On Tangent Spaces to Schubert Varieties

We prove the results on the tangent spaces to Schubert varieties announced in [V. Lakshmibai, Math. Res. Lett.2 (1995), 473–477] for G classical. We give two descriptions of the tangent space to a Schubert variety at id. The first description is in terms of the root system, and the second one is in terms of multiplicities of certain weights in the fundamental representations of G.