Lie Groups and Representation Theory

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

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)
[ 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) 15:40-16:10
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) 16:10-16:40
Holomorphic discrete series and Borel-de Siebenthal discrete series (ENGLISH)
[ Abstract ]
Let $G_0$ be a simply connected non-compact real simple Lie group with maximal compact subgroup $K_0$.
Let $T_0\\subset K_0$ be a Cartan subgroup of $K_0$ as well as of $G_0$. So $G_0$ has discrete series representations.
Denote by $\\frak{g}, \\frak{k},$ and $\\frak{t}$, the
complexifications of the Lie algebras $\\frak{g}_0, \\frak{k}_0$ and $\\frak{t}_0$ of $G_0, K_0$ and $T_0$ respectively.
There exists a positive root system $\\Delta^+$ of $\\frak{g}$ with respect to $\\frak{t}$, 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 $G_0/K_0$ is Hermitian symmetric, then $\\nu$ has coefficient $1$ in $\\mu$ and one can define holomorphic discrete series representation of $G_0$ using $\\Delta^+$.
If $G_0/K_0$ is not Hermitian symmetric, the coefficient of $\\nu$ in the highest root $\\mu$ is $2$.
In this case, Borel-de Siebenthal discrete series of $G_0$ is defined using $\\Delta^+$ in a manner analogous to the holomorphic discrete series.

Let $\\nu^*$ be the fundamental weight corresponding to $\\nu$ and $L_0$ be the centralizer in $K_0$ of the circle subgroup defined by $i\\nu^*$.
Note that $L_0 = K_0$, when $G_0/K_0$ is Hermitian symmetric. Otherwise, $L_0$ is a proper subgroup of $K_0$ and $K_0/L_0$ is an irreducible compact Hermitian symmetric space.
Let $G$ be the simply connected Lie group with Lie algebra $\\frak{g}$ and $K_0^* \\subset G$ be the dual of $K_0$ with respect to $L_0$ (or, the image of $L_0$ in $G$).
Then $K_0^*/L_0$ is an irreducible non-compact Hermitian symmetric space dual to $K_0/L_0$.
In this talk, to each Borel-de Siebenthal discrete series of $G_0$, a holomorphic discrete series of $K_0^*$ will be associated and occurrence of common $L_0$-types in both the series will be discussed.
Dipendra Prasad (Tata Institute of Fundamental Research) 16:50-17:40
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.