Root System Description of T(w, τ)
In this chapter we recall the construction due to Lakshmibai—MusiliSeshadri (cf. , ) of a special basis for the Weyl module associated to a fundamental weight in the case G is a classical group, and we compute the action of the Chevalley basis elements of g on this basis. Using these computations and a result of Polo (cf. )relating tangent spaces and Demazure modules, we give a description of the tangent space T(w, τ) τ≤w due to Lakshmibai ([98, 99, 100]) in terms of the root system. We also bring out the relationship between T(w, τ) and the multiplicities of certain weights in the fundamental representations of G. We have separated out the discussions of the case τ=id and the case of a general τ, since the results as well as the discussion for τ=id turn out to be more compact. We have also included a result of Polo (cf. ) describing the module structure of the tangent space at the identity. Also included are two criteria for smoothness due to Carrell—Kuttler ()
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