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CATEGORIES:Algebra seminar
SUMMARY:Periodic automorphisms of simple Lie algebras - Pa
ul Levy\, University of Lancaster
DTSTART:20100415T150000Z
DTEND:20100415T160000Z
UID:TALK2642AT
URL:/talk/index/2642
DESCRIPTION:Let &theta\; be an automorphism of order m of the
complex simple algebraic group _G_\, and let _g_ b
e the Lie algebra of G. Then there is a direct sum
decomposition _g_ = _g_(0)&plus\;_g_(1)&plus\;ยทยทยท
&plus\; _g_(_m_-1)\, where g(_j_) is the e^2&pi\;_
ij_/_m_^-eigenspace for the action of the differen
tial _d_&theta\; on _g_. In fact\, this is a _*Z*/
m*Z*_-grading: [_g_(_i_)\,_g_(_j_)]< _g_(_i+j_) (w
here _i_ and _j_ should be considered as integers
modulo _m_). Let _G_(0) be the connected component
of the fixed point subgroup for the action of &th
eta\; on _G_\; then _G_(0) is reductive\, Lie(_G_(
0)) = _g_(0) and _G_(0) stabilizes each of the sub
spaces _g_(_i_).\n\nThe first example to consider
is the case _m_=2\, that is\, where &theta\; is an
involution. In this case _G_(0) is commonly denot
ed _K_\, _g_(0) = _k_ and _g_(1) = _p_. Then it is
well known that the action of _K_ on _p_ shares m
any invariant-theoretic properties with the adjoin
t representation: closed orbits are orbits of semi
simple elements\, the invariants are polynomial\,
and so on. It is rather less well-known that (due
to the seminal work of Vinberg) most of these prop
erties also hold for the action of _G_(0) on _g_(1
) for arbitrary _m_. In this talk I will give an o
verview of Vinberg's results\, explain how they ca
n be extended to positive characteristic and discu
ss a long-standing conjecture of Popov concerning
the existence of an analogue of Kostant's slice to
the regular orbits in the adjoint representation.
LOCATION:Watson Building\, Lecture Room A
CONTACT:David Craven
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