Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022

• Self-dual cuspidal representations

  Speaker: Manish Mishra (IISER Pune) - https://sites.google.com/site/manishmishra/
  

  When: Wed, September 8, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

  View Abstract

  View Abstract

  Abstract: Let G be a connected reductive group defined over a finite or a non-archimedean local field F. We show that G(F) admits cuspidal representations when F is finite and supercuspidal representations when F is non-archimedean local. We also determine precisely when G(F) admits self-dual representations. For the results on self-duality, we assume some hypothesis on G. These hypotheses disallow G to have certain small rank factors when the field (in case F is finite) or the residue field (in case F is non-archimedean local) is of cardinality ≤ 5. When F is non-archimedean local and G is ramified, these hypotheses impose some additional restrictions on G. This is a joint work with Jeff Adler.
  
• Arthur and ABV packets

  Speaker: Jeffrey Adams (University of Maryland) -
  

  When: Wed, September 22, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and https://umd.zoom.us/j/96890967721
  

  View Abstract

  View Abstract

  Abstract: In 1983 Jim Arthur conjectured the existence of certain sets of representations
  of a reductive group G over a local field F, associated to homomorphisms from the Weil-Deligne group x SL(2) into the L-group of G. In the case of split classical groups he defined these "Arthur" packets in his 2010 book. For F=R Adams, Barbasch and Vogan gave a definition in their 1992 book, commonly referred to as ABV. Nicolas Arancibia and Paul Mezo, with some help from the speaker, have proved these two definitions agree when they are both defined.
  
  The talk will be available online at https://umd.zoom.us/j/96890967721
  
• Arthur and ABV packets II

  Speaker: Jeffrey Adams (University of Maryland) -
  

  When: Wed, September 29, 2021 - 2:00pm
  Where: Math 3206 and on zoom: https://umd.zoom.us/j/96890967721 (see seminar page for the password)
  
• The smooth locus of twisted affine Schubert varieties

  Speaker: Jiuzu Hong (UNC) - https://hong.web.unc.edu/
  

  When: Fri, October 22, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

  View Abstract

  View Abstract

  Abstract: Let G be a special parahoric group scheme of twisted type, excluding the absolutely special case for ramified odd unitary group. Using the methods and results of Zhu, we prove a duality theorem for G: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in the affine Schubert varieties of G. Along the way, we also establish the duality theorem for untwisted E_6. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of G. In particular, this confirms a conjecture of Haines and Richarz. This talk is based on the joint work with Marc Besson.
  
• A geometric approach to unipotent representations

  Speaker: Dmytro Matvieievskyi (Yale University) -
  

  When: Mon, November 22, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

  View Abstract

  View Abstract

  Abstract: For a real reductive group G, the set of unipotent representations is a finite set of irreducible representations that are supposed to be “building blocks” for all unitary representations of G. Moreover, unipotent representations are expected to be indexed by nilpotent G-orbits, possibly with some additional data. There were several attempts to give a definition of a unipotent representation, most notably the set of special unipotent representations, due to Barbasch-Vogan and Arthur. However, this notion does not include some interesting unitary representations, such as a metaplectic representation. In joint work with Ivan Losev and Lucas Mason-Brown, we propose a new definition of unipotent representations of a complex semisimple Lie group that extends the set of special unipotent ones. Using these definitions we are able to parameterize the unipotent representations and prove their key properties, including some that were not known for special unipotent representations before. In this talk, I will explain the definition and the parametrization of unipotent representations. Time permitting I will sketch what we expect to be a definition of a unipotent representation of a real semisimple Lie group, and our approach to studying them. The talk is based on arXiv:2108.03453.
  
• Derived Satake equivalence for Godement-Jacquet monoids, I

  Speaker: Jonathan Wang (Perimeter Institute) -
  

  When: Mon, December 13, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

  View Abstract

  View Abstract

  Abstract: Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. I will explain how the local unramified part of this theory can be geometrized to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. This confirms a particular case of a general conjecture of Ben-Zvi-Sakellaridis-Venkatesh. I will discuss how this equivalence fits into the general framework of (relative) Langlands duality and to conjectures of Braverman-Kazhdan and Ngo on constructions of general automorphic L-functions. In the second talk I will give an overview of the proof of the equivalence and some of its finer properties, which have connections to invariant theory on a "dual" variety related to the standard L-function. This is joint work with Tsao-Hsien Chen (in preparation).
  
• Derived Satake equivalence for Godement-Jacquet monoids, II

  Speaker: Jonathan Wang (Perimeter Institute) -
  

  When: Tue, December 14, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

  View Abstract

  View Abstract

  Abstract: Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. I will explain how the local unramified part of this theory can be geometrized to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. This confirms a particular case of a general conjecture of Ben-Zvi-Sakellaridis-Venkatesh. I will discuss how this equivalence fits into the general framework of (relative) Langlands duality and to conjectures of Braverman-Kazhdan and Ngo on constructions of general automorphic L-functions. In the second talk I will give an overview of the proof of the equivalence and some of its finer properties, which have connections to invariant theory on a "dual" variety related to the standard L-function. This is joint work with Tsao-Hsien Chen (in preparation).