Group and Lie Theory Seminar

Michigan Group and Lie Theory Seminar

Last changed: Nov 27, 1995

Fall 1995 schedule at a glance.

  • Sep 25 Robert Griess UofM
  • Oct 02 Guillaume Sanje-Mpacko UofM
  • Oct 09 No seminar
  • Oct 16 George Lusztig, MIT
  • Oct 23 Ian MacDonald
  • Oct 30 Jeffrey Adams, University of Maryland
  • Nov 06 Stephen Debacker UofChicago (graduate student)
  • Nov 13 Alex Ryba Marquette U
  • Nov 20 Timothy Hsu UofM
  • Nov 27 No seminar
  • Dec 04 Robert Griess UofM 
     Titles and Abstracts 
    


    
 

  •  Monday September 25
    • Speaker : Robert Griess UofM Title : A Survey of Infinite Dimensional Groups Abstract : We survey the basic results on Kac Moody groups and ones like them in infinite dimensional situations. Presentations. BN-like properties. Correspondence between subgroups and sublattices. Recognition theorems.
    

    
     

  •  Monday October 2
    • Speaker : Guillaume Sanje-Mpacko UofM Title : Hecke algebras and types for SL_{N}(F) Abstract : We will discuss a method for classifying smooth representations of connected reductive p-adic groups by restriction to compact open subgroups. As an application we give a classification of elliptic representations of SL_{N}(F) in case N is prime. We then show how these methods could be extended to yield a construction of types for unitary principal series.
    

    
     

  •  Monday October 9  No Seminar  


    
    

  •  Monday October 16
    • Speaker : George Lusztig, MIT Title : Total positivity in reductive groups Abstract : The theory of totally positive elements in GL_n(R) is classical; it is due to Schoenberg, Gantmacher and Krein. In this talk we will extend this theory to the case where GL_n(R) is replaced by any split reductive group over the reals. In particular we will define an interesting open semigroup inside such a group. 
    

    
    

  •  Monday October 23
    • Speaker : Ian MacDonald Title : Affine Hecke algebras and orthogonal polynomials 
    

    
    

  •  Monday October 30
    • Speaker : Jeffrey Adams, University of Maryland Title : Representations of Non-Linear Groups Abstract : The local Langlands conjecture describes the representations of an algebraic (linear) group in terms of geometry of the dual group. Non-linear groups play an important role in number theory, starting with the classical Shimura-Shintani theory of forms of half-integral weight. Small representations of groups often only occur on non-linear covering groups. The extension of local Langlands to non-linear groups is not well understood. I'll discuss some progress in this direction, including duality between orthogonal and metaplectic groups, and the theory of metaplectic forms of Kazhdan/Patterson/ Flicker.
    

    
    

  •  Monday November 6
    • Speaker : Stephen Debacker, UofChicago (graduate student) Title : Supercuspidal characters of $GL_{\ell}$ Abstract : Let $\ell$ be a prime. The only $p$-adic groups for which the characters of supercuspidal representations have been completely determined are $PGL_2$, $SL_2$, and $GL_2$ (Sally-Shalika, Shimizu, Silberger). The recent work of Corwin, Moy, and Sally provides us with explicit values for the supercuspidal characters of $GL_{\ell}$ on the elliptic set. There is much evidence to suggest that, off the elliptic set, supercuspidal characters of $GL_{\ell}$ have extremely simple behavior: the character is either the local character expansion or zero. This talk will first discuss the situation for unramified supercuspidal characters of $GL_{\ell}$ (completely worked out by Murnaghan) and then turn to the presenters own work on the ramified supercuspidal representations of $GL_{\ell}$.
    

    
    

  •  Monday November 13
    • Speaker : Alex Ryba, Marquette U Title : Modular Moonshine Abstract : The theme of modular moonshine is that properties of the Monster are reflected in the $p$-modular behaviour of its $pA-$centralizer. In particular, there are $p-$modular analogues of the Griess algebra, the Thompson series and the Monster vertex algebra. In this talk, I'll construct these $p-$modular structures.
    

    
    

  •  Monday November 20
    • Speaker : Timothy Hsu UofM Title : Quilts, the 3-string braid group, and some presentations of finite groups Abstract : This talk covers material in the speaker's thesis and related work. Among the topics discussed are finite simple groups, Moonshine, PSL_2(Z), and B_3 (the 3-string braid group) and its monodromy representations. There are two essentially independent parts: 
           I. "Quilts, PSL_2(Z), and B_3."  Quilts (developed by
              Norton, Conway, the speaker, and others) are diagrams
              which represent subgroups of either PSL_2(Z) or B_3.
              Quilts are defined, and their geometry is used to
              classify the way in which subgroups of PSL_2(Z) lift
              to subgroups of B_3.

    
          II. "T-systems and presentations of finite groups."
              Motivated by Moonshine's genus zero conjectures,
                    Norton proposed the study of a certain monodromy
              action of B_3 on pairs of generators of a (finite)
              group G.  An orbit of this action is called a T-system
              for G.  Having a T-system with a given shape turns
              out to be equivalent to satisfying a certain (finite)
              presentation, so given an interesting (i.e., simple)
              finite group G, one might hope to obtain a presentation
              of G by looking at its T-systems.  The results of I.
              are used to simplify the problem dramatically, and
              some examples (e.g., concise presentations for some
              Mathieu groups) and counterexamples are obtained as
              a result.

    


    
    

  •  Monday November 27  No Seminar  


    
    

  •  Monday December 4 

    •  Speaker  : Robert Griess  UofM 
 Title    : Conjugacy of embeddings in algebraic groups.



    
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