THE THEOREM OF THE HIGHEST WEIGHT

THE THEOREM OF THE HIGHEST WEIGHT ANKE D. POHL Abstract. Incomplete notes of the talk in the IRTG Student Seminar 07.06.06. This is a draft version and thought for internal use only. The Theorem of the Highest Weight states that the equivalence classes of irreducible finite-dimensional representations of a finite-dimensional complex semisimple Lie algebra g can be characterized by their highest weight. Moreover, it tells that the set of highest weight are identical with the set of dominant algebraically integral functionals on a Cartan subalgebra of g. In the first section we collect the definitions and propositions we need for the statement and (partial) proof of the Theorem of the Highest Weight. The omission of proofs does not mean that these are trivial. All proofs can be found in the book [Kna02] on which all of the talk is heavily based. The second section contains a statement of the Theorem of the Highest Weight. We will prove the stated properties of the highest weight and that the correspondence is injective. The proof of the surjectivity of the correspondence is very long hard work and can be found in [Kna02]. 1. Preliminaries A complex Lie algebra g is a complex algebra such that the C-bilinear product [, ]: g g g satisfies the following two properties for all X, Y, Z g (a) antisymmetry: [X, Y ] = [Y, X], (b) Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [Y, X]] = 0. Note that the antisymmetry implies that g is abelian if and only if [X, Y ] = 0 for all X, Y g. If V is a complex vector space, then End C (V ) endowed with the product [X, Y ] := X Y Y X is a complex Lie algebra. In what follows, g will always denote a complex Lie algebra and V a complex vector space. A representation π of g on V is a Lie algebra homomorphism π : g End C (V ). An invariant subspace for such a representation is a vector subspace U of V such that π(g)u U. A representation is called irreducible if the only invariant subspaces are 0 and V. 1

2 A. POHL Let π be a representation of g on V, and π a representation of g on V. Then π and π are called equivalent if there is a vector space isomorphism E : V V such that E π(x) = π (X) E for all X g. The equivalence classes of irreducible finite-dimensional representations of g are denoted by ĝ fin. We need to define three subclasses of Lie algebras: the solvable, the semisimple and the nilpotent ones. If a and b are subsets of g, then [a, b] := {[X, Y ] X a, Y b} denotes the Lie subalgebra of g spanned by the elements [X, Y ]. For n N we define g 0 := g and g n+1 := [g n, g n ]. The commutator series of g is the series g 0 g 1 g 2.... If there is an n N 0 such that g n = 0, then g is called solvable. The radical rad g is the largest solvable ideal in g, its existence can be proved for each Lie algebra g (cf. [Kna02, Prop 1.12]). Finally, g is called semisimple, if g has no nonzero solvable ideals, that is to say, rad g = 0. Next we define for n N the ideals g 0 := g and g n+1 := [g, g n ]. The series g 0 g 1 g 2... is called the lower central series for g. If there is an n N 0 such that g n = 0, then g is called nilpotent. Recall that the (infinitesimal) adjoint representation ad g : g End C (g) is a representation of g on itself. Proposition 1.1 ([Kna02, Prop 2.5]). Let g be a finite-dimensional complex Lie algebra and h a nilpotent Lie subalgebra. For α h = L(h, C) define Then we have g α := {X g H h n N: (ad g H α(h)1) n X = 0}. (a) g = α h g α, (b) h g 0, (c) [g α, g β ] g α+β. If g α 0, then g α is called a generalized weight space of g relative to ad g h. The item (c) shows that g 0 is a subalgebra. A nilpotent Lie subalgebra h of g is called a Cartan subalgebra if h = g 0. Of special importance is the next proposition. In general, for an infinitedimensional or real Lie algebra a Cartan subalgebra does not exist. Proposition 1.2 ([Kna02, Thm 2.9, Thm 2.15]). Any finite-dimensional complex Lie algebra g has a Cartan subalgebra, and this is unique up to (inner) automorphism.

THEOREM OF THE HIGHEST WEIGHT 3 If h is a Cartan subalgebra of g, then the generalized weight spaces of Proposition 1.1 are called root spaces, the nonzero weights are called roots, and = (g, h) denotes the (finite) set of roots. The decomposition g = h α g α is known as the root-space decomposition of g with respect to h. Elements of g α are called root vectors for the root α. If g is semisimple, then a Cartan subalgebra has special properties and the root spaces have a simple form. Proposition 1.3 ([Kna02, Prop 2.10, Prop 2.13, Cor 2.23, Prop 2.21]). Let g be a complex semisimple Lie algebra and h a Cartan subalgebra. Then h is abelian and ad g (h) is simultaneously diagonable. For each α, the root space g α is one-dimensional and g α = {X g H h: ad(h)x = α(h)x}. From now on let g be a complex semisimple finite-dimensional Lie algebra and h a Cartan subalgebra. Let π be a representation of g on V. For λ h = L(h, C) (complex linear functionals) we put V λ := {v V H h n N: (π(h) λ(h)1) n v = 0}. If V λ 0, then V λ is called a generalized weight space, the elements of V λ are called generalized weight vectors, and λ is a weight. The weight space belonging to a weight λ is {v V H h: π(h)v = λ(h)v}, which is obviously a subspace of V λ. Elements of the weight space are called weight vectors. For X, Y g we define B(X, Y ) := Tr(ad X ad Y ). Then B is a symmetric bilinear form on g which is called the Killing form. It has a couple of properties of which the following is important for us. Proposition 1.4 ([Kna02, Prop 2.17]). B h h is nondegenerate. Therefore to each element α h corresponds a unique H α h such that α(h) = B(H, H α ) for all H h. The Killing form B allows to define a dual bilinear form on h via ϕ, ψ := B(H ϕ, H ψ ) for ϕ, ψ h. A real Lie algebra g 0 is said to be a real form of g if g = g 0 ig 0 as real vector spaces. The next proposition tells us how to construct a real Lie algebra h 0 that is a real form of h such that h 0 = L(h 0, R) (real linear

4 A. POHL functionals) is the real form of h on which all the roots are real valued. The proof uses special properties of the Killing form. Proposition 1.5 ([Kna02, Cor 2.38]). Let W be the R linear span of in h. Then W is a real form of the vector space h, and the restriction of the bilinear form to W W is a positive-definite inner product. Moreover, if h 0 denotes the R linear span of all H α for α, then h 0 is a real form of the vector space h, the members of W are exactly those linear functionals that are real on h 0, and restriction of the operation of those linear functionals from h to h 0 is an R isomorphism of W onto h 0. Note that the previous proposition shows that the Killing form is positive definite on h 0. Three notions are absolutely important in the Theorem of Highest Weight. The first is, of course, highest weight which is defined via a total ordering on h 0. The second is algebraically integral functional, and the third is dominant functional. In fact, the Theorem of Highest Weight states (among more important things) that the highest weight is an algebraically integral dominant weight, and viceversa that every algebraically integral dominant weight occurs as a highest weight. An element λ of h is called algebraically integral if 2 λ, α / α 2 is an integer for each α. Total ordering on h 0 : The aim is to define a subset of h 0 to be the set of positive elements such that the notion of positivity on h 0 meets the following two properties: (P1) For each nonzero ϕ h 0, exactly one of ϕ and ϕ is positive, (P2) the sum of positive elements is positive, and any positive multiple of a positive element is positive. Then we define that ϕ > ψ or ψ < ϕ if ϕ ψ is positive. This way a total ordering of h 0 is established. If α h 0 is a root, it is called simple if α > 0 and if α does not decompose as α = β 1 + β 2 where β 1, β 2 are both positive roots. The set of positive roots is denoted +, and the subset of simple roots is denoted Π. An element λ of h is called dominant if λ, α 0 for all α +. One way to define such a notion of positivity is via a lexicographic ordering. One fixes a finite spanning set H 1,..., H m of h 0 as (real) vector space, e.g. a basis, and endows it with a total ordering, here 1,..., m. Then one defines ϕ h 0 to be positive if there exists a k {1,..., m} such that ϕ(h i ) = 0 for 1 i k 1 and ϕ(h k ) > 0. Proposition 1.8 states that each weight can be regarded as an element of h 0. The largest weight w.r.t. to the introduced ordering of h 0 is called the highest weight. Hence, a priori, the notion of highest weight heavily depends on the ordering.

THEOREM OF THE HIGHEST WEIGHT 5 Proposition 1.6 ([Kna02, Prop 2.49]). If l = dim R h 0, then there are l simple roots α 1,..., α l, and they are linearly independent. If β is a root and is written as β = x 1 α 1 + + x l α l, then all the x j have the same sign (0 is considered to be positive or negative, as needed), and all the x j are integers. The proof that each highest weight is dominant is based on the fact that g is spanned by embedded copies of sl(2, C). The Lie algebra sl(2, C) is generated by the vector space basis ( ) ( ) ( ) 1 0 0 1 0 0 h =, e =, f = 0 1 0 0 1 0 as a Lie subalgebra of the Lie algebra End C (C 2 ). The bracket relations are [h, e] = 2e, [h, f] = 2f, [e, f] = h. Let α be a root and let H α h be the vector defined in Proposition 1.4. Select nonzero elements E α g α and E α g α. Then [Kna02, Cor 2.25] states that the pair {E α, E α } can (and should) be normalized such that B(E α, E α ) = 1. If we set H α := 2 α(h α ) H α, E α := 2 α(h α ) E α, E α := E α, then we have the bracket relations [H α, E α] = 2E α, [H α, E α] = 2E α, [E α, E α] = H α. Hence {H α, E α, E α} spans a Lie aubalgebra sl α of g isomorphic to sl(2, C). The next proposition will guarantee that each highest weight is dominant. Proposition 1.7 ([Kna02, Cor 1.72]). Let π be a complex-linear representation of sl(2, C) on a finite-dimensional complex vector space V. Then π(h) is diagonable, all its eigenvalues are integers, and the multiplicity of an eigenvalue k equals the multiplicity of k. The following proposition is crucial in the proof of the Theorem of the Highest Weight. Proposition 1.8 ([Kna02, Prop 5.4]). Let h 0 be the real form of h. If π is a representation of g on the finite-dimensional complex vector space V, then (a) π(h) acts diagonably on V, so that every generalized weight vector is a weight vector and V is the direct sum of all the weight spaces, (b) every weight is real valued on h 0 and is algebraically integral, (c) roots and weights are related by π(g α )V λ V λ+α and π(h)v λ V λ. The universal enveloping algebra allows to extend a representation ϕ: g End C (V ) uniquely to a homomorphism of algebras U(g) End C (V ). This extension is the right framework for investigations on iterated actions of g on V.

6 A. POHL Proposition 1.9 ([Kna02, Prop 3.3, Remarks 2)]). Let g be a complex Lie algebra. There is a pair (U(g), ι) such that U(g) is an associative complex algebra with unit and ι: g U(g) is a linear mapping satisfying ι[x, Y ] = ι(x)ι(y ) ι(y )ι(x) for all X, Y g such that (U(g), ι) meets the following universal property: Whenever A is an associative complex algebra with unit and π : g A is a linear mapping satisfying π[x, Y ] = π(x)π(y ) π(y )π(x) for all X, Y g then there exists a unique algebra homomorphism π : U(g) A such that π(1) = 1 and the diagram commutes. U(g) ι g π As usual, one proves that U(g) is unique up to isomorphism and that ι is injective, therefore we will identify X and ι(x) for each X g. The algebra U(g) is called the universal enveloping algebra of g. The Poincaré-Birkhoff-Witt Theorem gives a vector space basis for U(g). Proposition 1.10 (Poincaré-Birkhoff-Witt, [Kna02, Thm 3.8]). Let {X i } i A be a basis of g, and suppose a total ordering has been imposed on the index set A. Then the set of all monomials π X j 1 i 1 X jn i n, where i 1 < < i n and all j k 0, is a basis of U(g). The following version of Schur s Lemma is needed in the proof that the correspondence that is stated in the Theorem of the Highest Weight is injective. Proposition 1.11 (Schur s Lemma, [Kna02, Prop 5.1]). Suppose ϕ and ϕ are irreducible representations of a finite-dimensional Lie algebra g on finite-dimensional vector spaces V and V, resp. If L: V V is a linear map such that ϕ (X)L = Lϕ(X) for all X g, then L is bijective or L = 0. A 2. The Theorem of the Highest Weight Let g be a complex semisimple Lie algebra, let h be a Cartan subalgebra, let = (g, h) be the set of roots, and let W ( ) be the Weyl group. Let B be the Killing form of g and let h 0 be the real form of h on which all the roots are real valued. Introduce an ordering in h 0. Let Π denote the resulting set of simple roots, and + that one of positive roots.

THEOREM OF THE HIGHEST WEIGHT 7 Theorem 2.1 (Theorem of the Highest Weight (algebraic), [Kna02, Thm 5.5]). The map ĝ fin {dominant algebraically integral linear functionals on h} [π] highest weight of π is a bijective correspondence. The highest weight λ of π λ has the additional properties: (a) λ depends only on the simple system Π and not on the ordering used to define Π, (b) the weight space V λ for λ is one-dimensional, (c) each root vector E α for arbitrary α + annihilates the members of V λ, and the members of V λ are the only vectors with this property, (d) every weight of π is of the form λ l i=1 n iα i with the integers n i 0 and the α i Π. References [Kna02] Anthony W. Knapp, Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002.