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Title:
Tilting modules and the p-canonical basis
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Abstract:
I will discuss a the rational representations of (connected) reductive algebraic groups in characteristic p. It is a classical observation that the action of wall crossing functors on the Grothendieck group of the principal block gives an action of the Weyl group. (This is analogous to the fact that wall-crossing functors on category O give a categorification of the regular representation.) This leads to a realization of the anti-spherical module for the affine Weyl group. We conjecture that this action may be lifted to an action of the Hecke category. This conjecture has remarkable consequences. For example, one obtains tilting character formulas in terms of the p-canonical basis and the Lusztig conjecture for large p essentially “for free”. To a certain extent it also tells us where to look to work out what happens for small p. (This is joint work with Simon Riche.)
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