2007 Fiscal Year Final Research Report Summary
The study of the representation theoretical aspect of generalized flag varieties
| Project/Area Number |
18540162
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATUMOTO Hisayosi The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (50272597)
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| Project Period (FY) |
2006 – 2007
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| Keywords | unitay representations / semisimple Lie groups / generalized Verma modules |
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| Research Abstract |
1 Homomorphisms between scalar generalized Verma modules
We had classified the homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras and explained how to use the operators constructed in the maximal case to get some operators in general. We conjectures that all the homomorphisms arise in this way.
We call a parabolic subalgebra normal, if each parabolic subalgebra which has a common Levi part is conjugate to each other under some inner automorphism. We had proved that for classical algebras and "almost half" of normal parabolic subalgebra, the above conjecture is affirmative for regular infinitesimal characters.
In the first year, the principal investigator gave geometrical proof of the above conjecture and removed the assumption "classical". In the second year, we investigated submodules of scalar generalized Verma modules of maximal Gelfand-Krillov dimensions.
2 Irreducibility of the space of continuous Whittaker vectors
The famous "multiplicity one theorem" tells us that the dimension of the space of continuous Whittaker vectors on an irreducible admissible representation of a quasi-split real linear Lie group is at most one. For non quasi-split groups the multiplicity one theorem fails. As a natural extension of the multiplicity one theorem to non quasi-split case, I propose the following conjecture.
The space of continuous Whittaker vectors is irreducible as a module over the finite W-algebra.
For example, we have an affirmative answer for the type A groups.
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