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Lie Groups and Representation Theory
Seminar information archive ～05/21｜Next seminar｜Future seminars 05/22～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|
2013/11/07
14:30-17:40 Room #000 (Graduate School of Math. Sci. Bldg.)
Vaibhav Vaish (the Institute of Mathematical Sciences) 14:30-15:20
Weightless cohomology of algebraic varieties (ENGLISH)
Visible actions on generalized flag varieties
--- Geometry of multiplicity-free representations of SO(N) (ENGLISH)
Holomorphic discrete series and Borel-de Siebenthal discrete series (ENGLISH)
Branching laws and the local Langlands correspondence (ENGLISH)
Vaibhav Vaish (the Institute of Mathematical Sciences) 14:30-15:20
Weightless cohomology of algebraic varieties (ENGLISH)
[ Abstract ]
Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups. These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.
The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel-Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.
Yuichiro Tanaka (the University of Tokyo) 15:40-16:10Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups. These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.
The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel-Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.
Visible actions on generalized flag varieties
--- Geometry of multiplicity-free representations of SO(N) (ENGLISH)
[ Abstract ]
The subject of study is tensor product representations of irreducible representations of the orthogonal group, which are multiplicity-free. Here we say a group representation is multiplicity-free if any irreducible representation occurs at most once in its irreducible decomposition.
The motivation is the theory of visible actions on complex manifolds, which was introduced by T. Kobayashi. In this theory, the main tool for proving the multiplicity-freeness property of group representations is the ``propagation theorem of the multiplicity-freeness property". By using this theorem and Stembridge's classification result, we obtain the following: All the multiplicity-free tensor product representations of SO(N) and Spin(N) can be obtained from character, alternating tensor product and spin representations combined with visible actions on orthogonal generalized flag varieties.
Pampa Paul (Indian Statistical Institute, Kolkata) 16:10-16:40The subject of study is tensor product representations of irreducible representations of the orthogonal group, which are multiplicity-free. Here we say a group representation is multiplicity-free if any irreducible representation occurs at most once in its irreducible decomposition.
The motivation is the theory of visible actions on complex manifolds, which was introduced by T. Kobayashi. In this theory, the main tool for proving the multiplicity-freeness property of group representations is the ``propagation theorem of the multiplicity-freeness property". By using this theorem and Stembridge's classification result, we obtain the following: All the multiplicity-free tensor product representations of SO(N) and Spin(N) can be obtained from character, alternating tensor product and spin representations combined with visible actions on orthogonal generalized flag varieties.
Holomorphic discrete series and Borel-de Siebenthal discrete series (ENGLISH)
[ Abstract ]
Let G0 be a simply connected non-compact real simple Lie group with maximal compact subgroup K0.
Let T0subsetK0 be a Cartan subgroup of K0 as well as of G0. So G0 has discrete series representations.
Denote by frakg,frakk, and frakt, the
complexifications of the Lie algebras frakg0,frakk0 and frakt0 of G0,K0 and T0 respectively.
There exists a positive root system Delta+ of frakg with respect to frakt, known as the Borel-de Siebenthal positive system for which there is exactly one non-compact simple root, denoted nu. Let mu denote the highest root.
If G0/K0 is Hermitian symmetric, then nu has coefficient 1 in mu and one can define holomorphic discrete series representation of G0 using Delta+.
If G0/K0 is not Hermitian symmetric, the coefficient of nu in the highest root mu is 2.
In this case, Borel-de Siebenthal discrete series of G0 is defined using Delta+ in a manner analogous to the holomorphic discrete series.
Let nu∗ be the fundamental weight corresponding to nu and L0 be the centralizer in K0 of the circle subgroup defined by inu∗.
Note that L0=K0, when G0/K0 is Hermitian symmetric. Otherwise, L0 is a proper subgroup of K0 and K0/L0 is an irreducible compact Hermitian symmetric space.
Let G be the simply connected Lie group with Lie algebra frakg and K∗0subsetG be the dual of K0 with respect to L0 (or, the image of L0 in G).
Then K∗0/L0 is an irreducible non-compact Hermitian symmetric space dual to K0/L0.
In this talk, to each Borel-de Siebenthal discrete series of G0, a holomorphic discrete series of K∗0 will be associated and occurrence of common L0-types in both the series will be discussed.
Dipendra Prasad (Tata Institute of Fundamental Research) 16:50-17:40Let G0 be a simply connected non-compact real simple Lie group with maximal compact subgroup K0.
Let T0subsetK0 be a Cartan subgroup of K0 as well as of G0. So G0 has discrete series representations.
Denote by frakg,frakk, and frakt, the
complexifications of the Lie algebras frakg0,frakk0 and frakt0 of G0,K0 and T0 respectively.
There exists a positive root system Delta+ of frakg with respect to frakt, known as the Borel-de Siebenthal positive system for which there is exactly one non-compact simple root, denoted nu. Let mu denote the highest root.
If G0/K0 is Hermitian symmetric, then nu has coefficient 1 in mu and one can define holomorphic discrete series representation of G0 using Delta+.
If G0/K0 is not Hermitian symmetric, the coefficient of nu in the highest root mu is 2.
In this case, Borel-de Siebenthal discrete series of G0 is defined using Delta+ in a manner analogous to the holomorphic discrete series.
Let nu∗ be the fundamental weight corresponding to nu and L0 be the centralizer in K0 of the circle subgroup defined by inu∗.
Note that L0=K0, when G0/K0 is Hermitian symmetric. Otherwise, L0 is a proper subgroup of K0 and K0/L0 is an irreducible compact Hermitian symmetric space.
Let G be the simply connected Lie group with Lie algebra frakg and K∗0subsetG be the dual of K0 with respect to L0 (or, the image of L0 in G).
Then K∗0/L0 is an irreducible non-compact Hermitian symmetric space dual to K0/L0.
In this talk, to each Borel-de Siebenthal discrete series of G0, a holomorphic discrete series of K∗0 will be associated and occurrence of common L0-types in both the series will be discussed.
Branching laws and the local Langlands correspondence (ENGLISH)
[ Abstract ]
The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and p-adic groups. We will survey some of the questions that have recently been considered drawing analogy with Compact Lie groups, and what it suggests in the context of real and p-adic groups via what is called the local Langlands correspondence.
The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and p-adic groups. We will survey some of the questions that have recently been considered drawing analogy with Compact Lie groups, and what it suggests in the context of real and p-adic groups via what is called the local Langlands correspondence.
