The q-Weyl dimension formula

1 The Weyl dimension formula and its q-analogue This material is taken from notes for a joint paper with Zajj Daugherty and Rahbar Virk. Everything here is " well known ". Proposition 1.1. Let g be a finite dimensional complex semisimple Lie algebra etc.etc. ...???? Let ν be a dominant integral weight so that the irreducible module L(ν) of highest weight ν is finite dimensional. Letˆκ 1 = ν, ν + 2ρ. (a) (id ⊗ tr L(ν))((1 2 ˆ κ 1 + γ) 0) = dim(L(ν)) = ev 0 (s ν) = α∈R + ν + ρ, α ρ, α , (b) (id ⊗ tr L(ν))((1 2 ˆ κ 1 + γ) 1) = 1 2 ˆ κ 1 dim(L(ν)). (c) (id ⊗ qtr L(ν))((R 21 R) 0) = dim q (L(ν)) = ev 2ρ (s ν) = α∈R + [ν + ρ, α] [ρ, α] , (d) [TW, Lemma 3.5.1] (id ⊗ qtr L(ν))((R 21 R) 1) acts on L(µ) by ev 2(µ+ρ) (s ν)id L(µ). Proof. (a) and (c) Let M be a U-module. On M ⊗ L(ν), (id ⊗ qtr L(ν))((R 21 R) 0) = (id ⊗ qtr L(ν))(id M ⊗L(ν)) = id M dim q (L(ν)), giving the first identity in (c). By (??) and the second identity in (??), v = e −hρ u, and dim q (L(ν))id M = (id ⊗ qtr L(ν))(id M ⊗L(ν)) = (id ⊗ tr L(ν))(1 ⊗ uv −1) = (id ⊗ tr L(ν))(1 ⊗ e hρ) = ev 2ρ (s ν)id M ,