2004 Fiscal Year Final Research Report Summary
Representations of solvable Lie groups and differential operators
| Project/Area Number |
14540194
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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 | Kinki University |
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Principal Investigator |
FUJIWARA Hidenori Kinki University, School of Humanity-Oriented Science and Engeneering, Professor, 産業理工学部, 教授 (50108643)
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| Project Period (FY) |
2002 – 2004
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| Keywords | nilpotent Lie group / solvable Lie group / unitary representation / irreducible decomposition / orbit method / multiplicity / invariant differential operator / monomial representation |
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| Research Abstract |
So-called "Polynomial conjecture" of Corwin-Greenleaf is a well known difficult conjecture for monomial representations of a connected and simply connected nilpotent Lie group. It has been a central aim of this research project. As there exists a strong parallelism for inducing and restricting representations, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A.Baklouti, G.Lion, J.Ludwig and B.Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.
1.Let χ be a unitary character of an analytic subgroup H of G. We consider the monomial representation τ induced by χ up to G. The algebra of invariant differential operators on the line bundle over G/H associated to these data is algebraic over a system of generators of the set of central elements of Corwin-Greenleaf if and only if τ is of finite multiplicities.
2.(Polynomial conjecture of Corwin - Greenleaf) Suppose that the monomial representation τ is of finite multiplicities. Then, the algebra of invariant differential operators on the line bundle over G/H associated to these data is isomorphic to the algebra of H-invariant polynomial functions on a certain affine subspace of the linear dual of the Lie algebra of G.
3. The above result 1 and the Frobenius reciprocity in distribution sense obtained in the previous research program have their counterpart for the restrictions. We formulated them for the restriction π|K of an irreducible unitary representation π of G to an analytic subgroup K. Then, we proved them in certain particular cases.
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Research Products
(10 results)
