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Anthony Joseph
Weizmann Institute
The injectivity theorem and KPRV determinants
Friday 5th December, 12:05-12:55pm,
Carslaw 173.
The fundamental Kostant separation theorem for the enveloping
algebra of a complex semisimple Lie algebra led Parthasarathy Rango-Rao
and Varadarajan to define and compute a family of determinants indexed
by the dominant weights. In joint work with G. Letzter and D. Todoric
we generalized these determinants to the parabolic setting and computed
them. This was a significantly more difficult problem and required
first setting up a framework in which these determinants were even
defined. To do this one must show that certain filtered over-algebras
of the corresponding quotients of the enveloping algebra are graded
injective as modules in the appropriate category. This in itself is
quite deep and a refinement of a result in algebraic geometry due to B.
Broer. Finally even when this is done the compution of the determinants
is much more difficult than for a Borel case owing to the existence of
multiple zeros. Here the Jantzen filtration technique is used as well
as the Bernstein-Gelfand equivalence of categories. These determinants
should give information on the open problem of determining the
Jordan-Holder series of primitive quotients.
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