## Highlights - the early work leading to the Kazhdan–Lusztig Conjecture and its proof around 1980\. (James Humphreys 12) - The emphasis here is on highest weight modules, starting with Verma modules and culminating in the determination of formal characters of simple highest weight modules in the setting of the Kazhdan–Lusztig Conjecture \(1979\)\. (James Humphreys 12) - The book ends with an introduction to the influential work of Beilinson, Ginzburg, and Soergel on Koszul duality\. (James Humphreys 13) - The basic object of study here is a semisimple Lie algebra g over a field of characteristic 0, having a Cartan subalgebra h which is split: the eigenvalues of ad h are in the field, for all h ∈ h\. (James Humphreys 14) - a good working knowledge of the structure of semisimple Lie algebras over an algebraically closed field of characteristic 0 such as C\. (James Humphreys 14) - Denote by Φ ⊂ h∗ the root system of g relative to h (James Humphreys 15) - To each root α ∈ Φ corresponds a nonzero 1-dimensional subspace of g called a root space: g α = {x ∈ g | [hx] = α\(h\)x for all h ∈ h}\. (James Humphreys 15) - simple system ∆ ⊂ Φ (James Humphreys 15) - positive system Φ+ ⊂ Φ (James Humphreys 15) - Cartan decomposition g = n− ⊕ h ⊕ n, where n := L α>0 gα and n− := α<0 gα \. (James Humphreys 15) - standard Borel subalgebra is b := h ⊕ n (James Humphreys 15) - b − := h ⊕ n− \. (James Humphreys 15) - These are maximal solvable subalgebras of g; more generally, any such subalgebra is called a Borel subalgebra\. (James Humphreys 15) - Any subalgebra p containing a Borel subalgebra is called parabolic; (James Humphreys 15) - The adjoint representation ad : g → Der g is a first example of a representation of g\. (James Humphreys 15) - The associated Killing form \(x, y\) := Tr\(ad x ad y\) is nondegenerate\. (James Humphreys 15) - The algebra g decomposes uniquely \(up to order of summands\) into the direct sum of simple ideals\. (James Humphreys 15) - In turn, the simple Lie algebras are uniquely determined by their \(irreducible\) root systems; these are classified explicitly as types A ` , B` , C` , D` , E6 , E7 , E8 , F4 , G2 \. (James Humphreys 15) - Subalgebras of type A1 in g play a special role\. Such an algebra is isomorphic to sl\(2, C\), which has a basis \(h, x, y\): 1 0 0 1 0 0 h := , x := , y := \. 0 −1 0 0 1 0 (James Humphreys 15) - Denote by Φ ⊂ h ∗ the root system of g relative to h (James Humphreys 15) - Denote by Φ ⊂ h ∗ the root system of g relative to h (James Humphreys 15) - the euclidean reflection sα defined by λ 7→ λ − 2\(λ, α\)/\(α, α\)α sends Φ to itself (James Humphreys 16) - the Cartan invariant hβ, α∨ i := 2\(β, α\)/\(α, α\) lies in Z for all α, β ∈ Φ\. Here α∨ := 2α/\(α, α\) is the coroot of α\. The Z-span Λr of Φ in E is called the root lattice\. (James Humphreys 16) - Each choice of simple system ∆ in Φ defines a partition into subsets of positive and negative roots \(denoted respectively Φ+ and Φ− (James Humphreys 16) - Here ∆ forms a basis of EP\(or a Z-basis of Λr \), while each β ∈ Φ+ can be written uniquelyPas β = α∈∆ cα α with cα ∈ Z + \. Define the height of β to be ht β := α∈∆ cα ; so ht β = 1 if and only if β ∈ ∆ (James Humphreys 16) - For any α, β ∈ Φ, the roots of the form α + kβ form an unbroken root string α − rβ, \. \. \. , α − β, α, α + β, \. \. \. , α + sβ, which involves at most four roots\. (James Humphreys 17) - If β is positive but not simple, there exists a simple root α for which hβ, α∨ i > 0; thus sα β ∈ Φ+ and ht sα β < ht β\. (James Humphreys 17) - The natural symmetry group attached to a root system Φ is its Weyl group W , the \(finite!\) subgroup of GL\(E\) generated by all reflections sα with α ∈ Φ \(or just the simple reflections sα with α ∈ ∆ when ∆ is a fixed simple system\)\. Evidently the root lattice Λr is stable under the action of W \. (James Humphreys 17) - Abstractly, W is a finite Coxeter group, having generators sα \(α ∈ ∆\) and defining relations of the form \(sα sβ \)m\(α,β\) = 1 (James Humphreys 17) - Moreover, W satisfies the crystallographic restriction m\(α, β\) ∈ {2, 3, 4, 6} when α 6= β\. (James Humphreys 17) - length function on W (James Humphreys 17) - Given a subset I ⊂ ∆, the subgroup WI it generates is called a “parabolic” subgroup of W (James Humphreys 17) - If S denotes the set of simple reflections sα \(with α ∈ ∆\), the set T of all reflections in W is defined by T := S w∈W wSw−1 \. This is known to consist of the reflections sα with α ∈ Φ+ (James Humphreys 18) - w0 ≤ w if and only if w0 occurs as a subexpression (James Humphreys 18) - This is an associative algebra with 1, infinite dimensional if a 6= 0 and noncommutative if a is not abelian\. The algebra U \(a\) is left and right noetherian and has no zerodivisors\. Any Lie algebra homomorphism a1 → a2 induces an associative algebra homomorphism between the enveloping algebras\. The adjoint action of a on itself induces an action of a on U \(a\) by derivations: if x ∈ a, then ad x sends u 7→ xu − ux for all u ∈ U \(a\) (James Humphreys 19) - Given an ordered basis \(x1 , \. \. \. , xn \) of a, the monomials xt 11 \. \. \. xtnn with t i ∈ Z + form a basis of U \(a\)\. This is called a PBW \(Poincaré–Birkhoff– Witt\) basis\. In particular, a may be identified with a subspace of U \(a\)\. (James Humphreys 19) - Then U \(g\) is the diP rect sum of subspaces U \(g\) ν with ν ∈ Λ (James Humphreys 19) - Denote the center of U \(g\) by Z\(g\) (James Humphreys 19) - the structure of Z\(g\): it turns out to be just a polynomial algebra in ` indeterminates, though this is far from obvious\. (James Humphreys 19) - In the classical theory one constructs directly a special element of Z\(g\) \(unique up to scalar multiples\) which is called a Casimir element (James Humphreys 19) - It plays a key role in algebraic proofs of Weyl’s complete reducibility theorem \(recalled below\) and related study of finite dimensional representations\. (James Humphreys 19) - g = sl\(2, C\), with standard basis \(h, x, y\), a Casimir element is h2 +2xy+2yx (James Humphreys 20) - Rewritten in a standard PBW ordering, it becomes h2 + 2h + 4yx\. (James Humphreys 20) - Later it will be shown that τ fixes Z\(g\) pointwise, using properties of the Harish-Chandra homomorphism (James Humphreys 20) - τ the transpose map, (James Humphreys 20) - There is a standard anti-involution τ : g → g which interchanges xα with yα for all α ∈ Φ+ and fixes all h ∈ h; it extends canonically to an anti-automorphism of U \(g\) (James Humphreys 20) - The representations we study will involve weights relative to the action of h, (James Humphreys 20) - There is a natural dual lattice in E to the root lattice, defined by Λ := {λ ∈ E | hλ, α∨ i ∈ Z for all α ∈ Φ}\. (James Humphreys 20) - We call Λ the integral weight lattice associated to Φ\. It lies in the Q-span E0 of the roots in h∗ and includes the root lattice Λr as a subgroup of finite index\. (James Humphreys 20) - When the simple system ∆ is fixed, there is a natural partial ordering on Λ defined by µ ≤ λ if and only if λ − µ ∈ Γ, where Γ ⊂ Λ is defined to be the set of all Z + -linear combinations of simple roots (James Humphreys 20) - The subset Λ+ := Z + $ 1 + · · · + Z + $` is called the set of dominant integral weights (James Humphreys 20) - ρ := $1 + · · · + $` ∈ Λ+ can also be characterized as half the sum of positive roots\. It satisfies hρ, α ∨ i = 1, or sα ρ = ρ − α, for all α ∈ ∆\. (James Humphreys 20) - ρ := $1 + · · · + $` ∈ Λ+ can also be characterized as half the sum of positive roots\. It satisfies hρ, α ∨ i = 1, or sα ρ = ρ − α, for all α ∈ ∆\. (James Humphreys 20) - For a simple system ∆, there is a natural fundamental domain C ⊂ E for the action of W \. Relative to the inner product on E, C := {λ ∈ E | \(λ, α\) > 0 for all α ∈ ∆}, (James Humphreys 21) - We call C a Weyl chamber in E\. The Weyl chambers are in natural bijection with simple systems in Φ and are permuted simply transitively by W \. (James Humphreys 21) - In the finite dimensional case, the theory of integral weights outlined above is sufficient (James Humphreys 21) - \. In the infinite dimensional setting, we have to broaden (James Humphreys 21) - λ ∈ Λ+ , all wλ ≤ λ for w ∈ W \. (James Humphreys 21) - the idea of “weight”\. If M is arbitrary, it still makes sense to define its weight spaces relative to the action of h\. For each λ ∈ h∗ , let M λ := {v ∈ M | h · v = λ\(h\)v for all h ∈ h}\. (James Humphreys 22) - If Mλ 6= 0, we say that λ is a weight of M \. The multiplicity of λ in M is then defined to be dim Mλ \(possibly ∞\)\. Define Π\(M \) := {λ ∈ h∗ | Mλ 6= 0}, the set of all weights occurring in M \. (James Humphreys 22) - It is easily checked that weight P vectors for distinct weights in M are linearly independent (James Humphreys 22) - one key fact due to Weyl has no analogue in the infinite dimensional theory and will only be quoted here\. It is usually referred to as Weyl’s Complete Reducibility Theorem : Every finite dimensional U \(g\)-module is isomorphic to a direct sum of simple modules, the multiplicities of the latter being uniquely determined (James Humphreys 22) - When dim M < ∞, M is always a weight module\. This follows from Weyl’s Theorem: more precisely, elements of h act on M via semisimple matrices while elements of n or n− act via nilpotent matrices (James Humphreys 22) - This generalizes the usual notion of Jordan decomposition, which arises intrinsically in g, independent of any specific linear realization\. All weights of M are in fact integral\. (James Humphreys 22) - When ∆ is fixed, the corresponding fundamental dominant weights in Λ form a convenient basis of h∗ (James Humphreys 22) - λ = ` P i ci $i (James Humphreys 22) - λ = ` P i ci $i (James Humphreys 22) - Thanks to Weyl’s theorem on complete reducibility, the problem is to classify and construct \(up to isomorphism\) all simple finite dimensional modules (James Humphreys 23) - Fix a standard basis \(h, x, y\) for g, with [hx] = 2x, [hy] = −2y, [xy] = h\. Since dim h = 1, weights λ ∈ h∗ may be identified with complex numbers\. In turn, the integral weight lattice Λ is identified with Z and Λr with 2Z (James Humphreys 23) - The simple modules are in natural bijection with dominant integral weights λ ∈ Λ+ and may be denoted L\(λ\)\. (James Humphreys 23) - Here L\(λ\) has 1dimensional weight spaces, with weights λ, λ−2, \. \. \. , −\(λ−2\), −λ\. (James Humphreys 23) - dim L\(λ\) = λ + 1\. (James Humphreys 23) - h · vi = \(λ − 2i\)vi , x · vi = \(λ − i + 1\)vi−1 , y · vi = \(i + 1\)vi+1 \. (James Humphreys 23) - v0 is \(up to scalars\) the unique weight vector in L\(λ\) killed by x (James Humphreys 23) - W has order 2 and permutes weights of L\(λ\) by sending λ−2i to its negative\. (James Humphreys 23) - In 1\.6 the finite dimensional simple modules are identified as those having dominant integral highest weights\. (James Humphreys 26) - Then in Chapter 2 we shall recover the classical theorems of Weyl (James Humphreys 26) - The most accessible infinite dimensional modules in O are the Verma modules \(1\.3\), which arise first as an auxiliary tool for the construction of simple modules (James Humphreys 26) - central characters (James Humphreys 26) - formal characters of weight modules (James Humphreys 26) - the study of subcategories Op attached to arbitrary parabolic subalgebras p \(not just the Borel subalgebra b\) (James Humphreys 26) - The BGG category O is defined to be the full subcategory of Mod U \(g\) whose objects are the modules satisfying the following three conditions: (James Humphreys 26) - all finite dimensional modules lie in O (James Humphreys 27) - the set Π\(M \) of all weights of M is contained in the union of finitely many sets of the form λ − Γ, where λ ∈ h∗ and Γ is the semigroup in Λr generated by Φ+ \. (James Humphreys 27) - a finite generating set in \(O1\) can always be taken to consist of weight vectors\. (James Humphreys 27) - Thanks to the PBW Theorem \(0\.5\), we can write U \(g\) = U \(n− \)U \(h\)U \(n\) (James Humphreys 27) - Now V is stable under U \(h\), while the action of U \(n− \) on V produces only weights lower than these (James Humphreys 27) - Theorem\. Category O satisfies: \(a\) O is a noetherian category, i\.e\., each M ∈ O is a noetherian U \(g\)module\. \(b\) O is closed under submodules, quotients, and finite direct sums\. \(c\) O is an abelian category\. \(d\) If M ∈ O and L is finite dimensional, then L ⊗ M also lies in O\. Thus M 7→ L ⊗ M defines an exact functor O → O\. \(e\) If M ∈ O, then M is Z\(g\)-finite: for each v ∈ M , the span of {z · v | z ∈ Z\(g\)} is finite dimensional\. \(f\) If M ∈ O, then M is finitely generated as a U \(n− \)-module\. (James Humphreys 27) - \(O1\) M is a finitely generated U \(g\)-module\. L \(O2\) M is h-semisimple, that is, M is a weight module: M = λ∈h ∗ Mλ \. \(O3\) M is locally n-finite: for each v ∈ M , the subspace U \(n\) · v of M is finite dimensional\. (James Humphreys 27) - All weight spaces of M are finite dimensional\. (James Humphreys 27) - work in the subcategory Oint whose objects all have integral weights: this encompasses for example all finite dimensional modules (James Humphreys 28) - we sometimes consider the larger category C of U \(g\)-modules whose objects are the weight modules with finite dimensional weight spaces (James Humphreys 28) - Since each v ∈ M is a sum of weight vectors, we may assume that v ∈ Mλ for some λ ∈ h∗ (James Humphreys 28) - a nonzero vector v + in a U \(g\)-module M to be a maximal vector of weight λ ∈ h∗ if v + ∈ Mλ (James Humphreys 28) - Exercise\. \(a\) If M ∈ O and [λ] := λ + Λr is any coset of h∗ modulo Λr , let M [λ] be the sum of all weight spaces Mµ for which µ ∈ [λ]\. Prove that M [λ] is a U \(g\)-submodule of M and that M is the direct sum of \(finitely many!\) such submodules\. \(b\) Deduce that all weights of an indecomposable module M ∈ O lie in a single coset of h∗ modulo Λr \. (James Humphreys 28) - and n · v + = 0 (James Humphreys 29) - highest weight module of weight λ if there is a maximal vector v + ∈ Mλ such that M = U \(g\) · v + (James Humphreys 29) - By the PBW Theorem, such a module satisfies M = U \(n− \) · v + \. (James Humphreys 29) - weight vectors having distinct weights are linearly independent (James Humphreys 29) - only finitely many choices of i1 , \. \. \. , im yield the same weight, by expressing sums of positive roots in terms of simple roots\. (James Humphreys 29) - gα maps Mµ into Mµ+α , (James Humphreys 29) - Theorem\. Let M be a highest weight module of weight λ ∈ h∗ , generated by a maximal vector v + \. Fix an ordering of the positive roots as α1 , \. \. \. , αm and choose corresponding root vectors yi in g−αi \. Then: \(a\) M is spanned by the vectors y1i1 \. \. \. ym im · v + with i ∈ Z+ , having j P respective weights λ − ij αj \. Thus M is a semisimple h-module\. \(b\) All weights µ of M satisfy µ ≤ λ: µ = λ − \(sum of positive roots\)\. \(c\) For all weights µ of M , we have dim Mµ < ∞, while dim Mλ = 1\. Thus M is a weight module, locally finite as n-module, and M ∈ O\. \(d\) Each nonzero quotient of M is again a highest weight module of weight λ\. \(e\) Each submodule of M is a weight module\. A submodule generated by a maximal vector of weight µ < λ is proper; in particular, if M is simple its maximal vectors are all multiples of v + \. \(f\) M has a unique maximal submodule and unique simple quotient\. In particular, M is indecomposable\. \(g\) All simple highest weight modules of weight λ are isomorphic\. If M is one of these, dim EndO M = 1\. (James Humphreys 29) - Here we start with the Borel subalgebra b corresponding to a fixed choice of positive roots, which in turn has an abelian quotient algebra b/n isomorphic to h (James Humphreys 30) - M \(λ\) := U \(g\) ⊗U \(b\) Cλ (James Humphreys 30) - denoted Cλ \. (James Humphreys 30) - Corollary\. Let M be any nonzero module in O\. Then M has a finite filtration 0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mn = M with nonzero quotients each of which is a highest weight module\. (James Humphreys 30) - This is called a Verma module and may also be written as Indgb Cλ (James Humphreys 31) - U \(g\) ∼ = U \(n− \) ⊗ U \(b\), which allows us to ∼ write M \(λ\) = U \(n \) ⊗ − Cλ as a left U \(n− \)-module (James Humphreys 31) - v + := 1 ⊗ 1 (James Humphreys 31) - v + is a maximal vector (James Humphreys 31) - and generates the U \(g\)-module M \(λ\) (James Humphreys 31) - the set of weights of M \(λ\) is visibly λ − Γ\. (James Humphreys 31) - M \(λ\) by generators and relations (James Humphreys 31) - the left ideal I of U \(g\) which annihilates v + is generated by n together with all h − λ\(h\) · 1 with h ∈ h\. Thus M \(λ\) ∼ = U \(g\)/I\. (James Humphreys 31) - Exercise\. Show that M \(λ\) has the following property: For any M in O, Hom U \(g\) M \(λ\), M = HomU \(g\) Indgb Cλ , M ∼ = g HomU \(b\) Cλ , Resb M , where Resgb is the restriction functor\. [Use the universal mapping property of tensor products\.] In the context of induced modules for group algebras of finite groups, this adjointness property is known as Frobenius reciprocity\. (James Humphreys 31) - Theorem\. Every simple module in O is isomorphic to a module L\(λ\) with λ ∈ h∗ and is therefore determined uniquely up to isomorphism by its highest weight\. Moreover, dim HomO L\(µ\), L\(λ\) = δλµ \. (James Humphreys 31) - Proposition\. Given λ ∈ h∗ and a fixed simple root α, suppose n := hλ, α∨ i lies in Z \. If v + is a maximal vector of weight λ in M \(λ\), then yαn+1 · v + is a maximal vector of weight µ := λ − \(n + 1\)α < λ\. Thus there exists a nonzero homomorphism M \(µ\) → M \(λ\) whose image lies in the maximal submodule N \(λ\)\. (James Humphreys 32) - Exercise\. When g = sl\(2, C\), show that M \(λ\) ⊗ M \(µ\) cannot lie in O\. (James Humphreys 33) - Corollary\. With λ as in the proposition, let v + be instead a maximal vector of weight λ in L\(λ\)\. Then yαn+1 · v + = 0\. (James Humphreys 33) - dim L\(λ\) < ∞ if and only if λ ∈ Z + \. In this case, the maximal submodule of M \(λ\) is isomorphic to L\(−λ − 2\)\. • M \(λ\) is simple if and only if λ is not in Z + \. (James Humphreys 33) - the integral weight lattice Λ is identified with Z and Λr with 2Z\. (James Humphreys 33) - dim h = 1, (James Humphreys 33) - Theorem\. The simple module L\(λ\) in O is finite dimensional if and only if λ ∈ Λ+ \. This is the case if and only if dim L\(λ\)µ = dim L\(λ\)wµ for all µ ∈ h∗ and all w ∈ W \. (James Humphreys 34) - λ\(h\)z · v + \. (James Humphreys 36) - z · v + = χλ \(z\)v + (James Humphreys 36) - pr : U \(g\) → U \(h\) (James Humphreys 36) - χλ \(z\) = λ\(pr\(z\)\) (James Humphreys 36) - λ : U \(h\) → C \. (James Humphreys 36) - λ : U \(h\) → C \. (James Humphreys 36) - χλ \(z\) = λ\(pr\(z\)\) (James Humphreys 36) - pr : U \(g\) → U \(h\) (James Humphreys 36) - z · v + = χλ \(z\)v + (James Humphreys 36) - λ\(h\)z · v + \. (James Humphreys 36) - For w ∈ W and λ ∈ h ∗ , define a shifted action of W \(called the dot action\) by w · λ = w\(λ + ρ\) − ρ (James Humphreys 37) - For w ∈ W and λ ∈ h ∗ , define a shifted action of W \(called the dot action\) by w · λ = w\(λ + ρ\) − ρ (James Humphreys 37) - he usual notion of regular weight for λ ∈ h∗ , requiring that the isotropy group of λ in W be trivial, or |W λ| = |W |, is replaced in the setting of linkage classes by a shifted notion of regular weight \(which might also be called dot-regular\): the weight λ ∈ h∗ is regular if |W · λ| = |W |, or in other words, hλ + ρ, α∨ i = 6 0 for all α ∈ Φ (James Humphreys 37) - χλ = χw·λ (James Humphreys 38) - χλ \(z\) = λ\(ξ\(z\)\) f (James Humphreys 38) - χλ \(z\) = λ\(ξ\(z\)\) f (James Humphreys 38) - χλ = χw·λ (James Humphreys 38) - Proposition\. If λ ∈ Λ and µ is linked to λ then χλ = χµ \. (James Humphreys 38) - Exercise\. Show that the homomorphism ψ is independent of the choice of a simple system in Φ\. [Any simple system has the form w∆ for some w ∈ W \.] (James Humphreys 39) - ξ\(z\) and w · ξ\(z\) agree on Λ\. (James Humphreys 39) - χλ \(z\) = \(λ + ρ\)\(ψ\(z\) (James Humphreys 39) - χλ \(z\) = \(λ + ρ\)\(ψ\(z\) (James Humphreys 39) - ξ\(z\) and w · ξ\(z\) agree on Λ\. (James Humphreys 39) - Theorem \(Harish-Chandra\)\. Let ψ : Z\(g\) → S\(h\) be the twisted HarishChandra homomorphism\. \(a\) The homomorphism ψ is an isomorphism of Z\(g\) onto S\(h\)W \. \(b\) For all λ, µ ∈ h∗ , we have χλ = χµ if and only if µ = w · λ for some w ∈ W\. \(c\) Every central character χ : Z\(g\) → C is of the form χλ for some λ ∈ h∗ \. (James Humphreys 39) - This is the key point (James Humphreys 39) - Exercise\. Prove that the transpose map τ \(see 0\.5\) fixes Z\(g\) pointwise\. [Check that τ commutes with the Harish-Chandra homomorphism ξ and use the fact that ξ is injective\.] (James Humphreys 41) - [M ] = [N ] if and only if M and N have the same composition factor multiplicities\. (James Humphreys 42) - each M ∈ O possesses a composition series with simple quotients isomorphic to various L\(λ\) (James Humphreys 42) - [M \(λ\) : L\(λ\)] = 1 in view of the fact that dim M \(λ\)λ = 1 (James Humphreys 42) - each M ∈ O can be written as a direct sum of indecomposable modules, with the summands being unique up to isomorphism and order (James Humphreys 42) - K\(O\) has a Z-basis consisting of the symbols [L\(λ\)], (James Humphreys 42) - the coefficient of [L\(λ\)] is the composition factor multiplicity [M : L\(λ\)]\. (James Humphreys 42) - the socle of a module M is the sum \(automatically direct\) of all simple submodules of M , denoted Soc M ; (James Humphreys 42) - it is the largest semisimple submodule of M \. (James Humphreys 42) - The radical of M is the intersection of all maximal submodules in M , denoted Rad M \. It may also be characterized as the smallest submodule for which M/ Rad M is semisimple; this quotient is called the head, denoted Hd M \. For example, Rad M \(λ\) = ∼ = N \(λ\) and Hd M \(λ\) ∼ = L\(λ\)\. (James Humphreys 42) - Exercise\. Fix a central character χ and let {V \(λ\) } be a collection of modules in Oχ indexed by the weights λ for which χ = χλ and satisfying: \(1\) \(λ\) λ = 1; \(2\) µ ≤ λ for all weights µ of V \(λ\) \(λ\) dim Vλ = 1; \(2\) µ ≤ λ for all weights µ of V \(λ\) \. Then the symbols [V \(λ\) ] form a Z-basis of the Grothendieck group K\(Oχ \)\. For example, take V \(λ\) = M \(λ\) or L\(λ\)\. (James Humphreys 43) - L O = χ Oχ i (James Humphreys 43) - Thanks to Harish-Chandra’s Theorem, each central character χ occurring in this way must be of the form χλ for some λ ∈ h∗ (James Humphreys 43) - Proposition\. Category O is the direct sum of the subcategories Oχ as χ ranges over the central characters of the form χλ \. Therefore each indecomposable module lies in a unique Oχ \. In particular, each highest weight module of weight λ lies in Oχλ \. (James Humphreys 43) - each indecomposable module lies in a single Oχ \. (James Humphreys 43) - O χ involves only finitely many simple modules and the corresponding Verma modules (James Humphreys 43) - each indecomposable module belongs to a single block (James Humphreys 44) - M decomposes uniquely as a direct sum of submodules, each belonging to a single block\. (James Humphreys 44) - Proposition\. If λ ∈ Λ, the subcategory Oχλ is a block of O\. (James Humphreys 44) - When χ = χ0 , Oχ is called the principal block (James Humphreys 44) - Exercise\. Suppose λ 6∈ Λ, so the linkage class W · λ is the disjoint union of its nonempty intersections with various cosets of Λr in h∗ \. Prove each M ∈ Oχλ has a corresponding direct sum decomposition M = Mi , in which all weights of Mi lie in a single coset\. [Recall Exercise 1\.1\(b\)\.] (James Humphreys 45) - Now we can define M M ∨ := Mλ∗ for any M ∈ C λ∈h ∗ and call this the dual in C\. (James Humphreys 63) - µ\(h\)f \(v\) = f \(µ\(h\)v\) = \(h · f \)\(v\) = λ\(h\)f \(v\)\. (James Humphreys 63) - Exercise\. Show that \(L ⊗ M \)∨ ∼ = L∨ ⊗ M ∨ if M ∈ O and dim L < ∞\. (James Humphreys 64) - What do we know at this point about the possible composition factors L\(µ\) of a Verma module M \(λ\)? The most obvious condition is \(1\) µ ≤ λ; in particular, µ must lie in the same coset of h∗ /Λr as λ\. (James Humphreys 65) - \(2\) µ = w · λ (James Humphreys 65) - Φ [λ] := {α ∈ Φ | hλ, α∨ i ∈ Z} (James Humphreys 65) - 3\.5\. Dominant and Antidominant Weights (James Humphreys 67) - Call λ ∈ h∗ antidominant if hλ + ρ, α∨ i 6∈ Z >0 for all α ∈ Φ+ \. For example, −ρ is antidominant\. Similarly, call λ dominant if hλ + ρ, α∨ i 6∈ Z <0 for all α ∈ Φ+ (James Humphreys 67) - Exercise\. Show that Φ[λ] ∩ Φ+ is a positive system in the root system Φ[λ] ; but the corresponding simple system \(call it ∆[λ] \) may be unrelated to ∆\. For a concrete example, take Φ to be of type B2 with short simple root α and long simple root β\. If λ := α/2, check that Φ[λ] contains just the four short roots in Φ\. (James Humphreys 67) - the subcategories corresponding to these orbits W[λ] · λ are precisely the blocks of O (James Humphreys 67) - Notice that any linkage class W · λ contains at least one antidominant weight: If a weight µ in the class is minimal relative to the standard partial ordering, but some hµ + ρ, α∨ i ∈ Z >0 , then the linked weight sα · µ < µ contradicts the minimality (James Humphreys 68) - Similarly, for any λ ∈ h∗ , the orbit W[λ] · µ contains at least one antidominant weight\. (James Humphreys 68) - Therefore there is a unique antidominant weight in the orbit W[λ] · λ\. (James Humphreys 68) - 3\.6\. Tensoring Verma Modules with Finite Dimensional Modules (James Humphreys 69) - We say that M ∈ O has a standard filtration \(also sometimes called a Verma flag\) if there is a sequence of submodules 0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mn = M for which each M i := Mi /Mi−1 \(1 ≤ i ≤ n\) is isomorphic to a Verma modul (James Humphreys 71) - the multiplicity with which each Verma module M \(λ\) occurs in a standard filtration of M by M : M \(λ\) , (James Humphreys 71) - [M : L\(λ\)] for the multiplicity of L\(λ\) in a Jordan–Hölder series of M (James Humphreys 71) - Exercise\. \(a\) If a module M has a standard filtration and there exists an epimorphism ϕ : M → M \(λ\), prove that Ker ϕ admits a standard filtration\. \(b\) Show by example when g = sl\(2, C\) that the existence of a monomorphism ϕ : M \(λ\) → M when M has a standard filtration fails to imply that Coker ϕ also has a standard filtration\. (James Humphreys 72) - O has enough projectives: for each M ∈ O, there is a projective object P ∈ O and an epimorphism P → M (James Humphreys 73) - projective if the left exact functor Hom\(P, ?\) is also right exact\. Equivalently, given an epimorphism π : M → N and any morphism ϕ : P → N , there is a lifting ψ : P → M such that πψ = ϕ\. Dually, an object Q is injective if the right exact functor Hom\(?, Q\) is also left exact\. (James Humphreys 73) - Here π is an epimorphism and is essential, meaning that no proper submodule of the projective module PM is mapped onto M (James Humphreys 75) - each M ∈ O has a projective cover π : PM → M \. (James Humphreys 75) - Each projective in O is a direct sum of copies of various P \(λ\), (James Humphreys 76) - Theorem \(BGG Reciprocity\)\. Let λ, µ ∈ h∗ \. Then P \(λ\) : M \(µ\) = [M \(µ\) : L\(λ\)], which is the same as [M \(µ\)∨ : L\(λ\)]\. (James Humphreys 78) - So far we have found five nonisomorphic indecomposable modules in Oχ : L\(λ\), L\(µ\) = M \(µ\), M \(λ\) = P \(λ\), M \(λ\)∨ = Q\(λ\), P \(µ\) = Q\(µ\)\. (James Humphreys 80) - Exercise\. Assume λ + ρ ∈ Λ+ \. \(a\) Prove that the unique simple submodule of M \(λ\) is isomorphic to M \(w◦ · λ\), where w◦ is the longest element of W \. \(b\) In case λ ∈ Λ+ , show that the inclusions obtained in the proposition are all proper\. (James Humphreys 89) - hλ k + ρ, ∨ αk+1 i = hw0 · λ + ρ, αk+1 ∨ i = hw0 \(λ + ρ\), αk+1 ∨ i= = hλ + ρ, \(w0 \)−1 αk+1 ∨ i = hλ + ρ, \(w0 \)−1 αk+1 ∨ i ∈ Z + \. (James Humphreys 89) - there is a sequence of embeddings M \(w · λ\) = M \(λn \) ⊂ M \(λn−1 \) ⊂ · · · ⊂ M \(λ0 \) = M \(λ\)\. (James Humphreys 89) - λ 0 := λ, (James Humphreys 89) - λ k := sk · λk−1 (James Humphreys 89) - w · λ = λn ≤ λn−1 ≤ · · · ≤ λ0 = λ, (James Humphreys 89) - λ + ρ ∈ Λ+ (James Humphreys 89) - M \(w · λ\) ⊂ M \(λ\) (James Humphreys 89) - Then M \( (James Humphreys 89) - [M \(λ\) : L\(w · λ\)] > 0\. (James Humphreys 89) - Example\. To illustrate the potential problem here, let g = sl\(3, (James Humphreys 90) - it is nontrivial to find an embedding M \(sγ · µ\) ,→ M \(µ\) (James Humphreys 90) - one gets embeddings M \(sγ · µ\) ,→ M \(sβ · λ\) ,→ M \(λ\)\. (James Humphreys 90) - µ := sα · λ\. (James Humphreys 90) - s γ · µ < µ\. (James Humphreys 90) - s γ = sα sβ sα (James Humphreys 90) - Theorem \(Verma\)\. Let λ ∈ h∗ \. Given α > 0, suppose µ := sα · λ ≤ λ\. Then there exists an embedding M \(µ\) ⊂ M \(λ\)\. (James Humphreys 92) - Exercise\. Work through the steps of this argument in the special case discussed in Example 4\.3\. (James Humphreys 93) - Proof \(Integral Case\)\. (James Humphreys 93) - Theorem\. The blocks of O are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by W[λ] to an antidominant weight λ\. Thus the blocks are in natural bijection with antidominant \(or alternatively, dominant\) weights\. (James Humphreys 96) - the blocks involving integral weights are the subcategories Oχλ containing the modules whose composition factors have highest weights linked to λ ∈ Λ (James Humphreys 96) - Exercise\. In the case of sl\(3, C\), what can be said at this point about Verma modules with a singular integral highest weight? Leaving aside the trivial case of −ρ, a typical linkage class has three elements: for example, if λ lies just in the α-hyperplane and is antidominant, the linked weights are λ, sβ · λ, sα sβ · λ\. (James Humphreys 98) - composition factor multiplicities of arbitrary Verma modules (James Humphreys 98) - [M \(w · λ\) : L\(λ\)] = 1 for all w\. (James Humphreys 98) - ch L\(sα · λ\) = ch M \(sα · λ\) − ch M \(λ\) (James Humphreys 98) - Since λ is antidominant, Theorem 4\.4 insures that M \(λ\) = L\(λ\), while no other M \(w · λ\) can be simple\. (James Humphreys 98) - L\(λ\) is the unique simple submodule of each M \(w · λ\) (James Humphreys 98) - M \(λ\) = M \(λ\)0 ⊃ M \(λ\)1 ⊃ M \(λ\)2 ⊃ \. \. (James Humphreys 108) - The linkage class of such a weight has size |W | and is indexed by its lowest weight λ, which is antidominant: (James Humphreys 108) - this means that hλ + ρ, α∨ i 6∈ Z >0 for all α > 0 (James Humphreys 108) - Exercise\. Let λ be regular, antidominant, and integral\. In the Jantzen filtration of M \(w · λ\), prove that the number n above is just `\(w\), so there are `\(w\) + 1 nonzero layers in the filtration \(as illustrated in 5\.4 below by sl\(3, C\)\)\. [Use 0\.3\(2\) to describe Φ+ w·λ \.] (James Humphreys 109) - The formal characters satisfy (James Humphreys 109) - M \(λ\)1 = N \(λ\), (James Humphreys 109) - contravariant form (James Humphreys 109) - M \(λ\)i /M \(λ\)i+1 has a nondegenerate co (James Humphreys 109) - M \(λ\)i := M \(λ\)i /M \(λ\)i+1 the ith filtration layer\. (James Humphreys 109) - the subset Φ[λ] of roots α for which hλ + ρ, α∨ i ∈ Z (James Humphreys 109) - this is a root system, with a positive system Φ [λ] ∩ Φ+ \. (James Humphreys 109) - Φ + λ for the subset of those α satisfying sα · λ < λ, (James Humphreys 109) - If M \(λ\)n 6= 0 but M \(λ\)n+1 = 0, the sum formula shows that n = |Φ+ λ| λ provided one knows that the unique simple submodule L\(µ\) of M \(λ\) occurs just once as a composition factor of M \(λ\) \(hence also of each submodule M \(sα · λ\)\) (James Humphreys 109) - Taking λ to be regular, antidominant, and integral, (James Humphreys 110) - It follows quickly that M \(w · λ\)2 = L\(λ\) and M \(w · λ\)i = 0 for i > 2\. (James Humphreys 110) - Similar reasoning then applies to the cases w = sβ sα and w = w◦ \. (James Humphreys 110) - require that µ − λ ∈ Λ; in this case we say that λ and µ are compatible (James Humphreys 143) - Φ [λ] = Φ[µ] and W[λ] = W[µ] (James Humphreys 143) - Now the W -orbit of ν := µ − λ contains a unique weight ν in Λ+ ; set L := L\(ν\) and write Tλµ for the resulting functor on O \(or on Oχλ \)\. We call Tλµ (James Humphreys 143) - M 7→ prµ L ⊗ \(prλ M \) (James Humphreys 143) - Ext nO \(Tλµ M, N \) ∼ = ExtnO O \(M, Tµλ N \)\. (James Humphreys 144) - λ, µ ∈ h∗ be compatible (James Humphreys 144) - the chambers are in natural bijection with simple systems in Φ (James Humphreys 145) - if C corresponds to ∆, then C lies on the positive side of all hyperplanes Hα with α ∈ ∆ \( (James Humphreys 145) - Exercise\. Let g = sl\(2, C\)\. Show that Tλµ need not take Verma modules to Verma modules\. [For example, let λ = 1 and µ = −3\.] (James Humphreys 145) - there is a \(shifted\) dominant chamber whose upper closure is itself: C ◦ := {λ ∈ E | hλ + ρ, α∨ i > 0 for all α ∈ ∆}\. (James Humphreys 146) - Let λ, µ ∈ h∗ be antidominant and compatible\. (James Humphreys 150) - Then Tλµ M \(w · λ\) ∼ = M \(w · µ\) for all w ∈ W[λ] \. Similarly, (James Humphreys 150) - Assume that λ \ lies in a facet F of E\(λ\) relative to the dot-action of W[λ] , while µ\ lies in F \. (James Humphreys 150) - Example\. Let g = sl\(2, C\) and consider just integral weights \(identified with integers\)\. In case λ ≥ 0, it lies inside the facet F := Z + \. Start with the exact sequence 0 → L\(−λ − 2\) → M \(λ\) → L\(λ\) → 0\. We know that L\(−λ − 2\) = M \(−λ − 2\)\. Now set µ = −1 \(that is, −ρ\), so M \(µ\) = L\(µ\)\. Here µ lies in F as required in the proposition\. By Theorem 7\.6, applying T λµ produces the exact sequence 0 → M \(µ\) → M \(µ\) → Tλµ L\(λ\) → 0\. This forces Tλµ L\(λ\) = 0\. (James Humphreys 151) - the blocks Oλ are parametrized (James Humphreys 151) - If w ∈ W[λ] , then Tλµ L\(w · λ\) is either 0 or else isomorphic to L\(w · µ\)\. (James Humphreys 151) - by antidominant weights λ ∈ h∗ : the objects in Oλ are the modules whose composition factors all have highest weights in W[λ] · λ\. (James Humphreys 152) - For an arbitrary λ ∈ h∗ , we know initially ∼ = L\(λ\) and Rad M \(λ\) = N \(λ\), while Soc M \(λ\) ∼ = that Hd M \(λ\) ∼ = L\(λ\) and Rad M \(λ\) = N \(λ\), while Soc M \(λ\) ∼ = L\(µ\), the simple module with antidominant highest weight in the block determined by λ\. (James Humphreys 188)