Twisted cells for real reductive lie groups

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Title Twisted cells for real reductive lie groups
Publication Type dissertation
School or College College of Science
Department Mathematics
Author McAfee, Sean
Date 2019
Description Given a real reductive linear Lie group G, consider the nite set D of Langlands parameters corresponding to (in nitesimal equivalence classes of) irreducible admissible representations of G having in nitesimal character . We may construct a free Z[q; q􀀀1]-moduleMwith two bases ft g; fc g (respectively the \standard" and \irreducible" basis of M), each indexed by elements 2 D, and with change of basis given by the so-called Kazhdan-Lustig-Vogan polynomials. This module M admits an action of the Hecke algebra H(W), where W is the complex Weyl group of G. The action of H(W) on the irreducible basis of M allows us to de ne a partial ordering on D; the equivalence classes C of D induced by this partial ordering are called cells of D. Now, let  be an involutive automorphism of G which preserves a xed Borel subgroup B  G. The action of  on G induces an action of  on both the set D and the complex Weyl group W, and we let D and W be the respective sets of xed points under this action. As above, we may construct a free Z[q; q􀀀1]-module M with two bases ft g; fc g (the \standard" and \irreducible" bases of M), each indexed by elements 2 D. The module M admits an action of the unequal parameter Hecke algebra H(W), and in the same fashion as above, the action of H(W) on the irreducible basis of M induces a partition of the set D into equivalence classes C called twisted cells. In this dissertation, we investigate the relationship between twisted and untwisted cells. In particular, we conjecture that each twisted cell C of D, when viewed as a subset of D, is completely contained in some untwisted cell C of D. We give partial results to support this conjecture in the case of an arbitrary real reductive linear Lie group G, and describe a method to prove the conjecture in the case of G = U(n; n).
Type Text
Publisher University of Utah
Dissertation Name Doctor of Philosophy
Language eng
Rights Management (c) Sean McAfee
Format application/pdf
Format Medium applcation/pdf
ARK ark:/87278/s690851v
Setname ir_etd
ID 1713436
Reference URL https://collections.lib.utah.edu/ark:/87278/s690851v
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