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CATEGORIES:Conferences/Seminars/Lectures
DESCRIPTION:Speaker: David Vogan (MIT)\n\nTitle: Restricting real group rep
 resentations to K\n\nAbstract: It's an old idea of Harish-Chandra that repr
 esentations of a real\nreductive G can be studied by understanding their re
 strictions to a\nmaximal compact K.  Traditionally this was done using the 
 Cartan-Weyl\nhighest weight theory for K representations.\n\nI will recall 
 a (closely related but very different) description of K\nrepresentations\, 
 in which the highest weight is replaced by the\nHarish-Chandra parameter of
  a discrete series on a Levi subgroup of G.\nUsing this new parameter\, one
  can define a very natural "height" of a\nrepresentation of K\, taking non-
 negative integer values. As evidence\nthat height is natural\, I offer\n\nC
 onjecture. Suppose G is split\, and Π1 and Π2 are two (minimal)\nprincipal 
 series representations of G. If Π1 and Π2 have the same\nrestriction to the
  center of G\, then the sets of heights of K-types of\nΠ1 and of Π2 are the
  same.\n\nFor example\, split G2 has principal series having two different\
 nrestrictions to K. In each\, the first heights appearing are\n\n0\, 3\, 6\
 , 9\, 10\,12\, 15...\n\nF4 has principal series representations having thre
 e different\nrestrictions to K. But in all three\, the first heights appear
 ing are\n\n0\, 8\, 11\, 15\,16\, 21\, 24\, 26\, 27\, 29\, 30\, 32\, 33\, 36
 ....\n\nThe numbers for split E8 (which also has three types of principal\n
 series) are\n\n0\, 29\, 46\, 57\, 58\, 68\, 75\, 84\, 87\, 91\, 92...\n\nIt
  seems more or less impossible to find a similar statement using\nhighest w
 eights. I will give some evidence for this conjecture.\n\nOf course there s
 hould be an elementary construction of these sequences\nof heights from the
  root system of G and the central character\; I have\nno idea how to make s
 uch a construction.\n\nIf there is time\, I will talk about the possibility
  of finding parallel\nstatements for p-adic groups.
DTEND:20240911T220000Z
DTSTAMP:20260417T121113Z
DTSTART:20240911T200000Z
GEO:42.358262;-71.090045
LOCATION:Building 2\, 2-142
SEQUENCE:0
SUMMARY:MIT Lie Groups Seminar
UID:tag:localist.com\,2008:EventInstance_47411850406196
URL:https://calendar.mit.edu/event/mit-lie-groups-seminar-5876
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