
Anthony Joseph
Weizmann Institute
The injectivity theorem and KPRV determinants
Friday 5th December, 12:0512:55pm,
Carslaw 173.
The fundamental Kostant separation theorem for the enveloping
algebra of a complex semisimple Lie algebra led Parthasarathy RangoRao
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 overalgebras
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 BernsteinGelfand equivalence of categories. These determinants
should give information on the open problem of determining the
JordanHolder series of primitive quotients.







