lecture notes9

Corollary 844 if and is not then the root string

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ght zero part of W is W0 = F hα . So, W contains only one irreducible V (even). Since we know that W contains Sα ∼ V (2), there cannot be any = other V (even). So, L2α = 0. In other words, twice a root cannot be a root. But then 1 α 2 also cannot be a root since twice of it is a root. And this implies that W1 = Lα/2 = 0. So, W = V (2) is irreducible. This also implies that Lα is one dimensional. ￿ Corollary 8.4.4. If α, β ∈ Φ and β is not ±α then the α-root string through β is irreducible and has the form: V (m) ∼ Lβ +qα ⊕ Lβ +(q−1)α ⊕ · · · ⊕ Lβ −rα = and m = q + r. And β (hα ) = q − r is an integer. (β (hα ) are the Cartan integers.) Proof. Let M be the α-root string through β . Then M0 = 0 since the root string does not go through 0. (The proof of the last proposition showed that Sα = Lα ⊕ F hα ⊕ L−α is the only α-root string through 0.) Therefore, M is a direct sum of V (odd)s. So, the α-root string has only Modd but a difference of 2 in hα -weights corresponds to a difference of α in roots since α(hα ) = 2. So, the root string contains only Lβ +kα for integer k . One of these is M1 and thus is 1-dimensional. So, M is irreducible. If M ∼ V (m) then Mm = Lβ +qα and M−m = Lβ −rα . The dimension of M is q + r + 1 = = m + 1. This implies that q + r = m = (β + q α)(hα ) = β (hα ) + 2q ￿ So, β (hα ) = r − q ∈ Z. 8.5. Inner product on H ∗ . We just proved that β (hα ) ∈ Z. We will rephrase this in terms of the inner product on H ∗ . t Recall that hα = κ(t2αα α ) . And,...
View Full Document

This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

Ask a homework question - tutors are online