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The Kazhdan-Lusztig basis seems to be quite tricky, is there a reason from representation theory or geometry of Schubert varieties to introduce these basis?

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12  
How could the answer be no?! –  Mariano Suárez-Alvarez May 3 '11 at 12:20
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Well, Kazhdan and Lusztig could have introduced their basis as a practical joke ("Let's see those suckers sweat on this!"). –  Angelo May 3 '11 at 13:59
    
Have you checked Wikipedia? en.wikipedia.org/wiki/Kazhdan-Lusztig_polynomial –  S. Carnahan May 3 '11 at 14:13
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This question seems rather odd. How would one come across these things without seeing them used or motivated? –  Yemon Choi May 3 '11 at 15:00
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The consensus seems to be that there isn't yet a real question being asked. It often helps to mention an explicit source and then formulate a definite question based on your reading of it. –  Jim Humphreys May 3 '11 at 17:32

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