By James E. Humphreys
Finite teams of Lie sort surround many of the finite easy teams. Their representations and characters were studied intensively for part a century, notwithstanding a few key difficulties stay unsolved. this is often the 1st finished remedy of the illustration concept of finite teams of Lie kind over a box of the defining top attribute. As a subtheme, the connection among traditional and modular representations is explored, within the context of Deligne-Lusztig characters. One objective has been to make the topic extra obtainable to these operating in neighboring elements of crew concept, quantity concept, and topology.
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Extra info for Modular representations of finite groups of Lie type
20]. Use of the Sum Formula rapidly gets more complicated, as the recursion begins to involve numerous cancellations; eventually multiple composition factors create too much ambiguity to go further. 10. 10. Generic Behavior of Weyl Modules We have outlined briefly the main tools developed by 1980 for the study of composition factor multiplicities in Weyl modules, in tandem with at least some of the submodule structure. At this point the overall picture may still strike the reader as hopelessly complicated.
Thanks to the Bruhat decomposition, it is enough to show that all w · v + lie in KV · v + , where w runs over representatives of the Weyl group in N . From the rank 1 case we see immediately when s = sα is a simple reflection that s · v + ∈ KVα · v + , using the assumption that λ ∈ Xr . For an arbitrary w ∈ W , write w = s1 · · · st in reduced form, and write Ui and Vi respectively for the root groups Uαi and Vαi . sk−1 Vk ⊆ V for each k ≤ t. This permits an easy inductive argument, based on the previous paragraph.
It can be normalized so that (v + , v + ) = 1 for a chosen maximal vector v + . After reduction modulo p the form becomes contravariant in the earlier sense relative to the action of G: (g · v, v ) = (v, τ (g) · v ) for all g ∈ G and v, v ∈ V (λ). As before, distinct weight spaces are orthogonal. But the form on V (λ) is typically degenerate. More precisely: Proposition. The contravariant form on a Weyl module V (λ) with λ ∈ X + has as radical the unique maximal submodule, thus inducing a nondegenerate contravariant form on the simple quotient L(λ).