A topological approach to Springer's representations

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Abstract

Let u be a unipotent element in a complex semisimple group G and let 3,, be the variety of Bore1 subgroups of G which contain u. In [4, 51, Springer has defined a representation of W, the Weyl group of G, on the homology of 2:, and showed how to decompose the W-representation in the top homology of .53'U so that all irreducible represen-tations of W are obtained. His mehod used etale cohomology of algebraic varieties in characteristic p. In this paper, we shall give an elementary construction (independent of &ale cohomology) of the Springer representation of W in the top homology of sU. At the same time, we shall construct a representation of W X W in the top homology of the variety Z of all triples (u, B, B'), where u runs through the set of unipotent elements in G and B E .-%'u, B' E AT',, . The variety Z has been studied by Steinberg [6] and Cross (unpublished Durham thesis). In a letter (1977) to one of us, Springer defined a representation of W X W on the homology groups of Z (using the methods of [4]); he conjec-tured that the representation in the top homology of 2 is the two-sided regular representation of W and showed how this could be used to prove the completeness of the set of W-representation in the top homologies of z#,,. 1. In this paper, (except in Section 4) all topological spaces are assumed to be locally compact and Hausdorff. For such a space X, we denote by H,(X)

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Kazhdan, D., & Lusztig, G. (1980). A topological approach to Springer’s representations. Advances in Mathematics, 38(2), 222–228. https://doi.org/10.1016/0001-8708(80)90005-5

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