A topological approach to Springer's representations

✍ Scribed by David Kazhdan; George Lusztig

• Publisher: Elsevier Science
• Year: 1980
• Tongue: English
• Weight: 355 KB
• Volume: 38
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(80)90005-5

✦ Synopsis

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 Wrepresentation in the top homology of .53'U so that all irreducible representations 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 The variety Z has been studied by Steinberg [6] and Cross (unpublished Durham thesis).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 conjectured 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#,,.