Generic and Mod p Kazhdan-Lusztig Theory for GL₂

2020/07/02 by Pépin, Cédric, Schmidt, Tobias
#11S37 #14M15 #19E20 #20C08 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2007.01364

Abstract

Let F be a non-archimedean local field with residue field \mathbbF_q and let G = GL_2/F. Let q be an indeterminate and let H⁽¹⁾(q) be the generic pro-p Iwahori-Hecke algebra of the group G(F). Let V_\widehatG be the Vinberg monoid of the dual group \widehatG. We establish a generic version for H⁽¹⁾(q) of the Kazhdan-Lusztig-Ginzburg spherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid V_\widehatG and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization q = q ∈ ℂ. At q = q = 0 ∈\mathbbF_q, the spherical map provides a dual parametrization of all the irreducible H⁽¹⁾_\mathbbF_q(0)-modules.

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