u′​m)(ufu^{\prime})(m)=u\cdot f(u^{\prime}m).)

These functors restrict to equivalences

Fμ:𝒪μ^⇆ℋλμ^:GμF_{\mu}:\mathcal{O}_{{\widehat{\mu}}}\leftrightarrows{}_{{\widehat{\mu}}}\mathcal{H}_{{\lambda}}:G_{\mu}

where Fμ(X)=Homℂ(Δλ,X)a​d​fF_{\mu}(X)=\operatorname{Hom}_{\mathbb{C}}(\Delta_{\lambda},X)_{adf} and Gμ​(Y)=Y⊗UΔλG_{\mu}(Y)=Y\otimes_{U}\Delta_{\lambda} which in turn restrict to equivalences

Fμ:𝒪μ⇆ℋλμ:Gμ.F_{\mu}:\mathcal{O}_{\mu}\leftrightarrows{}_{\mu}\mathcal{H}_{{\lambda}}:G_{\mu}.

Let κ:𝒪λ^​⟶∼​𝒪λ′\kappa:\mathcal{O}_{\widehat{\lambda}}\overset{\sim}{\longrightarrow}\mathcal{O}^{\prime}_{\lambda} denote the equivalence which is defined as the composition

(2.3) κ:𝒪λ^​→Fλ​ℋλλ^​→𝑠​ℋλ^λ​→Gλ​𝒪λ′.\kappa:\mathcal{O}_{\widehat{\lambda}}\overset{F_{\lambda}}{\to}{}_{{\widehat{\lambda}}}\mathcal{H}_{\lambda}\overset{s}{\to}{}_{\lambda}\mathcal{H}_{{\widehat{\lambda}}}\overset{G_{\lambda}}{\to}\mathcal{O}^{\prime}_{\lambda}.

Then we have

(2.4) κ​(Δx⋅λ)≅Δx−1⋅λ,x∈W.\kappa(\Delta_{x\cdot\lambda})\cong\Delta_{x^{-1}\cdot\lambda},\ x\in W.

2.6.3. Soergel Miličić’s equivalence

Assume now that Wf=WμW_{f}=W_{\mu}. Soergel and Miličić [13] constructed the equivalence

Gf:ℋμ^λ^→𝒩λ^f,X↦lim←n⁡X⊗UΔμn​(f).G_{f}:{}_{\widehat{{\lambda}}}\mathcal{H}_{{\widehat{\mu}}}\to\mathcal{N}^{f}_{\widehat{{\lambda}}},\ X\mapsto{\varprojlim}_{n}X\otimes_{U}\Delta^{n}_{\mu}(f).

Thus we get the equivalence σ=Gf∘s∘Fμ:𝒪μ^→𝒩λ^f\sigma=G_{f}\circ s\circ F_{\mu}:\mathcal{O}_{{\widehat{\mu}}}\to\mathcal{N}^{f}_{{\widehat{\lambda}}}. It is known that

(2.5) σ​(Δx⋅μ)≅Δx−1⋅λ​(f),x∈W.\sigma(\Delta_{x\cdot\mu})\cong\Delta_{x^{-1}\cdot\lambda}(f),\;x\in W.

The following lemma is easy to prove, for details see [9].

Lemma 2.

For M∈𝒪λM\in\mathcal{O}_{\lambda} we have a natural isomorphism El⊗Fλ​(M)​⟶∼​Fλ​(E⊗M)E^{l}\otimes F_{\lambda}(M)\overset{\sim}{\longrightarrow}F_{\lambda}(E\otimes M), e⊗ϕ↦{x↦e⊗ϕ​(x)}e\otimes\phi\mapsto\{x\mapsto e\otimes\phi(x)\}. It induces an isomorphism (p​rλ)l​(El⊗Fλ​(M))​⟶∼​Fλ​(p​rλ​(E⊗M))(pr_{\lambda})^{l}(E^{l}\otimes F_{\lambda}(M))\overset{\sim}{\longrightarrow}F_{\lambda}(pr_{\lambda}(E\otimes M)). Similarly, we have a natural isomorphism (Θλμ)l​Fλ​(M)≅Fμ​Θλμ​(M)(\Theta^{\mu}_{\lambda})^{l}F_{\lambda}(M)\cong F_{\mu}\Theta^{\mu}_{\lambda}(M).

3. The main results.

Throughout this section we assume that λ\lambda is a regular dominant integral weight and μ\mu a dominant integral weight. Let f:U​(𝔫)→ℂf:U(\mathfrak{n})\to\mathbb{C} be a character such that 𝔹f=𝔹μ\mathbb{B}_{f}=\mathbb{B}_{\mu}.

Proposition 3.

a.) For any projective functor 𝒫:ℳ→ℳ\mathcal{P}:\mathcal{M}\to\mathcal{M} there is a natural isomorphism Θ𝒫:𝒫∘Γ¯f→Γ¯f∘𝒫\Theta_{\mathcal{P}}:\mathcal{P}\circ\overline{\Gamma}_{f}\to\overline{\Gamma}_{f}\circ\mathcal{P} between functors from 𝒪\mathcal{O} to 𝒩f\mathcal{N}^{f}. b) For any morphism ϕ:𝒫→𝒫′\phi:\mathcal{P}\to\mathcal{P}^{\prime} of projective functors we have a commutative diagram

𝒫∘Γ¯f\textstyle{\mathcal{P}\circ\overline{\Gamma}_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕ∘Γ¯f\scriptstyle{\phi\circ\overline{\Gamma}_{f}}Θ𝒫\scriptstyle{\Theta_{\mathcal{P}}}𝒫′∘Γ¯f\textstyle{\mathcal{P}^{\prime}\circ\overline{\Gamma}_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Θ𝒫′\scriptstyle{\Theta_{\mathcal{P}^{\prime}}}Γ¯f∘𝒫\textstyle{\overline{\Gamma}_{f}\circ\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Γ¯f∘ϕ\scriptstyle{\overline{\Gamma}_{f}\circ\phi}Γ¯f∘𝒫′\textstyle{\overline{\Gamma}_{f}\circ\mathcal{P}^{\prime}}
Proof.

It is enough to prove a)a) and b)b) when 𝒫=TE\mathcal{P}=T_{E} and 𝒫′=TE′\mathcal{P}^{\prime}=T_{E^{\prime}} because then it follows that a)a) also holds for any direct summand in TET_{E} by taking ϕ:TE→TE\phi:T_{E}\to T_{E} to be an orthogonal projection onto this summand. Then the general case for b)b) follows as well.

We prove a) for 𝒫=TE\mathcal{P}=T_{E}. Since EE is finite dimensional we have E⊗M¯=E⊗M¯.\overline{E\otimes M}=E\otimes\overline{M}. Therefore

Γ¯f​(E⊗M)=Γf​(E⊗M¯)=Γf​(E⊗M¯).\overline{\Gamma}_{f}(E\otimes M)=\Gamma_{f}(\overline{E\otimes M})=\Gamma_{f}(E\otimes\overline{M}).

By Lemma 1 b) we conclude that E⊗Γf​(M¯)⊆Γf​(E⊗M¯)E\otimes\Gamma_{f}(\overline{M})\subseteq\Gamma_{f}(E\otimes\overline{M}). This inclusion is denoted ΘTE​(M)\Theta_{T_{E}}(M). In order to prove that it is an isomorphism, we proceed as follows: Let e1,…​ek{e}_{1},\ldots{e}_{k} be a basis for EE where ei{e}_{i} is a weight vector of weight λi\lambda_{i} ordered in such a way that λi<λj\lambda_{i}<\lambda_{j} implies i<ji<j. Let ∑i=1kei⊗mi∈Γf​(E⊗M¯)\sum^{k}_{i=1}{e}_{i}\otimes m_{i}\in\Gamma_{f}(E\otimes\overline{M}). We must prove that each mi∈Γf​(M¯)m_{i}\in\Gamma_{f}(\overline{M}). Pick n=n1>0n=n_{1}>0 such that for all α∈𝔹\alpha\in\mathbb{B} we have

0=(Eα−f​(Eα))n​(∑i=1kei⊗mi)=∑i=1k∑j=0n(nj)​Eαj​ei⊗(Eα−f​(Eα))n−j​mi.0=(E_{\alpha}-f(E_{\alpha}))^{n}(\sum^{k}_{i=1}{e}_{i}\otimes m_{i})=\sum^{k}_{i=1}\sum^{n}_{j=0}\binom{n}{j}E^{j}_{\alpha}{e}_{i}\otimes(E_{\alpha}-f(E_{\alpha}))^{n-j}m_{i}.

Note that Spanℂ⁡{e2,…,ek}\operatorname{Span}_{\mathbb{C}}\{e_{2},\ldots,e_{k}\} is a U​(𝔫)U(\mathfrak{n})-submodule of VV. Therefore it follows that e1⊗(Eα−f​(Eα))n​m1=0e_{1}\otimes(E_{\alpha}-f(E_{\alpha}))^{n}m_{1}=0 and hence that (Eα−f​(Eα))n​m1=0(E_{\alpha}-f(E_{\alpha}))^{n}m_{1}=0.

Note that Eαk​ei=0E^{k}_{\alpha}{e}_{i}=0 for all ii. Let n2=n+kn_{2}=n+k. Then (Eα−f​(Eα))n2​(e1⊗m1)=0(E_{\alpha}-f(E_{\alpha}))^{n_{2}}(e_{1}\otimes m_{1})=0 and therefore

(Eα−f​(Eα))n2​(∑i=2kei⊗mi)=0.(E_{\alpha}-f(E_{\alpha}))^{n_{2}}(\sum^{k}_{i=2}{e}_{i}\otimes m_{i})=0.

By repeating the above argument we conclude that (Eα−f​(Eα))n2​m2=0(E_{\alpha}-f(E_{\alpha}))^{n_{2}}m_{2}=0. Proceeding by induction we conclude that mi∈Γf​(M¯)m_{i}\in\Gamma_{f}(\overline{M}) for all ii.

We now prove b)b) for 𝒫=TE\mathcal{P}=T_{E} and 𝒫′=TE′\mathcal{P}^{\prime}=T_{E^{\prime}}. Let M∈𝒪M\in\mathcal{O} and ϕM:TE​(M)→TE′​(M)\phi_{M}:T_{E}(M)\to T_{E^{\prime}}(M) be the morphism given by ϕ:TE→TE′\phi:T_{E}\to T_{E^{\prime}}. We must show that the following diagram commutes:

TE​Γ¯f​(M)\textstyle{T_{E}\overline{\Gamma}_{f}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕΓ¯f​(M)\scriptstyle{\phi_{\overline{\Gamma}_{f}(M)}}ΘTE\scriptstyle{\Theta_{T_{E}}}TE′​Γ¯f​(M)\textstyle{T_{E^{\prime}}\overline{\Gamma}_{f}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ΘTE′\scriptstyle{\Theta_{T_{E^{\prime}}}}Γ¯f​TE​(M)\textstyle{\overline{\Gamma}_{f}T_{E}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Γ¯f​ϕM\scriptstyle{\overline{\Gamma}_{f}{\phi_{M}}}Γ¯f​TE′​(M)\textstyle{\overline{\Gamma}_{f}T_{E^{\prime}}(M)}

Let e1,…​ek{e}_{1},\ldots{e}_{k} and e1′,…,el′{e}^{\prime}_{1},\ldots,{e}^{\prime}_{l} be bases of EE and E′E^{\prime}. Let U¯=U/𝔞\overline{U}=U/\mathfrak{a} where 𝔞⊂Z\mathfrak{a}\subset Z is an ideal of finite codimension such that 𝔞​M=𝔞​TE​(M)=0\mathfrak{a}M=\mathfrak{a}T_{E}(M)=0. Then automatically 𝔞​Γ¯f​(M)=0\mathfrak{a}\overline{\Gamma}_{f}(M)=0. Pick ui​j∈U¯u_{ij}\in\overline{U} such that ϕU¯​(ei⊗1)=∑ej′⊗ui​j\phi_{\overline{U}}({e}_{i}\otimes 1)=\sum{e}^{\prime}_{j}\otimes u_{ij}. Then ϕA​(ei⊗a)=∑ej′⊗ui​j​a\phi_{A}({e}_{i}\otimes a)=\sum{e}^{\prime}_{j}\otimes u_{ij}a for any A∈U¯A\in\overline{U}-mod\operatorname{mod}, by functoriality. Hence we get for ei​⊗⁡m∈TE​Γ¯f​(M){e}_{i}\operatorname{\otimes}m\in T_{E}\overline{\Gamma}_{f}(M) that

Γ¯f​ϕM​(ΘTE​(ei​⊗⁡m))=Γ¯f​ϕM​(ei​⊗⁡m)=\overline{\Gamma}_{f}{\phi_{M}}(\Theta_{T_{E}}({e}_{i}\operatorname{\otimes}m))=\overline{\Gamma}_{f}{\phi_{M}}({e}_{i}\operatorname{\otimes}m)=
∑ej′​⊗⁡ui​j​m=ϕΓ¯f​(M)​(ei​⊗⁡m)=ΘTE′​(ϕΓ¯f​(M)​(ei​⊗⁡m)).\sum{e}^{\prime}_{j}\operatorname{\otimes}u_{ij}m=\phi_{\overline{\Gamma}_{f}(M)}({e}_{i}\operatorname{\otimes}m)=\Theta_{T_{E^{\prime}}}(\phi_{\overline{\Gamma}_{f}(M)}({e}_{i}\operatorname{\otimes}m)).

∎

A weaker version of a) above was proved in [6] Proposition 2.3.4. He showed that Γ¯f​(TE​(Δλ))≅TE​(Γ¯f​(Δλ))\overline{\Gamma}_{f}(T_{E}(\Delta_{\lambda}))\cong T_{E}(\overline{\Gamma}_{f}(\Delta_{\lambda})).

3.0.1.

Let ss be a simple reflection and assume that Wμ={e,s}W_{\mu}=\{e,s\}. Then there is the wallcrossing functor Θs:=Θμλ​Θλμ:𝒪λ^→𝒪λ^\Theta_{s}:=\Theta^{\lambda}_{\mu}\Theta^{\mu}_{\lambda}:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{O}_{\widehat{\lambda}}. Let Ψs\Psi_{s} denote the complex of functors I​d→ΘsId\to\Theta_{s} (given by the adjunction morphism). Then if w=s1​⋯​smw=s_{1}\cdots s_{m} is a reduced expression of an element in WW we get the Rouqier complex functor Ψw=Ψs1∘⋯∘Ψsm\Psi_{w}=\Psi_{s_{1}}\circ\cdots\circ\Psi_{s_{m}}, see [11].

Corollary 1.

Let x∈Wx\in W and let ss be a simple reflection such that x≺x​sx\prec xs. Then Θs​(Δx⋅λ​(f))=Θs​(Δx​s⋅λ​(f))\Theta_{s}(\Delta_{x\cdot\lambda}(f))=\Theta_{s}(\Delta_{xs\cdot\lambda}(f)) and the adjunction morphisms I​d→ΘsId\to\Theta_{s} and Θs→I​d\Theta_{s}\to Id define a short exact sequence

0→Δx⋅λ​(f)→Θs​(Δx⋅λ​(f))→Δx​s⋅λ​(f)→0.0\to\Delta_{x\cdot\lambda}(f)\to\Theta_{s}(\Delta_{x\cdot\lambda}(f))\to\Delta_{xs\cdot\lambda}(f)\to 0.

Hence the Rouquier complex Ψw​(Δx⋅λ​(f))\Psi_{w}(\Delta_{x\cdot\lambda}(f)) is an exact resolution of its 0’th cohomology for any x∈Wx\in W.

Proof.

It is well-known that the sequence 0→Δx⋅λ→Θs​(Δx⋅λ)→Δx​s⋅λ→00\to\Delta_{x\cdot\lambda}\to\Theta_{s}(\Delta_{x\cdot\lambda})\to\Delta_{xs\cdot\lambda}\to 0 is exact. Applying the functor Γ¯f\overline{\Gamma}_{f} we get from Propositions 1 and 3 the first two statements of the lemma. This implies formally that the above Whittaker Rouquier complex is exact as well. ∎

3.1. The functor τ:𝒪λ^→𝒩λ^f\tau:\mathcal{O}_{{\widehat{\lambda}}}\to\mathcal{N}^{f}_{{\widehat{\lambda}}}

Recall the equivalences

κ:𝒪λ^​⟶Fλ​ℋλλ^​⟶𝑠​ℋλ^λ​⟶Gλ​𝒪λ′​ and ​σ:𝒪μ^′​⟶Fμ​ℋλ^μ^​⟶𝑠​ℋμ^λ^​⟶Gf​𝒩λ^f.\kappa:\mathcal{O}_{{\widehat{\lambda}}}\overset{F_{{\lambda}}}{\longrightarrow}{}_{{\widehat{\lambda}}}\mathcal{H}_{{\lambda}}\overset{s}{\longrightarrow}{}_{{\lambda}}\mathcal{H}_{{\widehat{\lambda}}}\overset{G_{{\lambda}}}{\longrightarrow}\mathcal{O}^{\prime}_{\lambda}\hbox{ and }\sigma:\mathcal{O}^{\prime}_{{\widehat{\mu}}}\overset{F_{\mu}}{\longrightarrow}{}_{{\widehat{\mu}}}\mathcal{H}_{{\widehat{\lambda}}}\overset{s}{\longrightarrow}{}_{{\widehat{\lambda}}}\mathcal{H}_{{\widehat{\mu}}}\overset{G_{f}}{\longrightarrow}\mathcal{N}^{f}_{{\widehat{\lambda}}}.

Let τ\tau denote the composition

(3.1) τ=σ∘Θλμ∘κ:𝒪λ^​⟶𝜅​𝒪λ′​⟶Θλμ​𝒪μ^′​⟶𝜎​𝒩λ^f.\tau=\sigma\circ\Theta^{\mu}_{\lambda}\circ\kappa:\mathcal{O}_{{\widehat{\lambda}}}\overset{\kappa}{\longrightarrow}\mathcal{O}^{\prime}_{\lambda}\overset{\Theta^{\mu}_{\lambda}}{\longrightarrow}\mathcal{O}^{\prime}_{{\widehat{\mu}}}\overset{\sigma}{\longrightarrow}\mathcal{N}^{f}_{{\widehat{\lambda}}}.

It is evident that τ\tau is exact. We shall show in Theorem 1 that τ\tau is equivalent to Γ¯f\overline{\Gamma}_{f}. Since κ\kappa and σ\sigma are equivalences τ\tau is in reality determined by the projective functor Θλμ:𝒪λ′→𝒪μ^′\Theta^{\mu}_{\lambda}:\mathcal{O}^{\prime}_{\lambda}\to\mathcal{O}^{\prime}_{{\widehat{\mu}}}. We shall see in Lemma 3 below that the functor τ\tau commutes with projective functors (and with morphisms of projective functors). The functor Θλμ:𝒪λ′→𝒪μ^′\Theta^{\mu}_{\lambda}:\mathcal{O}^{\prime}_{\lambda}\to\mathcal{O}^{\prime}_{{\widehat{\mu}}} obviously doesn’t commute with projective functors, but after conjugating it with the equivalences σ\sigma and κ\kappa it does. The reason for this is essentially that the left and right action of UU on a Harish-Chandra bimodule commutes.

The left adjoint of the inclusion i:𝒪λ′→𝒪λ^′i:\mathcal{O}^{\prime}_{\lambda}\to\mathcal{O}^{\prime}_{{\widehat{\lambda}}} is ()⊗UUλ(\ )\otimes_{U}U_{\lambda} and its right adjoint is ()Jλ(\ )^{J_{\lambda}}, the functor of taking JλJ_{\lambda}-invariants. Therefore τ\tau has the left adjoint

(3.2) τL=κ−1∘Θμλ​()⊗UUλ∘σ−1.\tau^{L}=\kappa^{-1}\circ\Theta^{\lambda}_{\mu}(\ )\otimes_{U}U_{\lambda}\circ\sigma^{-1}.

and the right adjoint

(3.3) τR=κ−1∘Θμλ​()Jλ∘σ−1.\tau^{R}=\kappa^{-1}\circ\Theta^{\lambda}_{\mu}(\ )^{J_{\lambda}}\circ\sigma^{-1}.

3.1.1.

Expanding the maps in (3.1) we deduce a canonical equivalence of functors

(3.4) τ≅Gf∘(Θλμ)r∘Fλ:𝒪λ^​⟶Fλ​ℋλλ^​⟶(Θλμ)r​ℋμ^λ^​⟶Gf​𝒩λ^f.\tau\cong G_{f}\circ(\Theta^{\mu}_{\lambda})^{r}\circ F_{\lambda}:\mathcal{O}_{{\widehat{\lambda}}}\overset{F_{\lambda}}{\longrightarrow}{}_{{\widehat{\lambda}}}\mathcal{H}_{{\lambda}}\overset{(\Theta^{\mu}_{\lambda})^{r}}{\longrightarrow}{}_{{\widehat{\lambda}}}\mathcal{H}_{{\widehat{\mu}}}\overset{G_{f}}{\longrightarrow}\mathcal{N}^{f}_{{\widehat{\lambda}}}.

This formula is easier to work with.

Lemma 3.

a) For any projective functor 𝒫:ℳλ^→ℳλ^\mathcal{P}:\mathcal{M}_{\widehat{\lambda}}\to\mathcal{M}_{\widehat{\lambda}} there is a natural isomorphism η𝒫:𝒫∘τ→τ∘𝒫\eta_{\mathcal{P}}:\mathcal{P}\circ\tau\to\tau\circ\mathcal{P} between functors from 𝒪λ^\mathcal{O}_{\widehat{\lambda}} to 𝒩λ^f\mathcal{N}^{f}_{\widehat{\lambda}}. b) For any morphism ϕ:𝒫→𝒫′\phi:\mathcal{P}\to\mathcal{P}^{\prime} of projective functors we have a commutative diagram

𝒫​τ\textstyle{\mathcal{P}\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕτ\scriptstyle{\phi_{\tau}}η𝒫\scriptstyle{\eta_{\mathcal{P}}}𝒫′​τ\textstyle{\mathcal{P}^{\prime}\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}η𝒫′\scriptstyle{\eta_{\mathcal{P}^{\prime}}}τ​𝒫\textstyle{\tau\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ\scriptstyle{\tau}τ​𝒫′\textstyle{\tau\mathcal{P}^{\prime}}
Proof.

By similar arguments as were given in the beginning of the proof of Proposition 3 we may assume that 𝒫=p​rλ​TE​iλ\mathcal{P}=pr_{\lambda}T_{E}i_{\lambda} and 𝒫′=p​rλ​TE′​iλ\mathcal{P}^{\prime}=pr_{\lambda}T_{E^{\prime}}i_{\lambda}. Let VV be a finite dimensional irreducible representation with extremal weight μ−λ\mu-\lambda so that Θλμ=p​rμ​TV\Theta^{\mu}_{\lambda}=pr_{\mu}T_{V}. We use (3.4) as the definition of τ\tau.

a) Let M∈𝒪λ^M\in\mathcal{O}_{{\widehat{\lambda}}}. Let η𝒫=η𝒫​(M)\eta_{\mathcal{P}}=\eta_{\mathcal{P}}(M) denote the composition of the natural isomorphisms:

𝒫​τ​(M)=𝒫​Gf∘(Θλμ)r∘Fλ​(M)≅Gf∘𝒫l∘(Θλμ)r∘Fλ​(M)≅\mathcal{P}\tau(M)=\mathcal{P}G_{f}\circ(\Theta^{\mu}_{\lambda})^{r}\circ F_{\lambda}(M)\cong G_{f}\circ\mathcal{P}^{l}\circ(\Theta^{\mu}_{\lambda})^{r}\circ F_{\lambda}(M)\cong
Gf∘(Θλμ)r∘𝒫l∘Fλ​(M)≅Gf∘(Θλμ)r∘Fλ​(𝒫​M)=τ​𝒫​(M).G_{f}\circ(\Theta^{\mu}_{\lambda})^{r}\circ\mathcal{P}^{l}\circ F_{\lambda}(M)\cong G_{f}\circ(\Theta^{\mu}_{\lambda})^{r}\circ F_{\lambda}(\mathcal{P}M)=\tau\mathcal{P}(M).

Let us describe η𝒫\eta_{\mathcal{P}} explicitly. Let X∈𝒪λ^X\in\mathcal{O}_{\widehat{\lambda}} and Y∈ℋμ^λ^Y\in{}_{{\widehat{\lambda}}}\mathcal{H}_{{\widehat{\mu}}}. An element in 𝒫​X\mathcal{P}X is a (linear combination of elements of the form) e⊗xe\otimes x, e∈Ee\in E and x∈Xx\in X and an element in Fλ​XF_{\lambda}X can be represented by a ℂ\mathbb{C}-linear function ψ:Δn→X\psi:\Delta^{n}\to X (for nn sufficiently large). An element in (Θλμ)r​Y(⊆Vr⊗Y)(\Theta^{\mu}_{\lambda})^{r}Y\;(\subseteq V^{r}\otimes Y) can be written as vr⊗yv^{r}\otimes y for v∈Vv\in V, y∈Yy\in Y, and an element in Gf​(Y)G_{f}(Y) can be represented by y⊗U1¯∈Y⊗UΔμn​(f)y\otimes_{U}\overline{1}\in Y\otimes_{U}\Delta^{n}_{\mu}(f) (for nn sufficiently large). Here 1¯∈Δμn​(f)\overline{1}\in\Delta^{n}_{\mu}(f) is a generator.

Therefore an element in 𝒫​τ​(M)\mathcal{P}\tau(M) can be written as (a linear combination of elements)

e⊗((vr⊗ψ)⊗U1¯),e\otimes((v^{r}\otimes\psi)\otimes_{U}\overline{1}),

where ψ∈Homℂ⁡(Δn→M)\psi\in\operatorname{Hom}_{\mathbb{C}}(\Delta^{n}\to M), e∈Ee\in E and v∈Vv\in V. We then have

(3.5) η𝒫​(e⊗((vr⊗ψ)⊗U1¯))=(vr⊗(el⊗ψ))⊗U1¯,\eta_{\mathcal{P}}(e\otimes((v^{r}\otimes\psi)\otimes_{U}\overline{1}))=(v^{r}\otimes(e^{l}\otimes\psi))\otimes_{U}\overline{1},

where el⊗ψ∈Hom⁡(Δλ,E⊗M)e^{l}\otimes\psi\in\operatorname{Hom}(\Delta_{\lambda},E\otimes M) denotes the function d↦e⊗ψ​(d)d\mapsto e\otimes\psi(d).

b) Let M∈𝒪λ^M\in\mathcal{O}_{\widehat{\lambda}}. We want to establish the commutativity of the diagram

(3.6) 𝒫​τ​(M)\textstyle{\mathcal{P}\tau(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕτ​(M)\scriptstyle{\phi_{\tau(M)}}η𝒫\scriptstyle{\eta_{\mathcal{P}}}𝒫′​τ​(M)\textstyle{\mathcal{P}^{\prime}\tau(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}η𝒫′\scriptstyle{\eta_{\mathcal{P}^{\prime}}}τ​𝒫​(M)\textstyle{\tau\mathcal{P}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ​(ϕM)\scriptstyle{\tau(\phi_{M})}τ​𝒫′​(M)\textstyle{\tau\mathcal{P}^{\prime}(M)}

Let N>0N>0 be such that JλNJ^{N}_{\lambda} annihilates MM, τ​(M)\tau(M) and 𝒫​(M)\mathcal{P}(M) and define U¯=U/JλN\overline{U}=U/J^{N}_{\lambda}.

Let e1,…​ek{e}_{1},\ldots{e}_{k} and e1′,…,el′e^{\prime}_{1},\ldots,e^{\prime}_{l} be bases of EE and E′E^{\prime}, respectively. Then ϕU¯:𝒫​U¯→𝒫′​U¯\phi_{\overline{U}}:\mathcal{P}\overline{U}\to\mathcal{P}^{\prime}\overline{U} is given by ϕU¯​(ei⊗1)=∑jej′⊗ui​j\phi_{\overline{U}}({e}_{i}\otimes 1)=\sum_{j}e^{\prime}_{j}\otimes u_{ij} for some ui​j∈U¯u_{ij}\in\overline{U}. Therefore we get ϕA:𝒫​A→𝒫′​A\phi_{A}:\mathcal{P}A\to\mathcal{P}^{\prime}A is given by ϕA​(ei⊗a)=∑ej′⊗ui​j​a\phi_{A}({e}_{i}\otimes a)=\sum e^{\prime}_{j}\otimes u_{ij}a, for A∈U¯A\in\overline{U}-mod\operatorname{mod} by functoriality.
We now calculate

τ​(ϕM)∘η𝒫​(ei⊗((vr⊗ψ)⊗U1¯))=τ​(ϕM)​((vr⊗(eil⊗ψ))⊗U1¯)=\tau(\phi_{M})\circ\eta_{\mathcal{P}}(e_{i}\otimes((v^{r}\otimes\psi)\otimes_{U}\overline{1}))=\tau(\phi_{M})((v^{r}\otimes(e^{l}_{i}\otimes\psi))\otimes_{U}\overline{1})=
∑j(vr⊗(ej′⁣l⊗ui​j​ψ))⊗U1¯.\sum_{j}(v^{r}\otimes(e^{\prime l}_{j}\otimes u_{ij}\psi))\otimes_{U}\overline{1}.

On the other hand we have

η𝒫′∘ϕτ​(M)​(ei⊗((vr⊗ψ)⊗U1¯))=∑jη𝒫′​(ej′⊗ui​j​((vr⊗ψ)⊗U1¯))=\eta_{\mathcal{P}^{\prime}}\circ\phi_{\tau(M)}(e_{i}\otimes((v^{r}\otimes\psi)\otimes_{U}\overline{1}))=\sum_{j}\eta_{\mathcal{P}^{\prime}}(e^{\prime}_{j}\otimes u_{ij}((v^{r}\otimes\psi)\otimes_{U}\overline{1}))=
∑jη𝒫′​(ej′⊗((vr⊗ui​j​ψ)⊗U1¯))=∑j(vr⊗(ej′⁣l⊗ui​j​ψ))⊗U1¯.\sum_{j}\eta_{\mathcal{P}^{\prime}}(e^{\prime}_{j}\otimes((v^{r}\otimes u_{ij}\psi)\otimes_{U}\overline{1}))=\sum_{j}(v^{r}\otimes(e^{\prime l}_{j}\otimes u_{ij}\psi))\otimes_{U}\overline{1}.

∎

Theorem 1.

τ\tau is naturally equivalent to Γ¯f\overline{\Gamma}_{f}.

Proof.

Let P​r​o​j​(𝒪λ^)Proj(\mathcal{O}_{\widehat{\lambda}}) denote the full subcategory of projective objects in 𝒪λ^\mathcal{O}_{\widehat{\lambda}}. Let 𝒜{\mathcal{A}} be the full subcategory of 𝒪λ^\mathcal{O}_{\widehat{\lambda}} whose objects are of the form 𝒫​(Δλ)\mathcal{P}(\Delta_{\lambda}), where 𝒫:ℳλ^→ℳλ^\mathcal{P}:\mathcal{M}_{{\widehat{\lambda}}}\to\mathcal{M}_{{\widehat{\lambda}}} is a projective functor. Then the inclusion 𝒜→P​r​o​j​(𝒪λ^){\mathcal{A}}\to Proj(\mathcal{O}_{\widehat{\lambda}}) is an equivalence of categories. Let P=𝒫​(Δλ),P′=𝒫′​(Δλ)∈ℋP=\mathcal{P}(\Delta_{\lambda}),P^{\prime}=\mathcal{P}^{\prime}(\Delta_{\lambda})\in{\mathcal{H}} and let g:P→P′g:P\to P^{\prime} be a morphism. Then according to [4] there is a natural transformation ϕ:𝒫→𝒫′\phi:\mathcal{P}\to\mathcal{P}^{\prime} (not necessarily unique) such that ϕΔλ=g\phi_{\Delta_{\lambda}}=g. By Proposition 1 we have Γ¯f​(Δλ)≅Δλ​(f)\overline{\Gamma}_{f}(\Delta_{\lambda})\cong\Delta_{\lambda}(f) and using (2.4), (2.5) and Θλμ​(Δλ)≅Δμ\Theta^{\mu}_{\lambda}(\Delta_{\lambda})\cong\Delta_{\mu} we also obtain τ​(Δλ)≅Δλ​(f).\tau(\Delta_{\lambda})\cong\Delta_{\lambda}(f). Fix an isomorphism

ϵ:Γ¯f​(Δλ)​⟶∼​τ​(Δλ).\epsilon:\overline{\Gamma}_{f}(\Delta_{\lambda})\overset{\sim}{\longrightarrow}\tau(\Delta_{\lambda}).

Consider the diagram

Γ¯f​𝒫​(Δλ)\textstyle{\overline{\Gamma}_{f}\mathcal{P}({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Γ¯f​(ϕΔλ)\scriptstyle{\overline{\Gamma}_{f}(\phi_{{\Delta_{\lambda}}})}Θ𝒫−1\scriptstyle{\Theta^{-1}_{\mathcal{P}}}Γ¯f​𝒫′​(Δλ)\textstyle{\overline{\Gamma}_{f}\mathcal{P}^{\prime}({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Θ𝒫′−1\scriptstyle{\Theta^{-1}_{\mathcal{P}^{\prime}}}𝒫​Γ¯f​(Δλ)\textstyle{\mathcal{P}\overline{\Gamma}_{f}({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕΓ¯f​Δλ\scriptstyle{\phi_{\overline{\Gamma}_{f}{\Delta_{\lambda}}}}𝒫​(ϵ)\scriptstyle{\mathcal{P}(\epsilon)}𝒫′​Γ¯f​(Δλ)\textstyle{\mathcal{P}^{\prime}\overline{\Gamma}_{f}({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒫′​(ϵ)\scriptstyle{\mathcal{P}^{\prime}(\epsilon)}𝒫​τ​(Δλ)\textstyle{\mathcal{P}\tau({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕτ​Δλ\scriptstyle{\phi_{\tau{\Delta_{\lambda}}}}η𝒫\scriptstyle{\eta_{\mathcal{P}}}𝒫′​τ​(Δλ)\textstyle{\mathcal{P}^{\prime}\tau({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}η𝒫′\scriptstyle{\eta_{\mathcal{P}^{\prime}}}τ​𝒫​(Δλ)\textstyle{\tau\mathcal{P}({\Delta_{\lambda}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ​(ϕΔλ)\scriptstyle{\tau(\phi_{{\Delta_{\lambda}}})}τ​𝒫′​(Δλ)\textstyle{\tau\mathcal{P}^{\prime}({\Delta_{\lambda}})}

The middle square is obviously commutative and by Proposition 3 and Lemma 3 the top and the bottom squares are commutative as well. Thus the outer square is commutative and therefore we can define the natural transformation ψ:Γ¯f|𝒜→τ|𝒜\psi:\overline{\Gamma}_{f}|_{\mathcal{A}}\to\tau|_{\mathcal{A}} by

ψP=η𝒫∘𝒫​(ϵ)∘Θ𝒫−1​(Δλ):Γ¯f​(P)→τ​(P).\psi_{P}=\eta_{\mathcal{P}}\circ\mathcal{P}(\epsilon)\circ\Theta^{-1}_{\mathcal{P}}(\Delta_{\lambda}):\overline{\Gamma}_{f}(P)\to\tau(P).

Evidently ψ\psi is an equivalence and it induces an equivalence between the induced functors

Γ¯f,τ:Db​(𝒪λ^)≅Kb​(P​r​o​j​(𝒪λ^))≅Kb​(𝒜)→Db​(𝒩λ^f).\overline{\Gamma}_{f},\tau:D^{b}(\mathcal{O}_{\widehat{\lambda}})\cong K^{b}(Proj(\mathcal{O}_{{\widehat{\lambda}}}))\cong K^{b}({\mathcal{A}})\to D^{b}(\mathcal{N}^{f}_{{\widehat{\lambda}}}).

Since Γ¯f\overline{\Gamma}_{f} and τ\tau are exact and 𝒪λ^\mathcal{O}_{\widehat{\lambda}} is the heart of the standard tt-structure on Db​(𝒪λ^)D^{b}(\mathcal{O}_{\widehat{\lambda}}) the last equivalence restricts to an equivalence between the original functors Γ¯f,τ:𝒪λ^→𝒩λ^f\overline{\Gamma}_{f},\tau:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{N}^{f}_{{\widehat{\lambda}}}. ∎

Remark 1.

As a special case we see that Soergel’s functor 𝕍\mathbb{V} is equivalent to σ​Θλ−ρ​κ\sigma\Theta^{-\rho}_{\lambda}\kappa. This follows from Theorem 1 and Proposition 2.

Corollary 2.

The functor Γ¯f:𝒪λ^→𝒩λ^f\overline{\Gamma}_{f}:\mathcal{O}_{\widehat{\lambda}}\to\mathcal{N}^{f}_{{\widehat{\lambda}}} has a left adjoint Γ¯fL\overline{\Gamma}_{f}^{L} and a right adjoint Γ¯fR\overline{\Gamma}_{f}^{R}. The restricted functor Γ¯f:𝒪λ^→Γ¯f​(𝒪λ^)\overline{\Gamma}_{f}:\mathcal{O}_{\widehat{\lambda}}\to\overline{\Gamma}_{f}(\mathcal{O}_{\widehat{\lambda}}) is a quotient functor, i.e. the adjunction map V↦Γ¯f​Γ¯fL​VV\mapsto\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}V is an isomorphism for V∈Γ¯f​(𝒪λ^)V\in\overline{\Gamma}_{f}(\mathcal{O}_{\widehat{\lambda}}).

Proof.

The functors Γ¯fL\overline{\Gamma}_{f}^{L} and Γ¯fR\overline{\Gamma}_{f}^{R} are obtained by transporting the functors τL\tau^{L} and τR\tau^{R} from (3.2) and (3.3) by means of the equivalence Γ¯f≅τ\overline{\Gamma}_{f}\cong\tau. Let V=Γ¯f​(M)V=\overline{\Gamma}_{f}(M), M∈𝒪λ^M\in\mathcal{O}_{\widehat{\lambda}}. We must prove that the adjunction V→Γ¯f​Γ¯fL​VV\to\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}V is an isomorphism.

We first show that Δλ​(f)↦Γ¯f​Γ¯fL​Δλ​(f)\Delta_{\lambda}(f)\mapsto\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}\Delta_{\lambda}(f) is an isomorphism. For this purpose it is obviously enough to show that the adjunction Δλ​(f)​⟶∼​τ​τL​(Δλ​(f))\Delta_{\lambda}(f)\overset{\sim}{\longrightarrow}\tau\tau^{L}(\Delta_{\lambda}(f)) is an isomorphism. Recall that τ\tau equals the composition

𝒪λ^​→𝜅​𝒪λ′​⟶Θλμ​𝒪μ^′​→𝜎​𝒩λ^f.\mathcal{O}_{{\widehat{\lambda}}}\overset{\kappa}{\to}\mathcal{O}^{\prime}_{\lambda}\overset{\Theta^{\mu}_{\lambda}}{\longrightarrow}\mathcal{O}^{\prime}_{{\widehat{\mu}}}\overset{\sigma}{\to}\mathcal{N}^{f}_{{\widehat{\lambda}}}.

The left adjoint of 𝒪λ′​⟶Θλμ​𝒪μ^′\mathcal{O}^{\prime}_{\lambda}\overset{\Theta^{\mu}_{\lambda}}{\longrightarrow}\mathcal{O}^{\prime}_{{\widehat{\mu}}} is Θμλ​()⊗UUλ\Theta^{\lambda}_{\mu}(\ )\otimes_{U}U_{\lambda}, see (3.2). Since κ\kappa and σ\sigma are equivalences and σ−1​Δλ​(f)≅Δμ\sigma^{-1}\Delta_{\lambda}(f)\cong\Delta_{\mu} it suffices to show that the adjunction map Δμ↦Θλμ​(Θμλ​(Δμ)⊗UUλ)\Delta_{\mu}\mapsto\Theta^{\mu}_{\lambda}(\Theta^{\lambda}_{\mu}(\Delta_{\mu})\otimes_{U}U_{\lambda}) is an isomorphism. This is proved in Lemma 4 below.

Note that since Γ¯f\overline{\Gamma}_{f} commutes with projective functors and morphisms between them also its left and right adjoints do. Thus, for any projective functor 𝒫\mathcal{P}, the adjunction map 𝒫(Δλ(f))→Γ¯fΓ¯fL(𝒫(Δλ(f))\mathcal{P}(\Delta_{\lambda}(f))\to\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}(\mathcal{P}(\Delta_{\lambda}(f)) is equivalent to the map 𝒫[Δλ(f)→Γ¯fΓ¯fL(Δλ(f)]\mathcal{P}[\Delta_{\lambda}(f)\to\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}(\Delta_{\lambda}(f)] which is an isomorphism by the above.

Now pick a projective resolution 𝒫′​(Δλ)→𝒫​(Δλ)→M→0\mathcal{P}^{\prime}(\Delta_{\lambda})\to\mathcal{P}(\Delta_{\lambda})\to M\to 0. Applying Γ¯f\overline{\Gamma}_{f} we get the exact sequence

𝒫′​(Δμ)→𝒫​(Δμ)→V→0.\mathcal{P}^{\prime}(\Delta_{\mu})\to\mathcal{P}(\Delta_{\mu})\to V\to 0.

Applying the adjunction morphisms vertically we get a commutative diagram

𝒫′​(Δμ)\textstyle{\mathcal{P}^{\prime}(\Delta_{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}𝒫​(Δμ)\textstyle{\mathcal{P}(\Delta_{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}V\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}0\textstyle{0}Γ¯f​Γ¯fL​𝒫′​(Δμ)\textstyle{\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}\mathcal{P}^{\prime}(\Delta_{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Γ¯f​Γ¯fL​𝒫​(Δμ)\textstyle{\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}\mathcal{P}(\Delta_{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Γ¯f​Γ¯fL​V\textstyle{\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}0\textstyle{0}

Therefore V→Γ¯f​Γ¯fL​VV\to\overline{\Gamma}_{f}\overline{\Gamma}_{f}^{L}V is an isomorphism. ∎

Recall that the left adjoint of the functor 𝒪λ′​⟶Θλμ​𝒪μ^′\mathcal{O}^{\prime}_{\lambda}\overset{\Theta^{\mu}_{\lambda}}{\longrightarrow}\mathcal{O}^{\prime}_{{\widehat{\mu}}} is Θμλ​()⊗UUλ\Theta^{\lambda}_{\mu}(\ )\otimes_{U}U_{\lambda}.

Lemma 4.

The adjunction morphism π:Δμ↦Θλμ​(Θμλ​(Δμ)⊗UUλ)\pi:\Delta_{\mu}\mapsto\Theta^{\mu}_{\lambda}(\Theta^{\lambda}_{\mu}(\Delta_{\mu})\otimes_{U}U_{\lambda}) is an isomorphism.

Proof.

We shall prove that Θλμ​(Θμλ​(Δμ)⊗UUλ)≅Δμ\Theta^{\mu}_{\lambda}(\Theta^{\lambda}_{\mu}(\Delta_{\mu})\otimes_{U}U_{\lambda})\cong\Delta_{\mu}. From this it will follow that π\pi is an isomorphism because π\pi is nonzero (as it corresponds to the natural surjection Θμλ​Δμ↦(Θμλ​Δμ)⊗UUλ\Theta^{\lambda}_{\mu}\Delta_{\mu}\mapsto(\Theta^{\lambda}_{\mu}\Delta_{\mu})\otimes_{U}U_{\lambda} under adjunction) and any nonzero endomorphism of a Verma module is an isomorphism.

We have Θμλ​(Δμ)=Pw⋅λ\Theta^{\lambda}_{\mu}(\Delta_{\mu})=P_{w\cdot\lambda}, where ww is the longest element in WμW^{\mu}. Write P=Pw⋅λP=P_{w\cdot\lambda}. Let nx=(P:Δx⋅λ)=[Δx⋅λ:Lw⋅λ]n_{x}=(P:\Delta_{x\cdot\lambda})=[\Delta_{x\cdot\lambda}:L_{w\cdot\lambda}]. We shall show that PP is multiplicity free, i.e. that nx≤1n_{x}\leq 1 for all x∈Wx\in W. Since nx≤nen_{x}\leq n_{e} it suffices to show that ne=1n_{e}=1. Since Δλ\Delta_{\lambda} is projective we have

dimHom(Δλ,P)=[P:Lλ]=∑x∈W(P:Δx⋅λ)⋅[Δx⋅λ:Lλ]=(P:Δλ)=ne.\dim\operatorname{Hom}(\Delta_{\lambda},P)=[P:L_{\lambda}]=\sum_{x\in W}(P:\Delta_{x\cdot\lambda})\cdot[\Delta_{x\cdot\lambda}:L_{\lambda}]=(P:\Delta_{\lambda})=n_{e}.

But on the other hand

Hom⁡(Δλ,P)=Hom⁡(Δλ,Θμλ​Δμ)=Hom⁡(Θλμ​Δλ,Δμ)=Hom⁡(Δμ,Δμ)\operatorname{Hom}(\Delta_{\lambda},P)=\operatorname{Hom}(\Delta_{\lambda},\Theta^{\lambda}_{\mu}\Delta_{\mu})=\operatorname{Hom}(\Theta^{\mu}_{\lambda}\Delta_{\lambda},\Delta_{\mu})=\operatorname{Hom}(\Delta_{\mu},\Delta_{\mu})

and therefore ne=dimHom⁡(Δμ,Δμ)=1n_{e}=\dim\operatorname{Hom}(\Delta_{\mu},\Delta_{\mu})=1.

This implies that PJλ≅ΔλP^{J_{\lambda}}\cong\Delta_{\lambda}, see e.g. [2]. Therefore [PJλ:Δw0⋅λ]=1[P^{J_{\lambda}}:\Delta_{w_{0}\cdot\lambda}]=1.

We claim that this implies that [P/JλP:Δw0⋅λ]=1[P/J_{\lambda}P:\Delta_{w_{0}\cdot\lambda}]=1. Since we have a surjection P/Jλ​P→Δw⋅λP/J_{\lambda}P\to\Delta_{w\cdot\lambda} it is enough to show that [P/JλP:Δw0⋅λ]≤1[P/J_{\lambda}P:\Delta_{w_{0}\cdot\lambda}]\leq 1. Let I=P∗=Θμλ​(∇μ)I=P^{*}=\Theta^{\lambda}_{\mu}(\nabla_{\mu}) (= injective hull of Lw⋅λL_{w\cdot\lambda}) and T=Θμλ​(Δw0⋅μ)T=\Theta^{\lambda}_{\mu}(\Delta_{w_{0}\cdot\mu}) (a tilting module). Since (P/Jλ​P)∗=IJλ(P/J_{\lambda}P)^{*}=I^{J_{\lambda}} it is enough to show that [IJλ:Δw0⋅λ]≤1[I^{J_{\lambda}}:\Delta_{w_{0}\cdot\lambda}]\leq 1.

Now T⊆PT\subseteq P and hence TJλ⊆PJλT^{J_{\lambda}}\subseteq P^{J_{\lambda}} so that [TJλ:Δw0⋅μ]≤1[T^{J_{\lambda}}:\Delta_{w_{0}\cdot\mu}]\leq 1. Let K=Ker⁡(∇μ↠Δw0⋅μ)K=\operatorname{Ker}(\nabla_{\mu}\twoheadrightarrow\Delta_{w_{0}\cdot\mu}). Then we have [K:Δw0⋅μ]=0[K:\Delta_{w_{0}\cdot\mu}]=0 and therefore [ΘμλK:Δw0⋅λ]=0[\Theta^{\lambda}_{\mu}K:\Delta_{w_{0}\cdot\lambda}]=0. The exact sequence 0→Θμλ​K→I→T→00\to\Theta^{\lambda}_{\mu}K\to I\to T\to 0 gives the sequence

0→(Θμλ​K)Jλ→IJλ→TJλ.0\to(\Theta^{\lambda}_{\mu}K)^{J_{\lambda}}\to I^{J_{\lambda}}\to T^{J_{\lambda}}.

Thus, [IJλ:Δw0⋅μ]≤[TJλ:Δw0⋅μ]≤1[I^{J_{\lambda}}:\Delta_{w_{0}\cdot\mu}]\leq[T^{J_{\lambda}}:\Delta_{w_{0}\cdot\mu}]\leq 1. This proves the claim.

Consider now a Verma flag

0⊂F1⊂F2⊂…⊂FN=P,Fi/Fi−1≅Δxi⋅λ,0\subset F_{1}\subset F_{2}\subset\ldots\subset F_{N}=P,\ F_{i}/F_{i-1}\cong\Delta_{x_{i}\cdot\lambda},

so that N=[P:Δw0⋅λ]N=[P:\Delta_{w_{0}\cdot\lambda}]. Then we have xi⋅λ≥w⋅λx_{i}\cdot\lambda\geq w\cdot\lambda for each ii. Since [P/JλP:Δw0⋅λ]=1[P/J_{\lambda}P:\Delta_{w_{0}\cdot\lambda}]=1 we have [JλP:Δw0⋅λ]=N−1[J_{\lambda}P:\Delta_{w_{0}\cdot\lambda}]=N-1. Clearly Jλ​Fi⊆Fi−1J_{\lambda}F_{i}\subseteq F_{i-1} and the latter equality implies that the image of Jλ​FiJ_{\lambda}F_{i} in Fi−1/Fi−2≅ΔxiF_{i-1}/F_{i-2}\cong\Delta_{x_{i}} contains the copy of Δw0⋅λ\Delta_{w_{0}\cdot\lambda} for all i≥2i\geq 2. Let ti∈Fit_{i}\in F_{i} be such that its image in Fi/Fi−1F_{i}/F_{i-1} is a highest weight vector of weight w⋅λw\cdot\lambda. Pick si∈U​(𝔫−)​tis_{i}\in U(\mathfrak{n}_{-})t_{i} such that the image of sis_{i} in Fi/Fi−1F_{i}/F_{i-1} is a highest weight vector of weight w0⋅λw_{0}\cdot\lambda. Since the image of Jλ⋅si≠0J_{\lambda}\cdot s_{i}\neq 0 in Fi−1/Fi−2F_{i-1}/F_{i-2} it follows that also the image of Jλ⋅ti≠0J_{\lambda}\cdot t_{i}\neq 0 in Fi−1/Fi−2F_{i-1}/F_{i-2}. This shows that [JλP:Lw⋅λ]≥N−1[J_{\lambda}P:L_{w\cdot\lambda}]\geq N-1. This implies that [ΘλμJλP:Lμ]≥N−1[\Theta^{\mu}_{\lambda}J_{\lambda}P:L_{\mu}]\geq N-1.

On the other hand we have a short exact sequence

0→Θλμ​Jλ​P→Θλμ​P→Θλμ​(P/Jλ​P)→0.0\to\Theta^{\mu}_{\lambda}J_{\lambda}P\to\Theta^{\mu}_{\lambda}P\to\Theta^{\mu}_{\lambda}(P/J_{\lambda}P)\to 0.

It follows that Θλμ​Jλ​P\Theta^{\mu}_{\lambda}J_{\lambda}P contains a submodule of Θλμ​P=ΔμN\Theta^{\mu}_{\lambda}P=\Delta^{N}_{\mu} isomorphic to ΔμN−1\Delta^{N-1}_{\mu}. Since moreover Θλμ​(P/Jλ​P)↠Δμ\Theta^{\mu}_{\lambda}(P/J_{\lambda}P)\twoheadrightarrow\Delta_{\mu} we conclude that Θλμ​Jλ​P≅ΔμN−1\Theta^{\mu}_{\lambda}J_{\lambda}P\cong\Delta^{N-1}_{\mu} and that Θλμ​(P/Jλ​P)≅Δμ\Theta^{\mu}_{\lambda}(P/J_{\lambda}P)\cong\Delta_{\mu}. ∎

Remark 2.

Assume that ff is non-degenerate so that by Proposition 2 W​hf∘Γ¯f≅𝕍Wh_{f}\circ\overline{\Gamma}_{f}\cong\mathbb{V}. Hence the left adjoint of Γ¯f\overline{\Gamma}_{f} is Pw0⋅λ⊗ZW​h¯fP_{w_{0}\cdot\lambda}\otimes_{Z}\overline{Wh}_{f} in this case.

Remark 3.

If M∈𝒩λ^fM\in\mathcal{N}^{f}_{\widehat{\lambda}} (for arbitrary ff) is induced from a Whittaker module over the reductive Lie algebra 𝔩\mathfrak{l} (see Section 2.3.1), then the adjoints Γ¯fL​(M)\overline{\Gamma}_{f}^{L}(M) and Γ¯fR​(M)\overline{\Gamma}_{f}^{R}(M) can be calculated directly only using the definition of Γ¯f\overline{\Gamma}_{f} without using τL\tau^{L} and τR\tau^{R}.

3.2. Calculation of Γ¯fL\overline{\Gamma}_{f}^{L} and Γ¯fR\overline{\Gamma}_{f}^{R}.

We end this paper by calculating Γ¯fL\overline{\Gamma}_{f}^{L} and Γ¯fR\overline{\Gamma}_{f}^{R} on some standard Whittaker modules. Note that Γ¯fL:𝒩λ^f→𝒪λ^\overline{\Gamma}_{f}^{L}:\mathcal{N}^{f}_{{\widehat{\lambda}}}\to\mathcal{O}_{\widehat{\lambda}} is right exact and Γ¯fR:𝒩λ^f→𝒪λ^\overline{\Gamma}_{f}^{R}:\mathcal{N}^{f}_{{\widehat{\lambda}}}\to\mathcal{O}_{\widehat{\lambda}} is left exact. Recall that they commute with all projective functors.

We have

Γ¯fR​(Δx⋅λ​(f))≅τR​(Δx⋅λ​(f))≅κ−1∘(Θμλ​())Jλ∘σ−1​(Δx⋅λ​(f))≅κ−1∘Θμλ​(Δx⋅μ)Jλ.\overline{\Gamma}_{f}^{R}(\Delta_{x\cdot\lambda}(f))\cong\tau^{R}(\Delta_{x\cdot\lambda}(f))\cong\kappa^{-1}\circ(\Theta^{\lambda}_{\mu}(\ ))^{J_{\lambda}}\circ\sigma^{-1}(\Delta_{x\cdot\lambda}(f))\cong\kappa^{-1}\circ\Theta^{\lambda}_{\mu}(\Delta_{x\cdot\mu})^{J_{\lambda}}.

If x=ex=e then we have Θμλ​(Δμ)Jλ≅Δλ\Theta^{\lambda}_{\mu}(\Delta_{\mu})^{J_{\lambda}}\cong\Delta_{\lambda}, as explained in the proof of Lemma 4, and therefore Γ¯fR​(Δλ​(f))≅Δλ\overline{\Gamma}_{f}^{R}(\Delta_{\lambda}(f))\cong\Delta_{\lambda}. When x=w0x=w_{0} one can prove that T=Θμλ​(Δw0⋅μ)T=\Theta^{\lambda}_{\mu}(\Delta_{w_{0}\cdot\mu}) is a multiplicity free tilting module of highest weight w0​w⋅λw_{0}w\cdot\lambda and that in this case TJλ=Δw0​w⋅λT^{J_{\lambda}}=\Delta_{w_{0}w\cdot\lambda}, where ww is the longest element in WμW_{\mu}. Hence, Γ¯fR​(Δw0⋅λ​(f))=Δw−1​w0⋅λ\overline{\Gamma}_{f}^{R}(\Delta_{w_{0}\cdot\lambda}(f))=\Delta_{w^{-1}w_{0}\cdot\lambda}.

Similarly, we have

Γ¯fL​(Δx⋅λ​(f))≅κ−1∘(Θμλ​()⊗UUλ)∘σ−1​(Δx⋅λ​(f))≅κ−1∘(Θμλ​(Δx⋅μ)⊗UUλ).\overline{\Gamma}_{f}^{L}(\Delta_{x\cdot\lambda}(f))\cong\kappa^{-1}\circ(\Theta^{\lambda}_{\mu}(\ )\otimes_{U}U_{\lambda})\circ\sigma^{-1}(\Delta_{x\cdot\lambda}(f))\cong\kappa^{-1}\circ(\Theta^{\lambda}_{\mu}(\Delta_{x\cdot\mu})\otimes_{U}U_{\lambda}).

Now for x=ex=e we have Θμλ​(Δμ)⊗UUλ=Pw⋅λ/Jλ​Pw⋅λ\Theta^{\lambda}_{\mu}(\Delta_{\mu})\otimes_{U}U_{\lambda}=P_{w\cdot\lambda}/J_{\lambda}P_{w\cdot\lambda} so that Γ¯fL​(Δλ​(f))≅κ−1​Pw⋅λ/Jλ​Pw⋅λ\overline{\Gamma}_{f}^{L}(\Delta_{\lambda}(f))\cong\kappa^{-1}P_{w\cdot\lambda}/J_{\lambda}P_{w\cdot\lambda}. On the other hand, T/Jλ​T=∇w0​w⋅λT/J_{\lambda}T=\nabla_{w_{0}w\cdot\lambda} (as follows from the above and selfduality of TT) and therefore Γ¯fL​(Δw0⋅λ​(f))≅∇w​w0⋅λ\overline{\Gamma}_{f}^{L}(\Delta_{w_{0}\cdot\lambda}(f))\cong\nabla_{ww_{0}\cdot\lambda}.

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