Graduate Blog

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