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UID:19f3ffd75e7ef5b7a8600e39476a24e0
CATEGORIES:Algebra Seminar
CREATED:20241127T172349
SUMMARY:Modular Reduction of Nilpotent Orbits
LOCATION:H705
DESCRIPTION:Modular Reduction of Nilpotent Orbits(Jay Taylor, Dec. 4, 2024)\nThe genera
 l linear group G=GLn(k) over a field k acts on the space gl=gln of (nxn)-ma
 trices by conjugation. The set N(g) of nilpotent matrices is preserved by t
 his action and G acts with finitely many orbits. The Jordan normal form giv
 es a representative of each orbit that is contained in gln(Z). Importantly,
  the structure of the centralizer of this nilpotent matrix is independent o
 f k.\nIt is natural to ask to what extent this statement extends to aconnec
 ted reductive algebraic group G acting on its Lie algebra g or the dual spa
 ce g* via the (co-)adjoint representation. In general, thestructure of cent
 ralisers of nilpotent elements will depend on k butone can hope that the ce
 ntraliser dimension remains the same. In thistalk I will propose two varian
 ts of this idea, generalising theclassical situation above, and report on o
 n-going joint work with AdamThomas (Warwick) to establish the existence of 
 elements satisfying theproposed properties.
X-ALT-DESC;FMTTYPE=text/html:<pre><span style="font-size: large;" size="4"><strong>Modular Reduction of 
 Nilpotent Orbits(Jay Taylor, Dec. 4, 2024)</strong><br>The general linear g
 roup G=GL<sub>n</sub>(k) over a field k acts on the space <frak>gl=<frak>gl
 <sub>n</sub> of (nxn)-matrices by conjugation. The set N(g) of nilpotent ma
 trices is preserved by this action and G acts with finitely many orbits. Th
 e Jordan normal form gives a representative of each orbit that is contained
  in <frak>gl<sub>n</sub>(Z). Importantly, the structure of the centralizer 
 of this nilpotent matrix is independent of k.<br>It is natural to ask to wh
 at extent this statement extends to aconnected reductive algebraic group G 
 acting on its Lie algebra <frak>g or the dual space <frak>g* via the (co-)a
 djoint representation. In general, thestructure of centralisers of nilpoten
 t elements will depend on k butone can hope that the centraliser dimension 
 remains the same. In thistalk I will propose two variants of this idea, gen
 eralising theclassical situation above, and report on on-going joint work w
 ith AdamThomas (Warwick) to establish the existence of elements satisfying 
 theproposed properties.</frak></frak></frak></frak></frak></span></pre>
CONTACT:Jay Taylor
DTSTAMP:20260827T115847
DTSTART;TZID=America/New_York:20241127T080000
DTEND;TZID=America/New_York:20241127T170000
SEQUENCE:0
TRANSP:OPAQUE
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