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Journal of Lie Theory 25 (2015), No. 2, 377--393
Copyright Heldermann Verlag 2015

First extension groups of Verma modules and R-polynomials

Noriyuki Abe
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba / Meguro-ku, Tokyo 153--8914, Japan

We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of R-polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gabber and Joseph did not state it.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far the dimensions of extension groups from the coefficients of R-polynomials are.

Keywords: Verma module, Extension groups.

MSC: 17B10, 17B55

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