## Lie Groups and Representation Theory

Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

### 2019/10/30

16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Quentin Labriet (Reims University)
On holographic transform (English)
[ Abstract ]
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible
components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.

I will illustrate this construction by two examples :

- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.

- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.

I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.