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SUMMARY:Toshihisa Kubo (Ryukoku University)
DTSTART:20210610T120000Z
DTEND:20210610T124000Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/1
DESCRIPTION:Title: Differential symmetry breaking operators for $(O(n+1\,1)\, O(n\,
1))$ on differential forms\nby Toshihisa Kubo (Ryukoku University) as
part of Workshop on Branching Problems and Symmetry Breaking\n\n\nAbstract
\nLet $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take \
n$G' \\subset G$ to be a pair of Lie groups that act on $Y \\subset X$\, r
espectively. We call a differential operator $D$ between the spaces of smo
oth sections for a $G$-equivariant vector bundle over $X$ and that for a $
G'$-equivariant vector bundle over $Y$ a differential symmetry breaking op
erator (differential SBO for short) if $D$ is $G'$-intertwining.\n\nIn [Ko
bayashi-Kubo-Pevzner\, Lecture Notes in Math. 2170]\, we classified all th
e differential SBOs from the space of differential $i$-forms over the stan
dard Riemann sphere to that of differential $j$-forms over the totally geo
desic hypersphere. In this talk we shall discuss how we classify such oper
ators. This is a joint work with T. Kobayashi and M. Pevzner.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/1/
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SUMMARY:Ryosuke Nakahama (Kyushu University)
DTSTART:20210610T124500Z
DTEND:20210610T132500Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/2
DESCRIPTION:Title: Computation of weighted Bergman inner products on bounded symme
tric domains for $SU(r\,r)$ and restriction to subgroups\nby Ryosuke N
akahama (Kyushu University) as part of Workshop on Branching Problems and
Symmetry Breaking\n\n\nAbstract\nLet $D\\subset M(r\,\\mathbb{C})$ be the
bounded symmetric domain\, and we consider the weighted Bergman space $\\m
athcal{H}_\\lambda(D)$ on $D$. Then $SU(r\,r)$ acts unitarily on $\\mathca
l{H}_\\lambda(D)$. In this talk\, we compute explicitly the inner products
for some polynomials on $\\operatorname{Alt}(r\,\\mathbb{C})$\, $\\opera
torname{Sym}(r\,\\mathbb{C})\\subset M(r\,\\mathbb{C})$\, and prove that t
he inner products are given by multivariate hypergeometric polynomials whe
n the polynomials are some powers of the determinants or the Pfaffians. As
an application\, we present the results on the construction of symmetry b
reaking operators from $SU(r\,r)$ to $Sp(r\,\\mathbb{R})$ or $SO^*(2r)$.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/2/
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BEGIN:VEVENT
SUMMARY:Quentin Labriet (University of Reims)
DTSTART:20210610T140000Z
DTEND:20210610T144000Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/3
DESCRIPTION:Title: Symmetry breaking operators and orthogonal polynomials\nby Q
uentin Labriet (University of Reims) as part of Workshop on Branching Prob
lems and Symmetry Breaking\n\n\nAbstract\nSymmetry breaking operators are
intertwining operators for the restriction of an irreducible representatio
n. In some cases\, these are given by differential operators whose symbol
is related to some classical orthogonal polynomials. First\, I will descr
ibe the example of the Rankin-Cohen brackets which are symmetry breaking o
perators for the tensor product of two representations of the holomorphic
discrete series of $SL_2(\\mathbb R)$. I will explain how they are related
to Jacobi polynomials\, and to the classical Jacobi transform. In a secon
d part I will describe a link between orthogonal polynomials on the simple
x and symmetry breaking operators for the tensor product of multiple holom
orphic discrete series of $SL_2(\\mathbb R)$.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/3/
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SUMMARY:Ethan Shelburne (William & Mary / UNiversity of Bristish Columbia)
DTSTART:20210610T144500Z
DTEND:20210610T151500Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/4
DESCRIPTION:Title: Toward a holographic transform for the quantum Clebsch-Gordan Fo
rmula\nby Ethan Shelburne (William & Mary / UNiversity of Bristish Col
umbia) as part of Workshop on Branching Problems and Symmetry Breaking\n\n
\nAbstract\nA holographic transform is an equivariant map which increases
the number of variables in its domain\, a space of functions. The tensor p
roduct of two finite dimensional irreducible representations of the Lie al
gebra $\\mathfrak{sl}(2)$ decomposes into a direct sum of irreducible modu
les. In fact\, the tensor product of representations of $\\mathcal U_q(\\m
athfrak{sl}(2))$\, the quantum analogue of $\\mathfrak{sl}(2)$\, decompose
s in the same way. The purpose of this talk will be discussing the search
for explicit holographic transforms associated with these decompositions.\
n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuki Kannaka (RIKEN iTHEMS)
DTSTART:20210611T120000Z
DTEND:20210611T124000Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/5
DESCRIPTION:Title: The multiplicities of stable eigenvalues on compact anti-de Sitt
er $3$-manifolds\nby Kazuki Kannaka (RIKEN iTHEMS) as part of Workshop
on Branching Problems and Symmetry Breaking\n\n\nAbstract\nA \\textit{pse
udo-Riemannian locally symmetric space} is the quotient manifold $\\Gamma\
\backslash G/H$ of a semisimple symmetric space $G/H$ by a discontinuous g
roup $\\Gamma$. Toshiyuki Kobayashi initiated the study of spectral analys
is\nof \\textit{intrinsic differential operators} (such as the Laplacian)
of a pseudo-Rimannian locally symmetric space. Unlike the classical Rieman
nian setting\, the Laplacian of a pseudo-Rimannian locally symmetric space
is no longer an elliptic differential operator. In its spectral analysis\
, new phenomena different from those in the Riemannian setting have been d
iscovered in recent years\, following pioneering works by Kassel-Kobayashi
. For instance\, they studied the behavior of eigenvalues of intrinsic dif
ferential operators of $\\Gamma\\backslash G/H$ when deforming a discontin
uous group $\\Gamma$. As a special case\, they found infinitely many \\tex
tit{stable eigenvalues} of the (hyperbolic) Laplacian of a compact anti-de
Sitter $3$-manifold $\\Gamma\\backslash \\mathrm{SO}(2\,2)/\\mathrm{SO}(2
\,1)$ ([Adv.\\ Math.\\ 2016]). In this talk\, I would like to explain rece
nt results about the \\textit{multiplicities} of stable eigenvalues in the
anti-de Sitter setting.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/5/
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SUMMARY:Clemens Weiske (Paderborn University)
DTSTART:20210611T124500Z
DTEND:20210611T132500Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/6
DESCRIPTION:Title: Analytic continuation of unitary branching laws for real reducti
ve groups\nby Clemens Weiske (Paderborn University) as part of Worksho
p on Branching Problems and Symmetry Breaking\n\n\nAbstract\nLet $G$ be a
real reductive group\, $P$ a minimal parabolic and $H$ a reductive subgrou
p of $G$. Unitary branching laws describe how a unitary irreducible repres
entation of $G$ decomposes into a direct integral of unitary irreducible r
epresentations of $H$ when restricted to the subgroup $H$. If the represen
tation is in the unitary principal series and $H$ has an open orbit on the
flag manifold $G/P$\, Mackey theory reduces this problem to the Planchere
l formula of a homogeneous space for $H$\, which is known in many cases. I
n this case we show how to obtain branching laws for other unitary represe
ntations like complementary series representations from branching laws for
the unitary principal series by analytic continuation. We focus on the ex
emplary case of the rank-one pair $(O(1\,n+1)\,O(1\,n))$.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Genkai Zhang (Chalmers University)
DTSTART:20210611T140000Z
DTEND:20210611T144000Z
DTSTAMP:20260416T055109Z
UID:BranchingWorkshop/7
DESCRIPTION:Title: Induced representations of Hermitian Lie groups from Heisenberg
parabolic subgroups\nby Genkai Zhang (Chalmers University) as part of
Workshop on Branching Problems and Symmetry Breaking\n\n\nAbstract\nWe st
udy the induced representations of Hermitian Lie groups $G$ from Heisenbe
rg parabolic subgroups $P$. We find the composition series and complementa
ry series. For certain parameters of the representations the CR-Laplacian
on $G/P$ defines intertwining operator and we find its eigenvalues.\n
LOCATION:https://researchseminars.org/talk/BranchingWorkshop/7/
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