D-module Final Report I

D-module Final Report I

Hung Chiang

June 13, 2017

0.1 Borel-Weil-Bott Theorem

Let G/k be a connected semi-simple algebraic group, T be a maximal torus of G, B be a Borel subgroup of G containing T , N be the unipotent part of B, and X be the flag variety G/B. We thus has a choice of positive roots ∆⁺, simple roots Π ={α1,· · · , αl}, and Weyl vector ρ = α₁+· · · + αl

2 . Let P be the weight lattice.

For each λ∈ L = Homk(B/N, k^∗), we have a equivariant G-line bundle L(λ) on X (^[HTT] p.255). Set

P_sing ={λ ∈ P | ∃α ∈ ∆, ⟨λ − ρ, α^∨⟩ = 0}, P_reg = P − Psing.

Define a shifted action of W on P by

w⋆λ = w(λ − ρ) + ρ.

Theorem 1 (Borel-Weil-Bott,^[HTT] 9.11.2.). Assume λ∈ L ⊂ P .

(i) If ⟨λ, α^∨⟩ ≤ 0 for all α ∈ ∆⁺, then L(λ) is generated by global sections. That is, the natural morphism

OX ⊗kΓ(X,L(λ)) → L(λ)

is surjective.

(ii) L(λ) is ample if and only if ⟨λ, α^∨⟩ < 0 for all α ∈ ∆⁺. (iii) Assume char(k) = 0.

(a) If λ∈ Psing, then Hⁱ(X,L(λ)) = 0 for all i ≥ 0.

(b) Let λ∈ Preg and take w∈ W such that w⋆λ ∈ −P⁺. Then

Hⁱ(X,L(λ)) =





L⁻(w⋆λ) if i = l(w),

0 otherwise.

0.2 Berlinson-Bernstein Theorems

From now on we assume k = C. For every smooth variety Y and locally free OY-module of finite rank V, we consider the sheaf of diffenertial operators on V, D_Y^V ⊂ E ndCY(V). DY^V is isomorphic to V ⊗OY D_Y ⊗OY V^∗. There’s a natural filtration

F_p(D^V_Y) = 0 for all p < 0,

F_p(D^V_Y) ={P | fP − P f ∈ Fp−1(D_Y^V)∀f ∈ OY} for all p ≥ 0.

Assume K is alinear algebraic group acting on Y and V is a K-equivlent vector bundle. There is a natural morphism ∂ : U (k) to Γ(Y, D^V_Y). Let a ∈ k, then ∂a is defined by

(∂_as)(y) = d

dt(exp(ta)s(exp(−ta)y))|t=0 (s∈ V, y ∈ Y ).

Here exp is the exponential map w.r.t right invariant vector fields. Algebraically, let φ : p^∗₂V ∼= σ^∗V, then ∂a is determined by

ϕ((a⊗ 1) · φ⁻¹(σ^∗s)) = σ^∗(∂_as)

Here a is regarded as right invariant vector fields on K acting on k[K](^[HTT]Equation 11.1.7).

Consider X. Let D_λ := D^L(λ+ρ)_X . We have Φ_λ : U (g) → Γ(X, Dλ).

Definition 1. Let z be the center of U (g) = U (h) ⊕ (n⁻U (g) + U (g)n)(^[HTT]

Equation 9.4.7). Let p be the projection from U (g) to U (h). f be the automorphism of U (h) defined by f (h) = h− ρ(h)1 for h ∈ h. For each λ ∈ h^∗, define the central character

χ_λ(z) = (f◦ p(z))(λ) for all z ∈ z.

Proposition 1 (^[HTT]Theorem 11.2.2). Let λ∈ L. Then Φλ : U (g) → Γ(X, Dλ) is surjective. Let z be the center of U (g). Then Φ_λ(z) = χ_λ(z) for all z ∈ z. Moreover, ker(Φ_λ) = U (g)(ker(χ_λ)).

We assume the proposition.

Let Modqc(Dλ) be the abelian category of Dλ-modules which are quasi-coherent overOX and Mod(g) be the categoriy of U (g)-modules. We have additive functors

Γ(X,·) : Modqc(D_λ)→ Mod(g), D_λ ⊗U (g)(·) : Mod(g) → Modqc(D_λ).

We have adjointness

HomD_λ(Dλ⊗U (g)M,N ) ∼= HomU (g)(M, Γ(X,N )).

Let Mod(g, χ) be the category of U (g)-modules with central character χ and Mod_f(g, χ) be the full subcategory of Mod(g, χ) of finitely generated U (g)-modules. The propo- sition shows that Mod(g, χ_λ) ∼= Mod(Γ(X, Dλ)).

Theorem 2 (^[HTT] Theorem 11.2.3 & 11.2.4). Let λ∈ L.

1. Suppose

⟨λ, α^∨⟩ ≤ 0 for all α ∈ ∆⁺. (1) That is, λ∈ −P⁺. Then for all M ∈ Modqc(Dλ) we have H^k(X,M) = 0 for all k > 0.

2. Suppose

⟨λ, α^∨⟩ < 0 for all α ∈ ∆⁺. (2) Then for all M ∈ Modqc(D_λ), the natural morphism

D_λ⊗U (g)Γ(X,M) → M is surjective.

Proof. For v ∈ −P⁺, Borel-Weil-Bott theorem says Γ(X,L(v)) = H⁰(X,L(v)) = L⁻(v) and pv : OX ⊗CL⁻(v) → L(v) is surjective. Since H omOX(L(v), OX) = L(−v) and HomC(L⁻(v),C) = L⁺(−v), we have L(−v) ,→ OX ⊗CL⁺(−v). Apply L(v) ⊗OX (·), we have iv : OX ,→ L(v) ⊗CL⁺(−v). Since L(v) is a line bundle, ker(p_v) is a direct summand of OX ⊗CL⁻(v) as an OX-module locally. Therefore, im(iv) is a direct summand of L(v) ⊗CL⁺(−v) as an OX-module locally.

Let λ∈ L and M be a Dλ-module. Apply M ⊗OX (·), we get p_v :M ⊗CL⁻(v)↠ M ⊗OX L(v),

i_v :M ,→ M ⊗_OX L(v) ⊗_CL⁺(−v).

Proposition 2 (^[HTT] Proposition 11.4.1). (i) If λ satisfies (2), then ker(p_v) is a direct summand ofM ⊗CL⁻(v) as a sheaf of abelian groups.

(ii) If λ satisfies (1), then im(i_v) is a direct summand of M ⊗OX L(v) ⊗CL⁺(−v) as a sheaf of abelian groups.

Suppose λ satisfies (1). For all M ∈ Modqc(Dλ), H^k(X,M) = lim−→H^k(X,N )

where N runs over all coherent OX-submodule of M. It suffices to prove that the natural map H^k(X,N ) → H^k(X,M) is the zero map. Fix N . Borel-Weil- Bott theorem says L(v) is ample if and only if v satisfies (2). Hence there is a v ∈ L ∩ −P⁺ such that H^k(X,N ⊗OX L(v)) = 0 for all k > 0. For this v, consider the commutative diagram

H^k(X,N ) H^k(X,M)

H^k(X,N ⊗_OX L(v) ⊗_CL⁺(−v)) H^k(X,M ⊗_OX L(v) ⊗_CL⁺(−v)).

iv∗

i_v∗ is injective. On the other hand, H^k(X,N ⊗OXL(v)⊗CL⁺(−v)) = H^k(X,N ⊗OX

L(v)) ⊗_CL⁺(−v) = 0 for all k > 0. So H^k(X,N ) → H^k(X,M) is the zero map.

Suppose λ satisfies (2). For given M ∈ Modqc(D_λ), set M^′ be the image of D_λ⊗U (g)Γ(X,M) → M and M^′′ be the cokernel of it. If M^′′̸= 0, let N ⊂ M^′′ be a nonzero coherent OX-submodule. There is a v∈ L ∩ −P⁺ such that N ⊗OX L(v) is generated by global sections. In this case, Γ(X,N ⊗_OX L(v)) ̸= 0, neither is Γ(X,M^′′⊗OX L(v)) On the other hand,

p_v∗ : Γ(X,M^′′)⊗_CL⁻(v) = Γ(X,M^′′⊗_CL⁻(v))→ Γ(X, M^′′⊗_OX L(v)) is surjective. So Γ(X,M^′′)̸= 0. Consider the exact sequence

0→ Γ(X, M^′)→ Γ(X, M) → Γ(X, M^′′)→ 0.

By definition, Γ(X,M) = Γ(X, Dλ ⊗U (g) Γ(X,M)) ↠ Γ(X, M^′). So Γ(X,M^′) = Γ(X,M) and hence Γ(X, M^′′) = 0. SoM^′′must be 0. The isomorphism Γ(X,M) = Γ(X, Dλ⊗U (g)Γ(X,M)) is proved in the proof of Corollary 1.

0.3 Equivalences of Categories

For λ ∈ L satisfying (1), we denote Mod^eqc(D_λ) the full subcategory of Mod_qc(D_λ) consisting of objectsM satisfying that

(a) D_λ ⊗U (g)Γ(X,M) → M is surjective.

(b) For all nonzero subobject N ⊂ M in Modqc(Dλ), we have Γ(X,N ) ̸= 0.

Set Mod^e_c(D_λ) = Mod^e_qc(D_λ)∩ Modc(D_λ).

Corollary 1. Γ(X,·) induces equivlences of categories

Mod^e_qc(D_λ) ∼= Mod(g, χλ), Mod^e_c(D_λ) ∼= Modf(g, χ_λ).

Proof. We first prove that M → Γ(X, Dλ⊗U (g)M ) is an isomorphism for all M ∈ Mod(g, χ_λ). For given M ∈ Mod(g, χλ), consider an exact sequence

Γ(X, D_λ)^⊕I → Γ(X, Dλ)^⊕J → M → 0.

From Theorem 2.1, Γ(X,·) : Modqc(D_λ) → Mod(g, χλ) is an exact functor. So Γ(X, D_λ⊗U (g)(·)) is right exact. We have an commutative diagram with exact rows

Γ(X, D_λ)^⊕I Γ(X, D_λ)^⊕J M 0

Γ(X, D_λ)^⊕I Γ(X, D_λ)^⊕J Γ(X, D_λ⊗U (g)M ) 0.

id id

So M ∼= Γ(X, Dλ⊗U (g)M ).

Now we show that Γ(X,·) : Mod^e_qc(D_λ)→ Mod(g, χλ) is fully faithful. That is, for all M1,M2 ∈ Mod^eqc(Dλ),

Γ : Hom_Dλ(M1,M2)→ HomU (g)(Γ(X,M1), Γ(X,M2))

is an isomorphism. Since Dλ ⊗U (g)Γ(X,M1)→ M1 is surjective, we have Hom_Dλ(M1,M2) ,→ HomDλ(D_λ⊗U (g)Γ(X,M1),M2)

∼= HomU (g)(Γ(X,M1), Γ(X,M2)).

Assume ϕ ∈ HomU (g)(Γ(X,M1), Γ(X,M2)). Let K1 be the kernel of D_λ ⊗U (g)

Γ(X,M1)→ M1. Apply the exact functor Γ(X,·) on the exact sequenct 0→ K1 → Dλ⊗U (g)Γ(X,M1)→ M1 → 0

and we get the exact sequence

0→ Γ(X, K1)→ Γ(X, M1)→ Γ(X, M1)→ 0.

So Γ(X,K1) = 0. Let K2 be the image of

K1 D_λ⊗U (g)Γ(X,M1) ^1⊗ϕ D_λ⊗U (g)Γ(X,M2) M2.

Since Γ(X,·) is exact and Γ(X, K1) = 0, Γ(X,K2) = 0. So K2 = 0. hence we obtain ψ :M1 ∼= Dλ⊗U (g)Γ(X,M1)/K1 → M2 with Γ(ψ) = ϕ.

Next, we prove that Γ(X,·) : Mod^eqc(Dλ)→ Mod(g, χλ) is essentially surjective.

Given M ∈ Mod(g, χλ). Let L be a maximal element of the set of subobjects K of Dλ ⊗U (g)M in Modqc(Dλ) satisfying that Γ(X,K) = 0. Set M = Dλ⊗U (g)M /L.

Then Γ(X,M) = Γ(X, Dλ⊗U (g)M )/Γ(X,L) = M. Dλ⊗U (g)M → M is surjective.

For allN ⊂ M, the maximality of L shows that Γ(X, N ) ̸= 0. So M ∈ Mod^eqc(Dλ).

Finally, we have to show that Mod^e_c(D_λ) and Mod_f(g, χ_λ) correspond to each other. Let M ∈ Modf(g, χλ). Γ(X, Dλ) is left-noetherian. There is an exact se- quence

Γ(X, Dλ)^⊕I → Γ(X, Dλ)^⊕J → M → 0

with |I|, |J| < ∞. Apply the right exact functor Dλ⊗U (g)(·) on it and we get the exact sequence

D_λ^⊕I → D_λ^⊕J → Dλ⊗Ug M → 0.

We get D_λ⊗Ug M ∈ Mod^e_c(D_λ) and henceM = Dλ⊗UgM /L.

Conversely, let M ∈ Mod^ec(Dλ). Since Dλ⊗UgΓ(X,M) → M is surjective, M is locally generated by finitely many global sections. Since X is quasi-compact, M is globally generated by finitely many global sections. We have an exact sequence

D^⊕I_λ → M → 0

where|I| < ∞. Apply Γ(X, ·) on it and we get the exact sequence

Γ(X, D_λ)^⊕I → Γ(X, M) → 0 Hence Γ(X,M) is an finitely generated U(g)-module.

Suppose λ satisfies (2). Then Mod_qc(D_λ) = Mod^e_qc(D_λ). In this case, we have Corollary 2. Γ(X,·) induces equivlences of categories

Mod_qc(D_λ) ∼= Mod(g, χλ), Mod_c(D_λ) ∼= Modf(g, χ_λ).

Let K be a closed subgroup of G. We consider K-equivariant g-modules. That is, a g-module with a K-action satisfying that

k-actions obtained from the g-action and the K-action coincide. (3) k· (a · m) = Ad(k)(a) · (k · m) for all k ∈ K, a ∈ g, and m ∈ M. (4) We denote the full subcategory consisting of K-equivariant objects of Mod(g, χ) and Mod_f(g, χ) by Mod(g, χ, K) and Mod_f(g, χ, K), respectively.

We also introduce K-equivariant D-modules. Let K acts on Y . Consider mor- phisms p₂ : K× Y → Y , σ : K × Y → Y , m : K × K → K defined by p2(k, y) = y, σ(k, y) = ky, m(k1, k2) = k1, k2. A K-equivariant DY-module is a DY-module M with a isomorphism of D_K×Y-modules

φ : p^∗₂M ∼= σ^∗M satisfying the cocycle condition.

We consider categories Modqc(DY, K) and Modc(DY, K). For λ =−ρ, we have Mod(g, χ_−ρ) ∼= Modqc(D_X) and Mod_f(g, χ_−ρ) ∼= Modqc(D_X).

Theorem 3. For any closed subgroup K ≤ G, we have Mod(g, χ−ρ, K) ∼= Modqc(DX, K) and Mod_f(g, χ_−ρ, K) ∼= Modqc(D_X, K).

Proof. What we have to prove is K-equivariances defined on Mod(g, χ_−ρ) and Mod_qc(D_X) coincide.

Consider M ∈ Modqc(D_X). K, X and K × X are all D-affine. So DK×X- modules are Γ(K×X, DK×X) = Γ(K, D_K)⊗CΓ(X, D_X)-modules. Since Γ(K, D_K) ∼= Γ(K,OK)⊗_CU (k) and Γ(X, D_X) ∼= U (g)/U (g) ker(χ_−ρ), D_K×X-module structures are determined by actions of Γ(K,OK)⊗ 1, k ⊗ 1 and 1 ⊗ g.

Γ(K × X, p^∗₂M) ∼= Γ(K,OK)⊗_CΓ(X,M). For σ^∗M, consider isomorphisms ϵ₁ : K × X → K × X, ϵ2 : K × X → K × X defined by ϵ1(k, x) = (k, kx), ϵ₂(k, x) = (k, k⁻¹x). ϵ₁ = ϵ⁻¹₂ and σ = p₂◦ ϵ1. So

Γ(K× X, σ^∗M) ∼= Γ(K× X, ϵ^∗1p₂^∗M) ∼= Γ(K× X, (ϵ2)_∗p^∗₂M)

∼= Γ(K× X, p^∗2M) ∼= Γ(K,OK)⊗CΓ(X,M).

For given h ∈ Γ(X, OX) and m ∈ Γ(X, M), the element h ⊗ m ∈ Γ(K, OK)⊗C

Γ(X,M) corresponds to the global section h ◦ p1 ⊗ p⁻¹2 m of p₂^∗M = OK×X ⊗_p⁻¹_{2 OX} p⁻¹₂ M and the global section h ◦ p1 ⊗ σ⁻¹m : (k, x) 7→ (k, h(k)k⁻¹ · m(kx)) of σ^∗M = OK×X ⊗σ⁻¹OX σ⁻¹M. The Γ(K, DK)⊗CΓ(X, D_X)-action on p^∗₂M. is











(f ⊗ 1) · (h ⊗ m) = fh ⊗ m for all f ∈ Γ(K, OK), (a⊗ 1) · (h ⊗ m) = a · h ⊗ m for all a ∈ k,

(1⊗ p) · (h ⊗ m) = h ⊗ p · m for all p ∈ g.

Consider the Γ(K, D_K)⊗_CΓ(X, D_X)-action on σ^∗M. (f ⊗ 1) · (h ⊗ m) = fh ⊗ m for all f ∈ Γ(K, OK). (a⊗ 1) · (h ⊗ m) = a · h ⊗ m − h ⊗ a · m for all a ∈ k. Finally,

d

dt exp(tp)k⁻¹m(k exp(−tp)x)|t=0 = d

dtk⁻¹exp(t Ad(k)(p))m(exp(t Ad(k)(p))kx)|t=0. Let Ad(k)(p) =∑

ih_i(k)p_i. We have (1⊗ p) · (h ⊗ m) = ∑

ihh_i⊗ pi· m for all p ∈ g.

The K-equivariance of M is equivlent to an Γ(K, DK)⊗CΓ(X, D_X)-module isomorphism from Γ(K,OK)⊗CΓ(X,M) ∼= Γ(K× X, p^∗2M) to Γ(K × X, σ^∗M) ∼= Γ(K,OK)⊗CΓ(X,M) satisfying the cocycle condition. Since Γ(K, OK)-actions on both sides are the same, the condition is aC-module homomorphism

e

φ : Γ(X,M) = 1 ⊗CΓ(X,M) → Γ(K, OK)⊗CΓ(X,M)

satisfying the cocycle condition and e

φ((a⊗ 1) · (1 ⊗ m)) = (a ⊗ 1) · eφ(1⊗ m) for all a ∈ k, m ∈ Γ(X, M). (5) e

φ((1⊗ p) · (1 ⊗ m)) = (1 ⊗ p) · eφ(1⊗ m) for all p ∈ g, m ∈ Γ(X, M). (6) Let φ(m) =e ∑

jgj ⊗ mj. The cocycle condition is equivlent to a K-representation structure of Γ(X,M). (5) is

0 = ∑

j

a· gj ⊗ mj− a · m.

a : m7→∑

ja·gj⊗mj is the k-action obtained from the K-action while a : m7→ a·m is the k-action obtained from the g-action. So (5) is (3).

(6) is

∑

j

g_j⊗ p · mj =∑

i,j

h_ig_j ⊗ pi · mj, which is (4).

Theorem 4. Let Y be a smooth variety and K be a linear algebraic group action on Y . Suppose there are only finitely many K-orbits in Y . Then Mod_c(D_Y, K) ∼= Mod_rh(D_Y, K). Moreover, the simple objects in Mod_c(D_Y, K) is parametrized by Υ(Y, K), the set of pairs (O, L), where O⊂ Y is an irreducible K-orbit and L is a K-equivariant local system on O^an.

Proof. We use induction on the number of K-orbits of Y . Suppose Y is a ho- mogeneous K-space. Then Y ∼= K/K^′ for some K^′ ≤ K. Consider morphisms σ : K × Y → Y the natural action, p2 : K × Y → Y the second projection, l : K → Spec C, π : K → Y the quotient map, j : Spec(C) → K, j(x) = K^′ and i : K → K × Y , i(k) := (k⁻¹, kK^′). Then for any M ∈ Modc(D_Y, K), we have

π^∗M = (p2◦ i)^∗M = i^∗p^∗₂M ∼= i^∗σ^∗M = (σ ◦ i)^∗M

= (j◦ l)^∗M = l^∗j^∗M = OX ⊗C(j^∗M).

j^∗M is a finite dimensional C-vector space, so π^∗M ∈ Modrh(D_K). Since π is smooth,M ∈ Modrh(DY).

Now consider the general case. Let O be a closed K-orbit of Y and Y^′ = Y −O.

Suppose i : O ,→ Y and j :,→ Y . Then we have the distinguish triagle

∫

ii^†M M ∫

jj^†M ⁺¹ .

We have i^†M ∈ D^bc(D_O) and j^†M ∈ Dc^b(D_Y′). By induction hypothesis, i^†M ∈ D_rh^b (D_O) and j^†M ∈ D^brh(D_Y′). We conclude that ∫

ii^†M,∫

jj^†M ∈ Drh^b (D_Y) and henceM ∈ Modrh(D_Y).

By Riemann-Hilbert correspondence, Mod_rh(D_Y, K) ∼= Perv(CY, K), which is parametrized by Υ(Y, K).

In particular, B has only finite orbits in X. We conclude that simple objects in Mod_f(g, χ_−ρ, B) are parametrized by Υ(X, B).

Bibliography

[HTT] R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, Perverse Sheaves, and Rep- resentation Theory.

[S] T.A. Springer, Linear Algebraic Groups

D-module Final Report II

Hung Chiang

June 26, 2017

Proposition 1 (^[HTT] Proposition 9.4.5, proved in^[HC]). χλ = χµ if and only if λ and µ are in the same W -orbit.

⟨λ, α^∨⟩ ≤ 0 for all α ∈ ∆⁺. (1)

⟨λ, α^∨⟩ < 0 for all α ∈ ∆⁺. (2) For v ∈ −P⁺, Borel-Weil-Bott theorem says Γ(X,L(v)) = H⁰(X,L(v)) = L⁻(v) and p_v :OX⊗_CL⁻(v)→ L(v) is surjective. Since H om_OX(L(v), OX) = L(−v) and Hom_C(L⁻(v),C) = L⁺(−v), we have L(−v) ,→ OX⊗CL⁺(−v). Apply L(v)⊗OX(·), we have i_v :OX ,→ L(v) ⊗_CL⁺(−v). Since L(v) is a line bundle, ker(pv) is a direct summand of OX ⊗CL⁻(v) as an OX-module locally. Therefore, im(iv) is a direct summand of L(v) ⊗_CL⁺(−v) as an OX-module locally.

Let λ∈ L and M be a Dλ-module. Apply M ⊗OX (·), we get pv :M ⊗CL⁻(v)↠ M ⊗OX L(v),

iv :M ,→ M ⊗OX L(v) ⊗CL⁺(−v).

Proposition 2 (^[HTT] Proposition 11.4.1). (i) If λ satisfies (2), then ker(pv) is a direct summand ofM ⊗_CL⁻(v) as a sheaf of abelian groups.

(ii) If λ satisfies (1), then im(i_v) is a direct summand of M ⊗OX L(v) ⊗CL⁺(−v) as a sheaf of abelian groups.

Proof. Let

L⁻(v) = L¹ ⊃ L² ⊃ · · · ⊃ L^r= 0

be a filtration of B-modules of L⁻(v) satisfying that Lⁱ/Lⁱ⁺¹ is the character µ_i of B, µ₁ = v, and µ_i < µ_j only if i < j. Then we obtain corresponding filtrations

OX ⊗_CL⁻(v) = V¹ ⊃ V² ⊃ · · · ⊃ V^r = 0,

M ⊗_CL⁻(v) =V¹ ⊃ V² ⊃ · · · ⊃ V^r = 0, and

M ⊗O L(v) ⊗CL⁺(−v) = W^r ⊃ W^r−1 ⊃ · · · ⊃ W¹ = 0.

The corresponding composition factors are

Vⁱ/Vⁱ⁺¹ ∼=M ⊗_OX L(µi), Wⁱ⁺¹/Wⁱ ∼=M ⊗_OX L(v − µi).

SinceM ⊗OX L(µ) is a Dλ+µ-module, the action of z on it is χ_λ+µ. So we have

r−1

∏

i=1

(z− χλ+µi)(M ⊗_CL⁻(v)) = 0

and r∏−1

i=1

(z− χλ+v−µi)(M ⊗OX L(v) ⊗CL⁺(−v)) = 0.

Seen as sheaves of abelian groups, M ⊗CL⁻(v) and M ⊗OX L(v) ⊗CL⁺(−v) are equipped with locally finite z-actions and thus have decompositions into χ-primary parts:

M ⊗CL⁻(v) =⊕

χ

(M ⊗CL⁻(v))_χ, M ⊗OX L(v) ⊗CL⁺(−v) =⊕

χ

(M ⊗OX L(v) ⊗CL⁺(−v))χ.

The morphisms p_v and i_v are V¹ → V¹/V² and W² → W^r, respectively. It suffices to prove that

(i) If λ satisfies (2), then ker(p_v) = (M ⊗CL⁻(v))_χλ+v. That is, χ_λ+µi = χ_λ+v ⇔ i = 1.

(ii) If λ satisfies (1), then im(i_v) = (M⊗OXL(v)⊗CL⁺(−v))χλ. That is, χ_λ+v−µi = χλ ⇔ i = 1.

Suppose λ satisfies (2). If χ_λ+µi = χ_λ+v, then there is a w∈ W such that w(λ+µi) = λ + v. That is, (w(λ)− λ) + (w(µi)− v) = 0. Since ⟨λ, α⟩ < 0 for all α ∈ ∆⁺, w(λ)− λ ≥ 0 and the equality holds if and only if w = id. Since w(µi) is a weight of L⁻(v), w(µi)≥ v. So w = id and thus µi = v. That is, i = 1.

Suppose λ satisfies (1). If χ_λ+v−µi = χ_λ, then there is a w ∈ W such that w(λ) = λ + v− µi. That is, (w(λ)− λ) + (µi − v) = 0. Since ⟨λ, α⟩ ≤ 0 for all α∈ ∆⁺, w(λ)≥ λ. Also, µi ≥ v. So µi = v and thus i = 1.

Theorem 1 (^[HTT] Theorem 11.6.1). Let Y be a smooth variety and K be a linear algebraic group action on Y . Suppose there are only finitely many K-orbits in Y . Then Modc(DY, K) ∼= Modrh(DY, K). Moreover, the simple objects in Modc(DY, K) is parametrized by Υ(Y, K), the set of pairs (O, L), where O ⊂ Y is an irreducible K-orbit and L is a K-equivariant local system on O^an.

Proof. We use induction on the number of K-orbits of Y . Suppose Y is a ho- mogeneous K-space. Then Y ∼= K/K^′ for some K^′ ≤ K. Consider morphisms σ : K × Y → Y the natural action, p2 : K × Y → Y the second projection, l : K → Spec C, π : K → Y the quotient map, j : Spec(C) → K, j(x) = K^′ and i : K → K × Y , i(k) := (k⁻¹, kK^′). Then for any M ∈ Modc(D_Y, K), we have

π^∗M = (p2◦ i)^∗M = i^∗p^∗₂M ∼= i^∗σ^∗M = (σ ◦ i)^∗M

= (j◦ l)^∗M = l^∗j^∗M = OX ⊗C(j^∗M).

j^∗M is a finite dimensional C-vector space, so π^∗M ∈ Modrh(D_K). Since π is smooth,M ∈ Modrh(D_Y).

Now consider the general case. Let O be a closed K-orbit of Y and Y^′ = Y −O.

Suppose i : O ,→ Y and j :,→ Y . Then we have the distinguish triagle

∫

ii^†M M ∫

jj^†M ⁺¹ .

We have i^†M ∈ D^bc(DO) and j^†M ∈ Dc^b(DY^′). By induction hypothesis, i^†M ∈ D_rh^b (D_O) and j^†M ∈ D^b_rh(D_Y′). We conclude that ∫

ii^†M,∫

jj^†M ∈ D_rh^b (D_Y) and henceM ∈ Modrh(DY).

By Riemann-Hilbert correspondence, Mod_rh(D_Y, K) ∼= Perv(CY, K), which is parametrized by Υ(Y, K).

In particular, B has only finite orbits in X. We conclude that simple objects in Mod_f(g, χ_−ρ, B) are parametrized by Υ(X, B).

0.1 Highest Weight Module

Definition 1. Let λ∈ h^∗ and M be a g-module. If there exists 0 ̸= m ∈ M such that m∈ M , nm = 0, and M = U (g)m, then M is called a highest weight module

with highest weight λ. m is called a highest weight vector.

In this case, M = U (n⁻)m and M = ⊕

µ≤λM_µ. M_λ = C m. Since M is generated by m, M is the quotiend U (g)/N as a U (g)-module. The relation contains at least n and h− λ(h) for all h ∈ h.

Definition 2. The Verma module is defined as M (λ) := U (g)/(U (g)n +∑

h∈h

U (g)(h− λ(h)1)).

M (λ) is the unique maximal highest weight module. If M be a highest weight module, there is a unique surjective homomorphism f : M (λ) → M such that f (¯1) = m.

Lemma 1 (^[HTT] Lemma 12.1.3). M (λ) is a free U (n⁻)-module. In particular, we compute that

ch(M (λ)) =∑

µ

dim(M (λ)_µ)e^µ=∑

β≤0

dim(U (n⁻)_β)e^λ+β

=e^λ ∏

β∈∆⁺

(1 + e^−β + e^−2β+· · · )

= e^λ

∏

β∈∆⁺(1− e^−β). Proof. Let I = (U (g)n +∑

h∈hU (g)(h− λ(h)1)). We want to prove that U(g) = U (n⁻)⊕ I. By P BW theorem, we have a canonical isomorphism

U (n⁻)⊗ U(h) ⊗ U(n) ∼= U (g).

So we have

∑

h∈h

U (g)(h− λ(h)1)) =∑

h∈h

U (n⁻)U (h)U (n)(h− λ(h)1))

=∑

h∈h

U (n⁻)U (h)(C +U(n)n)(h − λ(h)1))

⊂U(n⁻)

(∑

h∈h

U (h)(h− λ(h)1)) )

+ U (g)n.

So I = U (n⁻)(∑

h∈hU (h)(h− λ(h)1)))

+ U (g)n. Finally we have the isomorphism

U (g) =U (n⁻)U (h)U (n)

=U (n⁻)U (h)(C ⊕U(n)n)

=U (n⁻)U (h)⊕ U(g)n

=U (n⁻) (

C ⊕∑

h∈h

U (h)(h− λ(h)1)) )

⊕ U(g)n

=U (n⁻)⊕ I.

Lemma 2 (^[HTT]Lemma 12.1.4). There is a unique maximal proper U (g)-submodule N ⊂ M(λ).

Proof. Any proper U (g)-submodule of M (λ) is a weight module whose weights < λ.

So the sum of them is also a proper U (g)-submodule.

Define L(λ) = M (λ)/N . L(λ) is the minimal highest weight module.

Problem 1. Compute ch(L(λ)).

Example 1. If λ∈ ∆⁺, then L(λ) = L⁺(λ). Weyl’s character formula says ch(L(λ)) =

∑

w∈W(−1)^l(w)w^{w(λ+ρ)−ρ}

∏

β∈∆⁺(1− e^−β)

= ∑

w∈W

(−1)^l(w)ch(M (w(λ + ρ)− ρ)).

Lemma 3 (^[HTT] Lemma 12.1.6). For z∈ z, zm = χλ+ρ(z)m.

Proof. We decompose z into u + v where z ∈ U(h) and v ∈ n⁻U (g) + U (g)n. Then zm = um = λ(u)m = χ_λ+ρ(z)m.

Proposition 3 (^[HTT] Proposition 12.1.7). Let M be a highest weight module with highest weight λ. The M has a decomposition series with finite length and each composition factor of it has the form L(µ) where µ≤ λ and µ + ρ ∈ W (λ + ρ).

Proof. If M is simple then we are done. If M is not simple, then we take a nonzero proper submodule N ⊂ M. Let µ be a maximal weight of N and 0 ̸= n ∈ Nµ, then U (g)m ⊂ N is a highest weight module with highest weight µ. χµ+ρ(z)n = zn = χ_λ+ρ(z)n for all z ∈ z. So χµ+ρ = χ_λ+ρ. We have µ < λ and µ + ρ ∈ W (λ + ρ).

Replace M by N and repeat the process. We can repeat only finitely many times and obtain a simple U (g)-module N₁, which is a highest weight module with highest weight µ1. N1 ∼= L(µ1). Replace M by M /N1 and repeat the process. We obtain a sequence

0 = N0 ⊂ N1 ⊂ N2 ⊂ · · · ,

the composition factors of which have the form L(µ) for some µ ≤ λ and µ + ρ ∈ W (λ + ρ). Since|W (λ + ρ)| < ∞ and L(µ) can occur no more than dim(Mµ) times, the sequence is finite.

Fix a equivlence class Λ = W (λ + ρ)− ρ. Let aµλ the the multiplicity of L(µ) appearing in the deconposition series of M (λ). We have a_µλ ̸= 0 only if µ ∼ λ and µ≤ λ. aλλ = 1. Let (b_µλ) be the inverse matrix of (a_µλ). Then b_µλ ∈ Z and

ch(M (λ)) =∑

µ∈Λ

a_µλch(L(µ)).

ch(L(λ)) =∑

µ∈Λ

b_µλch(M (µ)).

It suffices to compute bµλ.

0.2 Kazhdan-Lusztig Conjecture

The problem is answered when Λ = W (−ρ)−ρ. In this case, Λ ⊂ P . We are consid- ering objects M (−w(ρ) − ρ), L(−wρ − ρ) ∈ Modf(g, χ_ρ, B) = Mod_rh(D_X, B). Every object in M od_f(g, χ_ρ, B) has a composition series of finite length, which is proved similarly as in the proof above. We consider the Grothdieck group K(Mod_f(g, χ_ρ, B)).

We have

[L(−wρ − ρ)] = ∑

y∈W

b_yw[M (−yρ − ρ)].

[M (−wρ − ρ)] = ∑

y∈W

a_yw[L(−yρ − ρ)].

We want to compute byw.

Definition 3. The Hecke algebra H(W ) is the Z[q¹, q⁻¹] algebra which is freely generated by{Tw | w ∈ W } as a Z[q¹, q⁻¹]-module with multiplicative relations

T_yT_w = T_yw, if l(yw) = l(y) + l(w).

(T_s+ 1)(T_s− q) = 0, if s∈ W.

Proposition 4 (^[HTT] Proposition 12.2.3). There exists a unique family {Py,w(q)} of polynomials in Z[q] satisfying the following conditions:

P_y,w(q) = 0 if y̸≤ w, P_w,w(q) = 1, deg(P_y,w(q))≤ l(w)− l(y) − 1

2 if y < w,

∑

y≤w

Py,w(q)Ty = q^l(w)∑

y≤w

Py,w(q⁻¹)T_y⁻¹₋₁.

Conjecture 1 (Kazhdan-Lusztig).

b_y,w = (−1)^l(w)−l(y)P_y,w(1).

Definition 4. For each w∈ W we define

X_w = BwB/B.

Here w is seen as an element in W = N_G(H)/H. The Schubert variety is defined as X_w.

Proposition 5 (^[HTT] Theorem 9.9.4, 9.9.5). X is the disjoint union of{Xw | w ∈ W}. Each Xw is isomorphic to C^l(w). X_w =⨿

y≤wX_w.

We denote by IC(CXw) the intersection complex on X_w and set C^πXw = IC(CXw)[− dim(Xw)].

We’ll show that the Kazhdan-Lusztig conjecture is reduced the theorem below.

Theorem 2 (Kazhdan-Lusztig,^[HTT]Theorem 12.2.5). For any y, w∈ W , we have

∑

i

dim(Hⁱ(C^πXw)_yB)q^i/2 = P_y,w(q).

In particular, We have Hⁱ(C^πXw)_yB = 0 for all odd i and

∑

j

(−1)^jdim(H^j(C^πXw)_yB) = P_y,w(1).

Let Mw = D_X ⊗U (g)M (−w(ρ) − ρ), Lw = D_X ⊗U (g)L(−w(ρ) − ρ), and Nw =

∫

iw

OXw = (i_w)_∗(D_X←Xw ⊗D_Xw OXw).

Nw ∈ Modc(D_X, B) = Mod_rh(D_X, B).

Lemma 4 (^[HTT] Lemma 12.3.1). Let w∈ W . Then

(i) ch(Γ(X,Nw)) = ch(M (−w(ρ)−ρ)). In particular, [Mw] = [Nw] in K(Modrh(DX, B)).

(ii) The only D_X submodule of Nw whose support is contained in X_w− Xw is 0.

Proof. Define two subalgebras of g:

n₁ = ⊕

α∈∆⁺∩w(∆⁺)

g_−α, n₂ = ⊕

α∈∆⁺∩−w(∆⁺)

g_α.

Let the corresponding unipotent subgroup of G be N₁ and N₂ respectively. Define a morphism φ : N₁× N2 → X by

φ(n₁, n₂) = n₁n₂wB/B.

Then φ is an open embedding. φ({e} × N2) = Xw. Let V = im(φ), we have the commutative diagram

N1× N2 V X

N2 ={e} × N2 Xw φ

iw

So

Γ(X,Nw) =Γ(

X, (iw)_∗(

DX←Xw ⊗D_Xw OXw

))

=Γ(Xw, DX←Xw ⊗D_Xw OXw)

=Γ(Xw, DV←Xw ⊗D_Xw OXw)

∼=Γ(N2, DN1×N2←N2 ⊗D_N2 ON2).

D_N1×N2←N2 ∼= (

D_N1×{e}⊗O_N1×{e} C)

⊗C

( Ω^⊗−1_N

1×{e}⊗O_N1×{e} C)

⊗C⊗CΓ(N₂,N2).

First, D_N1 = U (n₁)⊗COX, so D_N1×e⊗O_N1×{e} C ∼= U (n1). Second,

Ω^⊗−1_N

1×{e}⊗O_N1×{e} C ∼=∧^dim(n1)(n1⊗COX)⊗OX C = ∧^dim(n1)n1.

Finally, the exponential map gives the isomorphism n₂ ∼= N2. Therefore, Γ(N₂,N2) ∼= Γ(n₂,On2) = S(n^∗₂).

ch(Γ(X,Nw)) = ch(U (n₁)) ch(∧^dim(n1)n₁) ch(S(n^∗₂)).

We compute that

ch(U (n₁)) = ∏

α∈∆⁺∩w(∆⁺)

(1 + e^−α+ e^−2α+· · · ) = ∏

α∈∆⁺∩w(∆⁺)

1 1− e^−α, ch(∧^dim(n1)) = e

∑

α∈∆+∩w(∆+)−α = e^{−w(ρ)−ρ}, and

ch(S(n^∗₂)) = ∏

α∈∆⁺∩−w(∆⁺)

(1 + e^−α+ e^−2α+· · · ) = ∏

α∈∆⁺∩−w(∆⁺)

1 1− e^−α. So

ch(Γ(X,Nw)) = e^{−w(ρ)−ρ}

∏

α∈∆⁺(1− e^−α) = ch(M (−w(ρ) − ρ)).

Set Z = X− V and j : V → X be the open embedding, we have a distinguished traingle

RΓ_Z(Nw) Nw j_∗(Nw) ⁺¹ .

By definition, Nw → j∗(Nw) is an isomorphism, so RΓZ(Nw) = 0. So ΓZ(Nw).

Hence the only D_X submodule of Nw whose support is contained in Z is 0. Since Xw− Xw ⊂ Z, the assertion follows.

LetL(Xw,OXw) be the minimal extension of the DXw-module Xw. L(Xw,OXw)∈ Mod_rh(D_X, B).

Proposition 6 (^[HTT] Lemma 12.3.2). Let w ∈ W . Then we have (i)

L =L(X ,O ).