Generalized Verma modules, b-functions of semi-invariants and duality for twisted D-modules on generalized flag manifolds

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Generalized Verma modules, b-functions of semi-invariants and duality for twisted D-modules on generalized flag manifolds

Corrado Marastoni

Dipartimento di Matematica Pura ed Applicata – Universit`a di Padova – Via Belzoni, 7 – I-35131 Padova (Italy) and R.I.M.S. – Kyoto University – Kitashirakawa, Sakyo-ku – Kyoto 606-8502 (Japan) E-mail: maraston@math.unipd.it

Abstract. Let G be a connected semisimple algebraic Lie group over C, P a parabolic subgroup, g and p their Lie algebras. We prove a microlocal version of Gyoja’s conjectures [2] about a relation between the irreducibility of generalized Verma modules on g induced from p and the zeroes of b-functions of P -semi-invariants on G. Our method uses a duality for twisted D-modules on generalized flag manifolds.

Modules de Verma généralisés, b-fonctions des quasi-invariants

et dualité pour les D-modules tordus sur les variétés de drapeaux généralisées

Résumé. Soient G un groupe algébrique sur C connexe semi-simple, P un sous-groupe parabolique, g et p leurs algèbres de Lie. On démontre une version microlocale des conjectures de Gyoja [2] sur une relation entre l’irreductibilité des modules de Verma généralisés sur g induits de p et les zéros des b-fonctions des quasi-invariants sur G par rapport

`

a P . Notre méthode utilise une dualité pour les D-modules tordus sur les variétés de drapeaux généralisées.

Version fran¸caise abrégée – Soient G un groupe algébrique sur C connexe semi-simple avec identité e, B un sous-groupe de Borel, T un tore maximal dans B, B⁻ le sous-groupe de Borel tel que B ∩ B⁻ = T et g, t, b, b⁻ leurs algèbres de Lie. Soient ∆ = ∆⁺ ∪ ∆⁻ ⊂ t^∗ le système des racines, {$_i : i ∈ I₀} ⊂ t^∗ les poids fondamentaux. Pour un i ∈ I₀, la fonction régulière f_i : G → C caractérisée par les propriétés fi(e) = 1 et fi(b⁰gb) = $i(b⁰)$i(b)fi(g) pour tout b⁰ ∈ B⁻et b ∈ B, est appelée “i-ème quasi-invariant” de G; en général, si ν =P

i∈I0νi$i ∈ t^∗, on d´efinit f^ν =Q

i∈I0f_i^νⁱ. Pour un I ( I0 fix´e, soient ∆_I =P

i∈IZαi∩ ∆, ρ_I= ¹₂P

α∈∆⁺\∆_Iα et l_I= t ⊕P

α∈∆Ig_α, n^±_I =P

α∈∆^±\∆_Ig_α, p^±_I = l_I⊕ n^±_I; consid´erons les sous-groupes connexes correspondants L_I, N_I^± et P_I^± de G.

Posons t^∗_I = {λ ∈ t^∗ : hλ, α^∨i ∈ Z^≥0 ∀α ∈ ∆⁺_I}. Pour un λ ∈ t^∗_I, on notera V (λ) la représentation irréductible de lI de plus haut poids λ, étendue à p^±_I avec l’action triviale de n^±_I. Le U (g)-module M_I⁺(λ) = U (g) ⊗_p⁺

I V (λ) est appelé “module de Verma généralisé”.

Les conjectures de Gyoja. L’action de P_I⁻ × P_I⁺ sur G donnée par (x, y)g = xgy⁻¹ est préhomogène avec orbite ouverte Ω_I = G \ f_I⁻¹(0), où f_I = Q

i∈I0\If_i. Soit N_I le D_G-module holonôme associé avec fI, et notons ΣI la variété caractéristique de NI (voir [3]). On a l’inclusion ΣI ⊂ S

OΛO, où O parcourt l’ensemble fini des orbites de P_I⁻× P_I⁺ dans G et ΛO est le fibré conormal à O dans G. Dans l’autre sens, Λ_O est dit “être une bonne Lagrangienne” (voir [8]) si ΛO ⊂ Σ_I et l’action induite de P_I⁻× P_I⁺ sur ΛO est préhomogène.

Consid´erons DG[s] = DG ⊗

C C[s], faisceau des opérateurs différentiels réguliers sur G avec paramètres complexes s = (s_i)_i∈I₀_\I. En idéntifiant s 'P

i∈I0\Is_i$_i, pour tout i ∈ I₀\ I l’id´eal B_I,i = {b(s) ∈ C[s] : P (s)f^s+$ⁱ = b(s)f^s ∃P (s) ∈ D_G[s]}

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est différent de zéro (voir [7]). D’autre part, si Λ₀est une bonne Lagrangienne, on peut microlocaliser la définition de B_I,i en Λ_O et obtenir un idéal différent de zéro et principal (voir [8]). Le générateur bI,ΛO,i(s) de cet idéal, qui divise tout élément de BI,i, est appelé “i-ème b-fonction locale” en ΛO.

Dans son travail [2] (voir 3.1 et 3.3), A. Gyoja avait formul´e les conjectures suivantes:

(C1) il existe une orbite O ⊂ G telle que Λ_O est une bonne Lagrangienne et b_I,Λ_O_,i(s) ∈ B_I,i pour tout i ∈ I₀\ I (donc, en particulier, B_I,i est un idéal principal engendré par la “i-ème b-fonction” (globale) bI,i(s) := bI,ΛO,i(s));

(C2) pour λ = λc ∈ P

i∈I0\IC$i, le module de Verma généralisé M_I⁺(λc) est irréductible si et seulement si b_I,i(λ_c− $_i− ν) 6= 0 pour tout i ∈ I₀\ I et ν ∈P

i∈I0\IZ≥0$_i.

Transformations intégrales pour les D-modules tordus. Considérons les variétés de drapeaux généralisées duales X_I^± = G/P_I^±, et notons π± : G → X_I^± les projections canoniques.

Soit µ ∈ t^∗_I. Pour tout ouvert U ⊂ X_I^± on pose O_X^±

I (µ)(U ) = {ϕ ∈ Γ(π_±⁻¹(U ), OG⊗ V (µ)) : _dt^dϕ(ge^tA)

t=0= ∓A · ϕ(g) ∀ g ∈ G, A ∈ lI}.

Si µ est int´egral, O_X^±

I (µ) est le faisceau des sections régulières du fibré vectoriel sur X_I^± associé à V (µ); en général, si on écrit µ = µ_c+ µ_d avec µ_c∈P

i∈I0\IC$i et µ_d∈P

i∈I0Z≥0$_i, alors O_X^±

I (µ) a une structure de faisceau sur X_I^± tordu par µc (voir [5]).

Pour un λ = λ_c ∈ P

i∈I0\IC$i donn´e, soient D_X^±

I ,λc l’anneau des op´erateurs diff´erentiels sur O_X^±

I (λ_c) (on note que D_X^±

I,λc est un faisceau d’op´erateurs diff´erentiels tordus sur X_I^±, voir [5]) et D^b(D_X^±

I,λc) la catégorie dérivée bornée des D_X^±

I,λc-modules à gauche. On considère les opérations d’image inverse D( · )^∗, image directe propre D( · )_!, produit tensoriel ⊗, produit ten-^D soriel extérieur ^D, au sens des D-modules tordus (voir [5]). En particulier, on peut construire un faisceau d’opérateurs différentiels tordus D_X⁺

I×X_I⁻,λc := D_X⁺

I,λc^DD_X−

I,λc sur X_I⁺× X_I⁻.

L’action diagonale de G sur X_I⁺× X_I⁻ définit un espace préhomogène qui a le même type de singularité que celle de l’espace préhomogène défini par l’action de P_I⁻× P_I⁺ sur G. Si O est une orbite de P_I⁻× P_I⁺ dans G, on note O⁰ l’orbite correspondante de G dans X_I⁺× X_I⁻, et Λ_O⁰ le fibré conormal à O⁰ dans X_I⁺× X_I⁻. En particulier, l’orbite ouverte Ω⁰_I ' G/L_I est une variété affine.

Soit B_Ω⁰

I,λc le faisceau sur X_I⁺× X_I⁻ des fonctions méromorphes à pôles sur le fermé (X_I⁺× X_I⁻) \ Ω⁰_I muni d’une structure de module sur D_X⁺

I×X_I⁻,λc; le dual (au sens des D-modules `a gauche) B^∗_Ω0 I,λc

sera donc un module sur D_X+

I×X_I⁻,−λc. Ces modules donnent deux foncteurs (que l’on appelle

“transformations intégrales” d’après [1]) par les formules suivantes, où X_I⁺← X^q¹ _I⁺× X_I⁻→ X^q² _I⁻sont les projections:

D^b(D_X⁺

I,−λc)

·^D◦ B^∗

Ω0I,−λc

// D^b(D_X⁻

I,λc)

B_Ω0

I,−λc D◦ ·

oo ,







·^D◦ B^∗_Ω0

I,−λc = Dq_2!(Dq1∗( · )⊗ B^D ^∗_Ω0 I,−λc) B_Ω⁰

I,−λc

D◦ · = Dq_1!(B_Ω⁰

I,−λc

⊗ DqD ₂^∗( · )).

A l’aide de [6, Lemma 2.2], on démontre que ces foncteurs sont en fait des équivalences de catégories quasi-inverses l’une de l’autre.

Si µ ∈ t^∗_Iest int´egral, le faisceau D_X^±

I,λc(µ) := D_X^±

I,λc⊗_O

X± I

O_X±

I (µ) est un D_X^±

I ,λc-module à gauche, et on peut donc considérer la transformée B_Ω⁰

I,−λc D◦ D_X⁻

I,λc(µ) dans D^b(D_X⁺

I,−λc).

Th´eor`eme. Il existe une et une seule orbite O_I de P_I⁻× P_I⁺ dans G telle que Λ_O_I ⊂ Σ_I et les projections naturelles Λ_O⁰

I → T^∗X_I^± soient surjectives au point g´en´erique. Pour une telle orbite, ΛOI est une bonne Lagrangienne et Λ_O⁰

I → T^∗X_I^± sont des applications birationnelles.

En outre, pour λ_c∈P

i∈I0\IC$i les propriétés suivantes sont équivalentes:

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(1) M_I⁺(λ_c) est un g-module irr´eductible;

(2) b_I,Λ_OI_,i(λ_c− $_i− ν) 6= 0 pour tout i ∈ I₀\ I et ν ∈P

i∈I0\IZ≥0$_i; (3) D_X⁺

I,−λc −→^∼ B_Ω⁰

I,−λc

◦ DD _X⁻

I ,λc(−2ρI) dans D^b(DXI,−λc).

Ce résultat montre que l’irréductibilité des modules de Verma généralisés est bien contrôlée (voir (C2)) par une famille de b-fonctions, décrites en tant que b-fonctions locales le long du fibré conormal à une orbite “privilégiée” O_I dans G. Cependant, nous ne savons pas si, en général, ces b-fonctions locales sont aussi globales (voir (C1)).

This note is organized as follows. After recalling Gyoja’s conjectures (Section 1), we intro- duce our approach based on an integral transform for twisted D-modules on dual generalized flag manifolds (Section 2) and then we state and comment our main result (Section 3).

1 Generalized Verma modules and b-functions of semi-invariants

Let G be a connected semisimple algebraic group over C with identity e, B a Borel subgroup of G, T a maximal torus contained in B, B⁻ the Borel subgroup such that B ∩ B⁻= T , and let g, t, b, b⁻ be their Lie algebras. Let ∆ = ∆⁺∪ ∆⁻⊂ t^∗ be the root system, and denote by r_α the reflection in t^∗ and by gα the root space in g associated to α ∈ ∆. Let {αi: i ∈ I0} ⊂ ∆⁺ be the simple roots, {$_i : i ∈ I₀} ⊂ t^∗ the fundamental weights, W = hr_α_i : i ∈ I₀i the Weyl group. For i ∈ I₀ let v_i be a highest weight vector in the irreducible g-module with highest weight $_i, and let v_i^∨ be a lowest weight vector in the dual g-module normalized so that hvi, v^∨_i i = 1. The i-th semi-invariant of G is the regular function f_i : G → C defined by fi(g) = hgv_i, v^∨_ii, which is also characterized by its properties f_i(e) = 1 and f_i(b⁰gb) = $_i(b⁰)$_i(b)f_i(g) for any b⁰ ∈ B⁻ and b ∈ B. More generally, for ν =P

i∈I0νi$i ∈ t^∗ we intend f^ν =Q

i∈I0f_i^νⁱ as a single valued branch of a multivalued function.

Let us fix I ( I0, and set ∆I=P

i∈IZαi∩ ∆, ρ_I= ¹₂P

α∈∆⁺\∆_Iα and WI = hrαi : i ∈ Ii ⊂ W ; let w_I be the longest element of W_I. We also set

l_I= t ⊕P

α∈∆Ig_α, n^±_I =P

α∈∆^±\∆_Ig_α, p^±_I = l_I⊕ n^±_I; let L_I, N_I^± and P_I^± be the associated connected subgroups of G.

The isomorphic classes of finite dimensional irreducible representations of l_I are parametrized by t^∗_I = {λ ∈ t^∗: hλ, α^∨i ∈ Z≥0 ∀α ∈ ∆⁺_I}

by associating to a representation its highest weight. For λ ∈ t^∗_I, we denote by V (λ) the irreducible representation of l_I with heighest weight λ, extended to p^±_I with the trivial action of n^±_I. The generalized Verma modules associated to the pair (I, λ) are the U (g)-modules with highest weight λ (resp. lowest weight w_Iλ) defined by

M_I^±(λ) = U (g) ⊗_p^±

I V (λ).

The action of P_I⁻ × P_I⁺ on G given by (x, y)g = xgy⁻¹ is prehomogeneous, with open orbit ΩI = P_I⁻P_I⁺, and one has ΩI = G \ f_I⁻¹(0), where fI = Q

i∈I0\Ifi (see [2, Lemma 2.6]). To the function f_I one associates a holonomic D_G-module N_I (see [8]); let Σ_I be the characteristic variety of N_I. One has the estimate Σ_I ⊂S

OΛ_O, where O ranges in the finite set of (P_I⁻× P_I⁺)-orbits in G and ΛO is the conormal bundle to O. Conversely, ΛO is called a “good Lagrangian” (see [8]) if Λ_O ⊂ Σ_I and the induced action of P_I⁻× P_I⁺ on Λ_O is prehomogeneous.

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Let D_G be the sheaf of rings of linear partial differential operators on G with regular coefficients.

Let s = (s_i)_i∈I₀_\Ibe complex variables, and set D_G[s] = D_G⊗

CC[s]. By identifying s 'P

i∈I0\Is_i$_i, the following ideal of C[s] is nontrivial for any i ∈ I0\ I (see [7]):

B_I,i= {b(s) ∈ C[s] : P (s)f^s+$ⁱ = b(s)f^s ∃ P (s) ∈ D_G[s]}.

On the other hand, if Λ_O is a good Lagrangian then the following ideal, where E_G is the sheaf of microdifferential operators of finite order on G, is nontrivial and principal for any i ∈ I0\ I (see [8]):

B_I,Λ_O_,i= {b(s) ∈ C[s] : Q(s)f^s+$ⁱ = b(s)f^s ∃ Q(s) ∈ E_G[s] generically invertible at Λ_O}.

The generator b_I,Λ_O_,i(s) of B_I,Λ_O_,i is called the “i-th local b-function” at Λ_O. In [2], A. Gyoja conjectured that:

Conjecture 1.1. ([2, 3.1]) There exists a orbit O ⊂ G such that Λ_O is a good Lagrangian and bI,ΛO,i(s) ∈ BI,i for any i ∈ I0\ I.

Since bI,ΛO,i(s) divides any element of BI,i, Conjecture 1.1 would imply that BI,i is a principal ideal generated by the “i-th (global) b-function” b_I,i(s) := b_I,Λ_O_,i(s).

Assuming Conjecture 1.1, Gyoja’s main conjecture is:

Conjecture 1.2. ([2, 3.3]) For λ = λ_c∈P

i∈I0\IC$i, the following statements are equivalent:

(1) the generalized Verma module M_I⁺(λ_c) is irreducible;

(2) bI,i(λc− $_i− ν) 6= 0 for any i ∈ I0\ I and ν ∈P

i∈I0\IZ≥0$i.

2 Duality for twisted D-modules on generalized flag manifolds We refer to [5] for the notions of twisted sheaves and twisted D-modules.

Consider the dual generalized flag manifolds X_I^± = G/P_I^±. The diagonal G-action on X_I⁺× X_I⁻ defines a prehomogeneous space whose singularity is of the same type than the one determined by the above prehomogeneous (P_I⁻× P_I⁺)-action on G. For a (P_I⁻× P_I⁺)-orbit O in G we denote by O⁰ the corresponding G-orbit in X_I⁺× X_I⁻, and by Λ_O⁰ the conormal bundle to O⁰ in X_I⁺× X_I⁻. Observe that the open orbit Ω⁰_I is an affine variety.

For λc ∈ P

i∈I0\IC$i we denote by D^b_λ_c(C_X^±

I ) the bounded derived category of λc-twisted sheaves on X_I^±, and we consider the operations of inverse image, direct image, tensor product etc.

in the framework of twisted sheaves. Any (µ_c, ν_c)-twisted sheaf on X_I⁺× X_I⁻ induces a (µ_c− ν_c)- twisted sheaf on Ω⁰_I ' G/L_I via the diagonal embedding lI → p⁺_I × p⁻_I.

Let us fix λc ∈P

i∈I0\IC$i, and denote by Ω⁰_I → X^j _I⁺× X_I⁻ the open embedding and by k_Ω⁰

I

the constant sheaf on Ω⁰_I with fiber C. To j!k_Ω⁰

I we can give a structure of (λc, λc)-twisted sheaf on X_I⁺× X_I⁻, and we denote it by CΩ⁰_I,λc; the dual C^∗_Ω⁰

I,λc = RHom (CΩ⁰_I,λc, C_X⁺

I×X_I⁻) is in degree zero and has a structure of (−λc, −λc)-twisted sheaf on X_I⁺× X_I⁻. By using CΩ⁰_I,λc and C^∗_Ω⁰

I,λc we can define “integral transforms” (see [1]) for twisted sheaves between X_I⁺ and X_I⁻ by the following functors, where X_I⁺← X^q¹ _I⁺× X_I⁻ → X^q² _I⁻ are the projections:

D^b_−λ_c(C_X⁺

I)

· ◦C^∗_Ω0

I,−λc

// D^b_λ_c(C_X⁻

I )

C_Ω0

I,−λc◦ ·

oo ,

( · ◦ C^∗_Ω⁰

I,−λc = Rq_2!(q₁⁻¹( · ) ⊗^LC^∗_Ω⁰

I,−λc) CΩ⁰_I,−λc ◦ · = Rq_1!(C^∗_Ω⁰

I,−λc ⊗^Lq⁻¹₂ ( · )). (2.1) Let π±: G → X_I^± be the canonical projections. Given µ ∈ t^∗_I, for any open U ⊂ X_I^± we define O_X^±

I (µ)(U ) = {ϕ ∈ Γ(π_±⁻¹(U ), OG⊗ V (µ)) : _dt^dϕ(ge^tA)

t=0 = ∓A · ϕ(g) ∀ g ∈ G, A ∈ lI}. (2.2)

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If µ is integral, V (µ) can be exponentiated to P_I^±: in this case, O_X^±

I (µ) is identified to the sheaf of regular sections of the vector bundle on X_I^± associated to V (µ). More generally, if µ = µ_c+ µ_dwith µc∈P

i∈I0\IC$i and µd∈P

i∈I0Z≥0$i, then O_X^±

I (µ) has a structure µc-twisted sheaf on X_I^±. The sheaf of rings D_X^±

I ,λc = {P ∈ E nd_Mod_λc_(C

X± I

)(O_X^±

I (λ_c)) : P is locally a differential operator}

is a sheaf of twisted differential operators on X_I^±; let D^b(D_X^±

I ,λc) the bounded derived category of left D_X^±

I,λc-modules. We consider the operations of inverse image D( · )^∗, proper direct image D( · )_!, tensor product⊗, external tensor product ^D ^D in the framework of twisted D-modules: in particular, we define a sheaf of twisted differential operators on X_I⁺× X_I⁻ by D_X⁺

I×X_I⁻,λc := D_X⁺

I,λc^DD_X− I ,λc. By using the Riemann-Hilbert correspondence (see [4]), from the twisted sheaf CΩ⁰_I,λc (resp. C^∗_Ω⁰

I,λc) we get a regular holonomic D_X⁺

I×X_I⁻,λc-module B_Ω⁰

I,λc (resp. a regular holonomic D_X⁺

I×X⁻_I,−λc- module B^∗_Ω0

I,λc). In fact, B_Ω⁰

I,λc is the sheaf of meromorphic functions with poles on (X_I⁺× X_I⁻) \ Ω⁰_I, endowed with the structure of module over D_X⁺

I×X_I⁻,λc, and B_Ω^∗0

I,λc is the dual of B_Ω⁰

I,λc in the sense of left D-modules. As in (2.1), we define functors of “integral transforms” for twisted D-modules:

D^b(D_X⁺

I,−λc)

·^D◦ B^∗

Ω0I,−λc

// D^b(D_X⁻

I ,λc)

B_Ω0

I,−λc D◦ ·

oo ,







·^D◦ B^∗_Ω0

I,−λc = Dq_2!(Dq1∗( · )⊗ B^D _Ω^∗0 I,−λc) B_Ω⁰

I,−λc

D◦ · = Dq_1!(B_Ω⁰

I,−λc

⊗ DqD ₂^∗( · )).

(2.3)

The geometric criterion of [6, Section 2] has a straightforward generalization to the twisted case and can be used to prove the following fact.

Proposition 2.1. The functors · ◦ C^∗_Ω⁰

I,−λc and CΩ⁰_I,−λc ◦ · (resp. ·^D◦ B^∗_Ω0

I,−λc and B_Ω⁰

I,−λc

D◦ ·) are quasi-inverse to each other, and thus they are equivalences between D^b_−λ

c(C_X⁺

I) and D^b_λ

c(C_X⁻

I ) (resp.

D^b(D_X+

I,−λc) and D^b(D_X⁻

I,λc)).

3 Local b-functions and irreducibility of generalized Verma modules

If µ ∈ t^∗_I is integral, the sheaf D_X±

I ,λc(µ) := D_X^±

I ,λc⊗_O

X± I

O_X± I (µ) has a natural structure of quasi-G-equivariant left D_X^±

I,λc-module (see [5]); in particular, we can apply the functor B_Ω⁰

I,−λc

D◦ · to D_X−

I,λc(µ). We can state our main result.

Theorem 3.1. (i) There exists a unique (P_I⁻× P_I⁺)-orbit OI ⊂ G such that Λ_O_I ⊂ Σ_I and the natural projections Λ_O⁰

I → T^∗X_I^± are generically surjective. For such an orbit, Λ_O_I is a good Lagrangian and the maps Λ_O⁰

I → T^∗X_I^± are birational.

(ii) For λ_c∈P

i∈I0\IC$i the following propositions are equivalent:

(1) M_I⁺(λ_c) is an irreducible g-module;

(2) b_I,Λ_OI_,i(λ_c− $_i− ν) 6= 0 for any i ∈ I₀\ I and ν ∈P

i∈I0\IZ≥0$_i; (3) D_X⁺

I,−λc −→^∼ B_Ω⁰

I,−λc

◦ DD _X−

I ,λc(−2ρ_I) in D^b(D_X⁺

I,−λc).

Let us sketch the proof. Part (i) follows from Proposition 2.1. In Part (ii), we prove that (1) and (2) are equivalent to (3). As for (1), there is an equivalence of categories (see [5]) between quasi-G- equivariant D_X+

I,−λc-modules and −λ_c-twisted (g, P_I⁺)-modules, which associates M = D_X+ I,−λc to

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M = M_I⁺(λc) and M⁰= B_Ω⁰

I,−λc D◦ D_X−

I ,λc(−2ρ_I) to M⁰ = M_I⁻(−w_Iλc)^∗ (here ( · )^∗ denotes duality).

Now, there exists a canonical nontrivial morphism of (g, P_I⁺)-modules φ : M → M⁰ (and hence a nontrivial morphism Φ : M → M⁰) which is an isomorphism if and only if M is irreducible.

The statement (2) is equivalent to the fact that the canonical global section f^λ^c of B_Ω⁰

I,−λc is a microlocal generator of B_Ω⁰

I,−λc on Λ_O⁰

I, and global sections of B_Ω⁰

I,−λc correspond (see [1]) to morphisms M → M⁰; in particular f^λ^c corresponds to Φ. Then the proof goes as in [6, Section 3.4].

Remarks 3.2.

(1) Unlike Example 3.3 and several other particular cases, the “privileged” orbit O_I is not always the closed (P_I⁻× P_I⁺)-orbit in G.

(2) Part (ii)(3) can be viewed as a quantization of the contact transformation between T^∗X_I⁺and T^∗X_I⁻ induced by Λ_O⁰

I (see also [1]).

(3) We do not know whether, in general, the ideals B_I,i (i ∈ I₀\ I) are principal or not, and in particular whether bI,Λ_OI,i(s) ∈ BI,i or not (see Conjecture 1.1).

(4) We have a partial “higher rank version” of Part (ii) (see also [2, p. 398]): for λ = λ_c+ λ_d∈ t^∗_I with λc ∈ P

i∈I0\IC$i and λd ∈ P

i∈I0Z≥0$i, the properties (1)’ M_I⁺(λ) is an irreducible g-module, and (3)’ D_X⁺

I,−λc(λ_d) −→^∼ B_Ω⁰

I,−λc D◦ D_X−

I ,λc(−w_Iλ_d−2ρ_I) in D^b(D_X⁺

I,−λc) are equivalent. However, we do not have a precise statement for (2)’ yet.

Example 3.3. (See [6]) Let G = SL(n + 1, C). Using the standard notations of Bourbaki, let p ∈ I0 = {1, . . . , n} and set I = Ip = I0\ {p}. In this case the b-function, which coincides with the local b-function on the conormal bundle to the closed orbit, is b_I_p(s) = Qp

j=1(s + j). Let λ = λ_c= r$_p, where r ∈ C. Then the above conditions are satisfied if and only if r /∈ Z≥−(p−1). Acknowledgments. This work has been prepared during a long-term stay at the Research Institute for Mathematical Sciences of Kyoto University supported by a JSPS fellowship. We are grateful to Masaki Kashiwara for his invaluable help.

References

[1] A. D’Agnolo and P. Schapira, “Leray’s quantization of Projective Duality” Duke Math. Journal 84, no.

2 (1996), 453–496.

[2] A. Gyoja, “Highest weight modules and b-functions of semi-invariants”. Publications R.I.M.S., Kyoto University 30, no. 3 (1994), 353–400.

[3] M. Kashiwara, “b-functions and holonomic systems”. Invent. Math. 38 (1976/77), 33–53.

[4] M. Kashiwara, “The Riemann-Hilbert problem for holonomic systems”. Publications R.I.M.S., Kyoto University 20, no. 2 (1984), 319–365.

[5] M. Kashiwara, “Representation theory and D-modules on flag varieties”. Ast´erisque S.M.F. 173-174 (1989), 55–109.

[6] C. Marastoni, “Grassmann duality for D-modules”. Ann. Sci. ´Ecole Norm. Sup. 31 (1998), 459–491.

[7] C. Sabbah, “Proximit´e ´evanescente II”. Compositio Math. 64 (1987), 213–241.

[8] M. Sato, M. Kashiwara, T. Kimura and T. Oshima, “Microlocal analysis of prehomogeneous vector spaces”. Invent. Math. 62 (1980), 117–179.