Monomial representation of solvable Lie groups
| Project/Area Number |
11640189
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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 Kyushu School of Engeneering, Professor, 九州工学部, 教授 (50108643)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000)
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| Keywords | nilpotent Lie group / solvable Lie group / unitary representation / irreducible decomposition / orbit method / multiplicity / invariant differential operator / monomial representation / 既約公解 |
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| Research Abstract |
It is well known that there exists a strong parallelism for inducing and restricting representations. In this research, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A. Baklouti, G. Lion and B. Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.
1. (Commutativity conjecture of Duflo, Corwin Greenleaf) Let χ be a unitary character of an analytic subgroup H of G. Then, the monomial representation τ induced by χ up to G is of finite multiplicities if and only if the algebra of invariant differential operators on the line bundle over G/H associated to these data is commutative.
2. (Frobenius reciprocity) Let π be an irreducible unitary representation of G. The multiplicity of π in the canonical central decomposition of τ is equal to the dimension of the space of (H, χ ) semi-invariant generalized vectors.
3. The above result 1 has its counterpart for the restrictions. Namely, let's restrict an irreducible unitary representation of G to an analytic subgroup K. Then, this restriction is of finite multiplicities if and only if the associated algebra of K invariant differential operators is commutative.
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Report
(4 results)
Research Products
(16 results)
