Thursday, June 23, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Stefan Dawydiak

Supervisor: Alexander Braverman

Thesis title: Three pictures of Lusztig’s asymptotic Hecke algebra

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Let W ̃ be an extended affine Weyl group, **H** be the its Hecke algebra over the ring Z[**q**, **q**−1] with standard basis {T_w}w∈W ̃ , and *J* be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis. 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 (**q**−1)-adic completion of **H** via Lusztig’s map φ. 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 q, 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 W ̃. In type A ̃_n and type (C_2 ) ̃, we show that the denominators all divide a power of the Poincaré 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 **H**-modules at roots of the Poincaré 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 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 ⊂ *J* , the based ring of the lowest two sided cell of W ̃, 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 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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A draft of the thesis can be found here: Stefan-Dawydiak-Thesis-v4.2