Unicity of types for supercuspidal representations of p-adic SL2

Peter Latham

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We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of SL2 over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the SL2-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in SL2 is restricted from one in GL2. From this it follows that any archetype must be induced from a Bushnell-Kutzko type. Given a supercuspidal representation π of SL2(F), we give an additional explicit description of the number of archetypes admitted by π in terms of its ramification. We also describe a relationship between archetypes for GL2 and SL2 in terms of L-packets, and deduce an inertial Langlands correspondence for SL2.

Original languageEnglish
Pages (from-to)376-390
Number of pages15
JournalJournal of Number Theory
Publication statusPublished - May 2016


  • Bushnell-Kutzko types
  • Langlands correspondence
  • P-adic groups
  • Special linear group

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