Three Pictures of Lusztig’s Asymptotic Hecke Algebra

Let $\Waff$ be an extended affine Weyl group, $\HH$ be its Hecke algebra over the ring $\Z[\bq, \bq^{-1}]$ with standard basis $\{T_w\}_{w\in\Waff}$, and $J$ be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}_{w\in\Waff}$. This thesis studies the algebra $J$ from several perspectives, proves theorems about various incarnations of $J$, and provides tools to be applied for future work. We prove three types of results. In the second and third chapters, we investigate $J$ as a subalgebra of the $(\bq^{-1})$-adic completion of $\HH$ via Lusztig's map $\phi$. In the second chapter, we use Harish-Chandra's Plancherel formula for $p$-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\bq$, depending only on the two-sided cell containing $w$, with no poles outside of a finite set of roots of unity that depends only on $\Waff$. In type $\tilde{A}_n$ and type $\tilde{C}_2$, we show that the denominators all divide a power of the Poincar\'{e} polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification of $\HH$-modules at roots of the Poincar\'{e} polynomial than is currently known. In the third chapter, we reprove the results of the second chapter without using any tools from harmonic analysis in the special case $\G=\SL_2$. In this case we also prove a positivity property for the coefficients of $T_x$ in $t_w$, that we conjecture holds in general. We also produce explicit formulas for the action of $J$ on the Iwahori invariants $\mathcal{S}^I$ of the Schwartz space of the basic affine space. In the fourth chapter, we give a triangulated monoidal category of coherent sheaves whose Grothendieck group surjects onto $J_0\subset J$, the based ring of the lowest two sided cell of $\Waff$, equipped with a monoidal functor from the category of coherent sheaves on the derived Steinberg variety. We show that this partial categorification acts on natural coherent categorifications of $\mathcal{S}^I$. In low rank cases, we construct complexes lifting the basis elements $t_w$ of $J_0$ and their structure constants.

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