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Algebra Seminar
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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.