A Kazhdan-Lusztig Algorithm for Whittaker Modules

Abstract

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted $\mathcal {D}$-modules on the associated flag variety using Beilinson–Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan–Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm for computing multiplicities of composition factors of Verma modules using Kazhdan–Lusztig polynomials.

Appendix: Geometric Preliminaries

In this appendix we record some some fundamental results about functors between categories of modules over twisted sheaves of differential operators which play a critical role in the arguments of Sections 4 and 5. For a detailed treatment of this subject, see [7, 14, 15].

A.1 Twisted Sheaves of Differential Operators

Let X be a smooth complex algebraic variety of dimension n. Denote by $\mathcal {O}_{X}$ the structure sheaf of X, $\mathcal {D}_{X}$ the sheaf of differential operators on X, $\mathcal {T}_{x}$ the tangent sheaf on X, Ω_X the cotangent sheaf on X, and ω_X the invertible $\mathcal {O}_{X}$-module of differential n-forms on X. Denote by $i_{X}:\mathcal {O}_{X} \rightarrow \mathcal {D}_{X}$ the natural inclusion. A twisted sheaf of differential operators on X is a pair $(\mathcal {D}, i)$ of a sheaf $\mathcal {D}$ of associative $\mathbb {C}$-algebras with identity on X and a homomorphism $i:\mathcal {O}_{X} \rightarrow \mathcal {D}$ of sheaves of $\mathbb {C}$-algebras with identity that is locally isomorphic to the pair $(\mathcal {D}_{X}, i_{X})$.

For $f:Y \rightarrow X$ a morphism of smooth algebraic varieties and $\mathcal {D}$ a twisted sheaf of differential operators on X, we define

$$ \mathcal{D}_{Y \rightarrow X} = \mathcal{O}_{Y} \otimes_{f^{-1}\mathcal{O}_{X}} f^{-1} \mathcal{D}. $$

Then $\mathcal {D}_{Y \rightarrow X}$ is a left $\mathcal {O}_{Y}$-module for left multiplication and a right $f^{-1}\mathcal {D}$-module for right multiplication on the second factor. Denote by $\mathcal {D}^{f}$ the sheaf of differential $\mathcal {O}_{Y}$-module endomorphisms of $\mathcal {D}_{Y \rightarrow X}$ which are also $f^{-1}\mathcal {D}$-module endomorphisms. There is a natural morphism of sheaves of algebras $i_{f}: \mathcal {O}_{Y} \rightarrow \mathcal {D}^{f}$, and the pair $(\mathcal {D}^{f}, i_{f})$ is a twisted sheaf of differential operators on Y.

Let $\mathcal {D}$ be a twisted sheaf of differential operators on X and ${\mathcal{L}}$ an invertible $\mathcal {O}_{X}$-module. The twist of $\mathcal {D}$ by ${\mathcal{L}}$ is the sheaf $\mathcal {D}^{{\mathcal{L}}}$ of differential $\mathcal {O}_{X}$-module endomorphisms of ${\mathcal{L}} \otimes _{\mathcal {O}_{X}} \mathcal {D}$ that commute with the right $\mathcal {D}$-action. Because ${\mathcal{L}} \otimes _{\mathcal {O}_{X} } \mathcal {D}$ is an $\mathcal {O}_{X}$-module for left multiplication, there is a natural homomorphism $i_{{\mathcal{L}}}:\mathcal {O}_{X} \rightarrow \mathcal {D}^{{\mathcal{L}}}$, and $(\mathcal {D}^{{\mathcal{L}}}, i_{{\mathcal{L}}})$ is a twisted sheaf of differential operators on X. If $f:Y \rightarrow X$ is a morphism of smooth algebraic varieties as above, $(\mathcal {D}^{{\mathcal{L}}})^{f} = (\mathcal {D}^{f})^{f^{*}({\mathcal{L}})}$.

If X is a homogeneous space for a group G with Lie algebra $\mathfrak {g}$, then a homogeneous twisted sheaf of differential operators on X is a triple $(\mathcal {D}, \gamma , \alpha )$, where $\mathcal {D}$ is a twisted sheaf of differential operators on X, γ is the algebraic action of G on X, and $\alpha :\mathcal {U}(\mathfrak {g}) \rightarrow {\Gamma }(X,\mathcal {D})$ is a morphism of algebras such that the following three conditions are satisfied:

• (i) the multiplication in $\mathcal {D}$ is G-equivariant;

• (ii) the differential of the G-action on $\mathcal {D}$ agrees with the action T↦[α(ξ),T] for $\xi \in \mathfrak {g}$ and $T \in \mathcal {D}$; and

• (iii) the map $\alpha : \mathcal {U}(\mathfrak {g}) \rightarrow {\Gamma }(X, \mathcal {D})$ is a morphism of G-modules.

For x ∈ X, denote by B_x the stabilizer of x in G and $\mathfrak {b}_{x}$ its Lie algebra. For each B_x-invariant linear form $\lambda \in \mathfrak {b}_{x}^{*}$ one can construct a homogeneous twisted sheaf of differential operators $\mathcal {D}_{X,\lambda }$ [7, App. A §1] and all homogeneous twisted sheaves of differential operators on X occur in this occur in this way.

If $\mathcal {A}$ is a sheaf of $\mathbb {C}$-algebras on X, we denote by $\mathcal {A}^{\circ }$ the opposite sheaf of $\mathbb {C}$-algebras on X. Then if $(\mathcal {D}, i)$ is a twisted sheaf of differential operators on a smooth algebraic variety X, $(\mathcal {D}^{\circ }, i)$ is also a twisted sheaf of differential operators on X. In particular, the pair $(\mathcal {D}_{X}^{\circ }, i_{X})$ is a twisted sheaf of differential operators, and it is naturally isomorphic to $(\mathcal {D}_{X}^{\omega _{X}}, i_{\omega _{X}})$. If X is a homogeneous space and δ is the B_x-invariant linear form which is the differential of the representation of B_x on the top exterior power of the cotangent space at x, then $(\mathcal {D}_{X,\lambda })^{\circ }$ is naturally isomorphic to $\mathcal {D}_{X, -\lambda + \delta }$.

A.2 Modules over Twisted Sheaves of Differential Operators

Let $\mathcal {D}$ be a twisted sheaf of differential operators on a smooth complex algebraic variety X. For a category ${\mathcal{M}}(\mathcal {D})$ of $\mathcal {D}$-modules, we denote ${\mathcal{M}}_{qc}(\mathcal {D})$ (resp. ${\mathcal{M}}_{coh}(\mathcal {D})$) the corresponding category of quasicoherent (resp. coherent) $\mathcal {D}$-modules. We can view left $\mathcal {D}$-modules as right right $\mathcal {D}^{\circ }$-modules and vice-versa. In other words, the category ${\mathcal{M}}^{L}_{qc}(\mathcal {D})$ of quasicoherent left $\mathcal {D}$-modules on X is isomorphic to the category the category ${\mathcal{M}}_{qc}^{R}(\mathcal {D}^{\circ })$ of quasicoherent right $\mathcal {D}^{\circ }$-modules on X. This relationship allows us to freely use right or left modules depending on the particular situation, and because of this, we frequently drop the exponents ‘L’ and ‘R’ from our notation.

For a coherent $\mathcal {D}$-module $\mathcal {V}$, we can define the characteristic variety Ch$\mathcal {V}$ of $\mathcal {V}$ in the same way as the non-twisted case [13, Ch. III §3]. Because this construction is local, the results in the non-twisted case carry over to our setting. In particular, we have the following structure:

• (i) Ch$\mathcal {V}$ is a conical subvariety of the cotangent bungle T^∗(X).

• (ii) dim$(\text {Ch}\mathcal {V})\geq \dim (X)$.

If dim$(\text {Ch}\mathcal {V})= \dim (X)$, we say that $\mathcal {V}$ is a holonomic $\mathcal {D}$-module. Holonomic $\mathcal {D}$-modules form a thick subcategory ${\mathcal{M}}_{hol}(\mathcal {D})$ of ${\mathcal{M}}_{coh}(\mathcal {D})$. If $\mathcal {V}$ is coherent as an $\mathcal {O}_{X}$-module, we call $\mathcal {V}$ a connection. Connections are locally free as $\mathcal {O}_{X}$-modules and their characteristic variety is the zero section of T^∗(X), so they are holonomic.

For an invertible $\mathcal {O}_{X}$-module ${\mathcal{L}}$ and a twisted sheaf $\mathcal {D}$ of differential operators on X, we define the twist functor from ${\mathcal{M}}_{qc}^{L}(\mathcal {D}^{{\mathcal{L}}})$ by

$$ \mathcal{V}\mapsto (\mathcal{L} \otimes_{\mathcal{O}_{X}} \mathcal{D}) \otimes_{\mathcal{D}} \mathcal{V} $$

for $\mathcal {V} \in {\mathcal{M}}_{qc}^{L}(\mathcal {D})$. The twist functor is an equivalence of categories.

For an abelian category $\mathcal {C}$, we use the notation $D(\mathcal {C})$ and $D^{b}(\mathcal {C})$ to refer to the derived category and bounded derived category of $\mathcal {C}$, respectively. We identify $\mathcal {C}$ with its image in $D(\mathcal {C})$ (resp. $D^{b}(\mathcal {C})$) under the natural embedding.

For a morphism $f: Y \rightarrow X$ of smooth algebraic varieties and a twisted sheaf $\mathcal {D}$ of differential operators on X, we define the inverse image functor $f^{+}:{\mathcal{M}}_{qc}^{L}(\mathcal {D}) \rightarrow {\mathcal{M}}_{qc}^{L}(\mathcal {D}^{f})$ by

$$ f^{+}(\mathcal{V})=\mathcal{D}_{Y \rightarrow X} \otimes_{f^{-1}\mathcal{D}}f^{-1}\mathcal{V} $$

for $\mathcal {V} \in {\mathcal{M}}_{qc}^{L}(\mathcal {D})$. In general f⁺ is right exact with left derived functor Lf⁺. If f is an open immersion, then f⁺ is exact and $f^+(\mathcal {V}) = \mathcal {V}|_{Y}$. If f is a submersion, then f⁺ is exact. We define the extraordinary inverse image functor $f^{!}:D^{b}({\mathcal{M}}_{qc}^{L}(\mathcal {D}))\rightarrow D^{b}({\mathcal{M}}_{qc}^{L}(\mathcal {D}^{f}))$ by

$$ f^!=Lf^{+}\circ [\dim Y-\dim X]. $$

If f is an immersion then f^! is the right derived functor of the left exact functor $L^{\dim Y - \dim X} f^{+}:{\mathcal{M}}_{qc}^{L}(\mathcal {D})\rightarrow {\mathcal{M}}_{qc}^L(\mathcal {D}^f)$. In this setting, we refer to the functor $L^{\dim Y - \dim X} f^{+}$ as f^!, and for $\mathcal {V} \in {\mathcal{M}}_{qc}(\mathcal {D})$, we refer to the k^th-cohomology modules $H^{k}f^!(\mathcal {V})$ as $R^{k}f^!(\mathcal {V})$.

We define the direct image functor $f_{+}:D^{b}({\mathcal{M}}_{qc}^{R}(\mathcal {D}^{f})) \rightarrow D^{b}({\mathcal{M}}_{qc}^{R}(\mathcal {D}))$ by

$$ f_{+}(\mathcal{W}^{\cdot}) = Rf_{\bullet}(\mathcal{W}^{\cdot} \otimes^{L}_{\mathcal{D}^{f}} \mathcal{D}_{Y \rightarrow X}), $$

for $\mathcal {W}^{\cdot } \in D^{b}({\mathcal{M}}^{R}(\mathcal {D}^{f}))$. Here Rf_∙ is the right derived functor of the sheaf-theoretic direct image functor f_∙. If f is an immersion, f₊ is the right derived functor of the left exact functor $H^{0} \circ f_{+} \circ D: {\mathcal{M}}_{qc}^{R}(\mathcal {D}^{f}) \rightarrow {\mathcal{M}}_{qc}^{R}(\mathcal {D})$, where D is the natural embedding of ${\mathcal{M}}_{qc}^{R}(\mathcal {D}^{f})$ into the derived category $D({\mathcal{M}}_{qc}^{R}(\mathcal {D}^{f}))$. In this setting, we refer to H⁰ ∘ f₊ ∘ D by f₊. If f is an open immersion, then f₊ = Rf_∙ is the sheaf-theoretic direct image. If f is affine, then f₊ is exact.

The relationship between the twist functor and the direct image functor is the following.

Proposition 1

(Projection Formula) Let $f:Y \rightarrow X$ be a morphism of smooth complex algebraic varieties, $\mathcal {D}$ a twisted sheaf of differential operators on X, and ${\mathcal{L}}$ an invertible $\mathcal {O}_{X}$-module. Then the following diagram commutes.

For a module $\mathcal {V} \in {\mathcal{M}}^{R}_{qc}(\mathcal {D})$, and a smooth subvariety Y ⊂ X, denote by ${\Gamma }_{Y}(\mathcal {V})$ the $\mathcal {D}$-module of local sections Y. The functor ${\Gamma }_{Y}:{\mathcal{M}}^{R}_{qc}(\mathcal {D})\rightarrow {\mathcal{M}}^{R}_{qc}(\mathcal {D})$ is a left-exact functor, and we denote by $R{\Gamma }_{Y}:D^{b}({\mathcal{M}}_{qc}^{R}(\mathcal {D}))\rightarrow D^{b}({\mathcal{M}}_{qc}^{R}(\mathcal {D}))$ its right derived functor. The following equivalence of categories is very useful in computations.

Theorem 1 (Kashiwara)

If Y is a closed smooth subvariety of a smooth algebraic variety X, $i:Y \rightarrow X$ the natural immersion, and $\mathcal {D}$ a twisted sheaf of differential operators on X, then the functor

$$ i_{+}:\mathcal{M}_{qc}^{R}(\mathcal{D}^{i}) \rightarrow \mathcal{M}^{R}_{qc}(\mathcal{D}) $$

establishes an equivalence of categories between ${\mathcal{M}}^{R}_{qc}(\mathcal {D}^{i})$ and the full subcategory ${\mathcal{M}}^{R}_{qc,Y}(\mathcal {D})$ of supported in Y. The quasiinverse of i₊ is i^!. In particular, if $\mathcal {V}$ is a quasicoherent $\mathcal {D}^{i}$-module, then $i^!(i_{+}(\mathcal {V}))=\mathcal {V}$, and if $\mathcal {U}$ is a $\mathcal {D}^{i}$-module, then $i^!(i_{+}(\mathcal {V}))=\mathcal {V}$, and if $\mathcal {U}$ is a quasicoherent $\mathcal {D}$-module, then $i_{+}(i^!(\mathcal {U}))={\Gamma }_{Y}(\mathcal {U})$.

Let $i:Y \rightarrow X$ be the immersion of a closed subvariety. If $\mathcal {J}_{Y}$ is the ideal of $\mathcal {O}_{X}$ consisting of germs vanishing on Y, we can define an filtration of $\mathcal {D}_{Y \rightarrow X}$ by (left $\mathcal {D}^{i}$, right $i^{-1}\mathcal {O}_{X}$)-modules by

$$ F_{p}\mathcal{D}_{Y \rightarrow X} =\{T \in \mathcal{D}_{Y \rightarrow X}|T \varphi = 0 \text{ for } \varphi \in (\mathcal{J}_{Y})^{p+1}\}, $$

for $p \in \mathbb {Z}_{+}$. We call this filtration the filtration by normal degree. By Kashiwara’s theorem, it induces a natural $\mathcal {O}_{X}$-module filtration on supported on Y. Namely, if $\mathcal {W} \in {\mathcal{M}}^{R}_{qc}(\mathcal {D}^{i})$,

$$ F_{p}i_{+}(\mathcal{W})=i_{\bullet}(\mathcal{W} \otimes_{\mathcal{D}^{i}} F_{p}\mathcal{D}_{Y \rightarrow X}). $$

The associated graded module has the form

$$ Gr i_{+}(\mathcal{W})=i_{\bullet}(\mathcal{W} \otimes_{\mathcal{O}_{Y}} S(\mathcal{N}_{X|Y})), $$

(1)

where $\mathcal {N}_{X|Y} = i^{*}(\mathcal {T}_{X})/\mathcal {T}_{Y}$ denotes the normal sheaf of Y, and $S(\mathcal {N}_{X|Y})$ is the corresponding sheaf of symmetric algebras [7, App. A §3.3].

The interaction between $\mathcal {D}$-module functors and fiber products is captured by base change.

Theorem 2 (Base Change Formula)

Let $f: X \rightarrow Z$ and $g: Y \rightarrow Z$ be morphisms of smooth complex algebraic varieties such that the fiber product X ×_ZY is a smooth algebraic variety, and let $\mathcal {D}$ be a twisted sheaf of differential operators on Z. Then the commutative diagram

determines an isomorphism

$$ g^! \circ f_{+} = q_{+} \circ p^! $$

of functors from $D^{b}({\mathcal{M}}(\mathcal {D}^{f}))$ to $D^{b}({\mathcal{M}}(\mathcal {D}^{g}))$.

A.3 Beilinson–Bernstein Localization

A key ingredient in this story is the localization theory of Beilinson and Bernstein, which we briefly review here. Full details can be found in [2, 14]. For the remainder of this appendix, let $\mathfrak {g}$ be a complex reductive Lie algebra, $\mathfrak {h}$ the abstract Cartan subalgebra of $\mathfrak {g}$ [15, §2], and X the flag variety of $\mathfrak {g}$. Fix $\lambda \in \mathfrak {h}^{*}$, and let 𝜃 be the Weyl group orbit of λ in $\mathfrak {h}^{*}$. In [2], Beilinson and Bernstein construct a twisted sheaf of differential operators $\mathcal {D}_{\lambda }$ on X for each $\lambda \in \mathfrak {h}^{*}$. (In the notation of Section Appendix A.1, $\mathcal {D}_{\lambda } = \mathcal {D}_{X, \lambda + \rho }$.) They show that for any μ in the Weyl group orbit 𝜃 of λ, the global sections ${\Gamma }(X, \mathcal {D}_{\mu })$ of $\mathcal {D}_{\mu }$ are equal to $\mathcal {U}_{\theta }$, which is the quotient of $\mathcal {U}(\mathfrak {g})$ by the ideal in $\mathcal {Z}(\mathfrak {g})$ corresponding to 𝜃 under the Harish-Chandra homomorphism. This implies that the global sections functor Γ maps quasicoherent $\mathcal {D}_{\lambda }$-modules into $\mathcal {U}(\mathfrak {g})$-modules with infinitesimal character χ_λ; that is, there is a left exact functor

$$ {\Gamma}: \mathcal{M}_{qc}(\mathcal{D}_{\lambda}) \rightarrow \mathcal{M}(\mathcal{U}_{\theta}). $$

Beilinson and Bernstein define a localization functor

$$ {\Delta}_{\lambda}: \mathcal{M}(\mathcal{U}_{\theta}) \rightarrow \mathcal{M}_{qc}(\mathcal{D}_{\lambda}) $$

by ${\Delta }_{\lambda }(V) = \mathcal {D}_{\lambda } \otimes _{\mathcal {U}_{\theta }}V$ for $V \in {\mathcal{M}}(\mathcal {U}_{\theta })$. The localization functor is right exact and is a left adjoint to Γ. In [2] it is shown that for antidominant regular $\lambda \in \mathfrak {h}^{*}$, Δ_λ is an equivalence of categories, and its quasi-inverse is Γ.

A.4 Translation Functors

Fix $\lambda \in \mathfrak {h}^{*}$, and let $\mathcal {D}_{\lambda }$ be the corresponding homogeneous twisted sheaf of differential operators. Any μ in the weight lattice $P({\Sigma }) = \{\lambda \in \mathfrak {h}^{*} | \alpha ^{\vee }(\lambda ) \in \mathbb {Z} \text { for all } \alpha \in {\Sigma } \}$ naturally determines a $G=\text {Int}{\mathfrak {g}}$-equivariant invertible $\mathcal {O}_{X}$-module $\mathcal {O}(\mu )$ on X. Twisting by $\mathcal {O}(\mu )$ defines a functor

$$ -(\mu): \mathcal{M}(\mathcal{D}_{\lambda}) \rightarrow \mathcal{M}(\mathcal{D}_{\lambda + \mu}) $$

by $\mathcal {V}(\mu ) = \mathcal {O}(\mu ) \otimes _{\mathcal {O}_{X}} \mathcal {V}$ for $\mathcal {V} \in {\mathcal{M}}(\mathcal {D}_{\lambda })$. We call this functor the geometric translation functor. It is evidently an equivalence of categories, and it also induces an equivalence of categories on ${\mathcal{M}}_{qc}(\mathcal {D}_{\lambda })$ (resp. ${\mathcal{M}}_{coh}(\mathcal {D}_{\lambda })$) with ${\mathcal{M}}_{coh}(\mathcal {D}_{\lambda })$ (resp. ${\mathcal{M}}_{coh}(\mathcal {D}_{\lambda +\mu })$).