Lecture 16 - Weyl s Character Formula I: The Weyl Function and the Kostant Partition Function

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1 Lecture 16 - Weyl s Character Formula I: The Weyl Function and the Kostant Partition Function March 22, 2013 References: A. Knapp, Lie Groups Beyond an Introduction. Ch V Fulton-Harris, Representation Theory. Ch 24, 25 R. Cahn, Semi-Simple Lie Algebras and Their Representations. Ch XIII 1 Formal Characters and Characters We defined a formal character to be any element in the group algebra over the weight lattice. We express such elements as formal linear combinations over e(λ), where the λ lie in the weight lattice in g. Given an irreducible representation V Λ (with, of course, Λ dominant integral), we define the representation s formal character to be χ Λ = λ N λ e(λ) (1) where λ ranges over the weights in the weight scheme of V Λ. Humphrey s handles formal characters in an awkward way, which we will not adopt. But to explain, for Humphreys the formal symbols e(λ) is interpreted to be a function on g, namely the characteristic function of λ. That is, e(λ)(x) = 0 if x λ, and e(λ)(λ) = 1. However this does not commute with the usual multiplication of formal characters (namely algebra multiplication in the group algebra), so Humphreys switches to a convolution-type multiplication. We will not do this. Instead, we treat e(λ) as the formal symbol e 2πiλ or Exp(2πiλ), and interpret it as an operator (nonlinear) on g as follows e 2πiλ (x) = e 2πi(λ, x) (2) 1

2 where the inner product is the Killing inner product. Formal characters are then functions on g, given by ch Λ (x) = λ N λ e 2πi(λ, x) (3) This interpretation is natural in the following sense. Given a (real) Lie group G with maximal torus H, we saw that H preserves the weight-space decomposition. Given a weight λ, we have a map, χ λ : H S 1 C, given (implicitly) by X.v λ = ξ λ (X) v λ. (4) for X H. This is called the multiplicative character of the representation. Now we have already see that H = Exp(2πi h), so if X = Exp(2πih) for some h h, then Then the formal character is ξ λ (h) = e 2πiλ(h) = e 2πi(λ, h ). (5) ( ch Λ (h ) = T r V Λ e 2πih ). (6) 2 Weyl s Function and Alternating Functions Given an algebra g with base and positive roots Φ +, consider the Weyl function Q Q = (e(α/2) e( α/2)) = e(δ) (1 e( α)) This is an alternating function, in the sense that if σ W, then σ Q Q σ = (det σ) Q. (8) To see this, note that if σ = σ β, β is a basic reflection, then σ β sends β to β and permutes the remaining roots. Then Q(σ β x) = (e ) πi(α, σβx) e πi(α, σ βx) = = (e πi(σ βα, x) e πi(σ βα, x) ) (e πi(σ ββ, x) e πi(σ ββ, x) ) = (e πi(β, x) e πi(β, x)) = Q(x) \{β} \{β} (e πi(σ βα, x) e πi(σ βα, x) ) (e πi(α, x) πi(α, e x)) (7) (9) 2

3 Now given any function T on g we can alternate it as follows. Define an operator on functions A = σ W (det σ) σ (10) (where the factor acts on functions via external multiplication) and compose with T to obtain We check T (σ α x) = σ W(det σ)t (σσ α x) T = AT. (11) = σ W(det σ α σ)t (σx) (12) = (det σ α ) σ W(det σ)t (σx) = (det σ α ) T (x). We compute a second useful expressions for Q. Proposition 2.1 (Weyl Denominator Formula) We have that Q is the alternation of e(δ). Specifically Q = A e(δ) Q(x) = = σ W(det σ)e(σδ) (det σ) e 2πi(σδ, x) = σ W σ W (det σ) e 2πi(δ, σx) (13) Pf. The highest weight exponential in Q is clearly e 1 2 α>0 α = e δ, while the lowest weight exponential is e( δ). Thus Q is the alternation of a sum of exponentials e(ρ) that can be expressed as follows: Q = A β e(δ β) (14) where the sum ranges over those β that are sums of positive roots. Since every root is W- conjugate to a dominant root, we only need consider those β so that δ β has non-negaitve Dynkin coefficients. However, if ρ has any Dynkin coefficient zero, then Ae(ρ) = 0. To see this, note that if (ρ, α i ) = 0, then σ i ρ = ρ where σ i is the reflection in α i. Then on the one hand σ i Ae(ρ) = Ae(ρ), and on the other σ i Ae(ρ) = Ae(σ i ρ) = Ae(ρ). Thus Ae(ρ) = 0. Therefore Q = Ae(δ) (15) 3

4 3 The Kostant Partition Formula If λ is any weight in the root lattice, meaning λ = α<0 n αα where n α Z, define P (λ) (16) to be the number of ways of expressing λ as a sum of positive roots (not counting multiplicities or order, so for instance 2α + β, β + α + α, and α + β + α count as just one way). Obviously if γ is a positive sum of negative roots, then P (γ) = 0. One use of the partition function is in expressing the character of a Verma module. We have that U(n ) is (vector-space) isomorphic to the polynomial generated by the y α for α Φ +, and that V Λ is a vector space over U(n ).v + where v + is any highest weight vector. Thus χ V Λ = µ P (Λ µ)e(µ) = e(λ) ν P (ν) e( ν) (17) where µ ranges over all points in the weight lattice and ν ranges over all points that are finite sums of roots with non-negative integral coefficients. Define K, which we will call the Kostant character, to be K = ν P (ν)e( ν) (18) so that χ V Λ = e(λ) K. Lemma 3.1 We have K = α>0 (1 + e( α) + e( 2α) +... ) (19) Pf. If ν is a positive sum of weights, then e( ν) is in the product on the right. Further, its coefficient is precisely the number of ways that e( ν) can be expressed as a product e( α)... e( ζ) for α,..., ζ Φ +. The following lemma gives the relation between the Kostant partition function and Q. Namely, K is just Q with a phase shift. Lemma 3.2 We have Q K = e(δ) (20) or Q(x) K(x) = e 2πi(δ, x) (21) 4

5 Pf. Since Q = (e(α/2) e( α/2)) = e δ e( α)) (22) α>0 α>0(1 we have e δ Q K = α>0(1 e( α)) (1 + e( α)... ) = 1. (23) Therefore Q ch V Λ = e(λ + δ). 5