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Seminars

Computation of weighted Bergman inner products on bounded symmetric domains for U(r,r) and restriction to subgroups

Hold Date
2021-07-01 16:00〜2021-07-01 18:00
Place
Zoom
Object person
 
Speaker
Ryosuke Nakahama (IMI Kyushu University, JSPS-PD)

Let $D\subset M(r,\mathbb{C})$ be the bounded symmetric domain, and we consider the weighted Bergman space $\mathcal{H}_\lambda(D)$ on $D$. Then $U(r,r)$ acts unitarily on $\mathcal{H}_\lambda(D)$. In this talk, we compute explicitly the inner products for some polynomials on $M(r',\mathbb{C})\oplus M(r'',\mathbb{C})$, $\operatorname{Alt}(r,\mathbb{C})$, $\operatorname{Sym}(r,\mathbb{C})\subset M(r,\mathbb{C})$, and prove that the inner products are given by multivariate hypergeometric polynomials when the polynomials are some powers of the determinants or the Pfaffians. As an application, we present the results on the construction of symmetry breaking operators from $U(r,r)$ to $U(r',r'')\times U(r'',r')$, $Sp(r,\mathbb{R})$ or $SO^*(2r)$.