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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
392a9c33265e9ac24001d3296fd0e6c9abb7e7d7 | subsection | 155 | 285 | Body | Also, when d=4, s^{\prime }=3, under the identification \operatorname{SO}^*(6)\simeq \operatorname{SU}(1,3) up to covering, we haveV_{(k_1+m_2+m_3,k_2+m_1+m_3, k_3+m_1+m_2)}^{(3)\vee } \\
\qquad {} \simeq V_{(0;-k_2-k_3-2m_1-m_2-m_3, -k_1-k_3-m_1-2m_2-m_3, -k_1-k_2-m_1-m_2-2m_3)}^{(1,3)\vee } \\
\qquad {} \simeq V_{(m_... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
4002206df91b8322c9268d159965a85452e911b9 | subsection | 156 | 285 | Body | \end{array}\right.}Therefore by and (REF ), we have&F_{\tau \rho }(x_{12};w_{11},w_{22})=\mathrm {K}(x_{12})\bigl \langle {\rm e}^{(y_{11}|w_{11})_{\mathfrak {p}^+_{11}}}I_{W_{11}},
{\rm e}^{(y_{11}|Q(x_{12})w_{22})_{\mathfrak {p}^+_{11}}}I_{W_{11}}\bigr \rangle _{\hat{\rho }_{11},y_{11}}\\
&={\left\lbrace \begin{array... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.0... |
2347f568aa68ce855b0b42066710fa00db80dab2 | subsection | 157 | 285 | Body | Also, for d=6, \mathbf {Pf}(x_{12}) is defined in (REF ). By Theorem REF , by substituting w_{11}, w_{22} with \overline{\frac{\partial }{\partial x_{11}}}, \overline{\frac{\partial }{\partial x_{22}}}, we get the intertwining operator from (\mathcal {H}_1)_{\tilde{K}_1} to \mathcal {H}_{\tilde{K}}, and by Theorem REF ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1506.05919",
"end": 613,
"openalex_id": "https://openalex.org/W1193240370",
"raw": "Nakahama R., Norm computation and analytic continuation of vector valued holomorphic discrete series representations, J. Lie Theory 26 (2016)... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
e437f8eaddd31eb01a8ddb561e3fa453f130bb5c | subsection | 158 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{\lambda +k}\big (D_{11},V_{\langle l\rangle }^{(s^{\prime })\vee }\big )_{\tilde{K}_{11}}\boxtimes \mathcal {O}_{\lambda }(D_{22},V_{(\underbrace{\scriptstyle k+1,\ldots ,k+1}_l,\underbrace{\scriptstyle k,\ldots ,k}_{s^{\prime }-l},
\underbrace{\scr... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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-0... |
5907a4a66da6c94fee743133ceacddd3258d399f | subsection | 159 | 285 | Body | Let (G,G_1)=(U(q,s), U(q^{\prime },s^{\prime })\times U(q^{\prime \prime },s^{\prime \prime })) with q=q^{\prime }+q^{\prime \prime }, s=s^{\prime }+s^{\prime \prime }, q^{\prime }\le s^{\prime \prime }.
Let k\in \mathbb {Z}_{\ge 0}, and \mathbf {l}\in \mathbb {Z}_{++}^{\min \lbrace q^{\prime \prime },s^{\prime }\rbrac... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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-... |
af1dd474162a4261913c631bf433dd0b946dbba3 | subsection | 160 | 285 | Body | If q^{\prime }=s^{\prime \prime } or k=0 or \mathbf {l}=(0,\ldots ,0) or “s^{\prime }\ge q^{\prime \prime } and \mathbf {l}=(l,\ldots ,l)”
or “\lambda _1+\lambda _2+k+l_{s^{\prime }}>q^{\prime }+s^{\prime }-1 and \lambda _1+\lambda _2+l_{q^{\prime \prime }}>q^{\prime \prime }+s^{\prime \prime }-1”,
then this extends to... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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-0.008111519739031792,
... |
205b35aebf775edb714748c2d55633073a348f54 | subsection | 161 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{\lambda +2k}\big (D_{11},V_{(l,0,\ldots ,0)}^{(s^{\prime })\vee }\big )_{\tilde{K}_{11}}\boxtimes \mathcal {O}_\lambda \big (D_{22},V_{(\underbrace{\scriptstyle k+l,k,\ldots ,k}_{s^{\prime }},0,\ldots ,0)}^{(s^{\prime \prime })\vee }\big )_{\tilde{K... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.020833326503634453,
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0.025381555780768394,
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... |
b305fea58fcd4add0452bbc374135ae246cf0a9e | subsection | 162 | 285 | Body | If s^{\prime }=s^{\prime \prime } or k=0 or “s^{\prime \prime }=s^{\prime }+1 and l=0”
or \lambda >2s^{\prime \prime }-3 or “s^{\prime \prime }=s^{\prime }+1 and \lambda +k>2s^{\prime \prime }-3”,
then this extends to the map between the spaces of all holomorphic functions.
Let (G,G_1)=(\operatorname{SO}^*(2s), \opera... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.0050244806334376335,
-0.00846951175481081,... |
19a022600b58d03a0be940f818d1f1c98f859a44 | subsection | 163 | 285 | Body | Then the linear maps
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{\lambda +2k}\big (D_{11},V_{(l,\ldots ,l,0)}^{(s^{\prime })\vee }\big )_{\tilde{K}_{11}}\boxtimes \mathcal {O}_\lambda \big (D_{22},V_{(\underbrace{\scriptstyle k+l,\ldots ,k+l,k}_{s^{\prime }},0,\ldots ,0)}^{(s^{\prime \prime })\vee }\big )_{\tild... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.019423354417085648,
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-0.01650908961892128,
-0.009650645777583122,
... |
8469a4b18a2fe9af04c87ae317f5f4e0caa9c9c4 | subsection | 164 | 285 | Body | Here if s^{\prime } is odd, then for \mathbf {l}=(l_1,\ldots ,l_{\lfloor s^{\prime }/2\rfloor },l_{\lceil s^{\prime }/2\rceil })\in (\mathbb {Z}_{\ge 0})^{\lceil s^{\prime }/2\rceil },
we write \mathbf {l}^{\prime }=(l_1,\ldots ,l_{\lfloor s^{\prime }/2\rfloor })\in (\mathbb {Z}_{\ge 0})^{\lfloor s^{\prime }/2\rfloor }... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.026187969371676445,
0.050514087080955505,
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0.0110108507797122,
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-0.0204192902892828,
... |
117c9ac8c514536e5ca521a30309fd9f4e675c2d | subsection | 165 | 285 | Body | \frac{1}{(\lambda +(k_1+k_2,k_1+k_3,k_2+k_3))_{(m_3,m_2,m_1),2}} \\
\hphantom{\mathcal {F}_{\lambda ,k_1,k_2,k_3}f\begin{pmatrix}x_{11}&x_{12}\\-{}^t\hspace{-1.0pt}x_{12}&x_{22}\end{pmatrix}=}{} \times \mathrm {K}_{(k_1,k_2,k_3)}^{(2)}(x_{12})\mathbf {K}_{\mathbf {m},\mathbf {k}}^{(2)\prime }
\left(x_{12}\frac{\partial... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.022798387333750725,
0.041079822927713394,
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-0.006092540919780731,
... |
450c2f07e5ac53556efcedb1434b84cd95ca9b6e | subsection | 166 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k_1,k_2}\colon \ \mathcal {O}_{\lambda +k_1+k_2}(D_{11})\mathbin {\hat{\boxtimes }}\mathcal {O}_\lambda \big (D_{22},V_{(0;-k_2,-k_2,-k_1,-k_1,-k_1-k_2)}^{(1,5)\vee }\big ) \longrightarrow \mathcal {O}_\lambda (D),\\
\mathcal {F}_{\lambda ,k_1,k_2}f(x_{11},x_{12},x_{22})\\
\q... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.041073281317949295,
0.023359283804893494,
-0.026288731023669243,
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0.01946861296892166,
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0.004783236887305975,
-... |
fbae88e6b97c797cc6e4cd78c0eb4a45571b094a | subsection | 167 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k_1,k_2}\colon \ \mathcal {O}_{\lambda +k_1+k_2}(D_{11})_{\tilde{K}_{11}}\\
\qquad {} \boxtimes \mathcal {O}_{\lambda +\frac{k_1+k_2}{2}}\Bigl (D_{22},
V_{\left(\frac{k_1+k_2}{2},\frac{k_1-k_2}{2},\frac{k_1-k_2}{2},\frac{k_1-k_2}{2},\frac{k_1-k_2}{2}\right)}^{[10]\vee }\Bigr ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.025689195841550827,
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0... |
e2225294fc05d0d30f646d37cd5e5f07faea3b03 | subsection | 168 | 285 | Body | If k_2=0 or \lambda >9,
then this extends to the map between the spaces of all holomorphic functions.When d=4 and s^{\prime }=3, for (k_1,k_2,k_3)=(k+l,k,k) and m,l_1,l_2\in \mathbb {Z}_{\ge 0}, l_1+l_2=l we have\frac{1}{(\lambda +2k+(l,0))_{(m+l_1-l,l_2),4}}\mathbf {K}_{m,(l_1,l_2)}^{(4)}(z_{11};y_{11}) \\
\qquad {} =... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.030935103073716164,
0.03816549852490425,
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0.0015873688971623778,
0.007871061563491821,
... |
322a3db3e21e17a5598f0893580f9433f6c10ce1 | subsection | 169 | 285 | Body | Similarly, for (k_1,k_2,k_3)=(k+l,k+l,k) and m,l_1,l_2\in \mathbb {Z}_{\ge 0}, l_1+l_2=l we have\frac{1}{(\lambda +2k+2l)_{m-l_1}(\lambda +2k+l-1)_{l-l_2}}\mathbf {K}_{m,-(l_1,l_2)}^{(4)}(z_{11};y_{11}) \\
\qquad {} =\frac{1}{(\lambda +(2k+2l,2k+l,2k+l))_{(m-l_1,l-l_2,0),2}}\mathbf {K}_{(0,l-l_2,m-l_1),(k+l,k+l,k)}^{(2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02062823809683323,
0.024961387738585472,
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0.00338336662389338,
-0.02950814552605152,
0.02... |
b615727530752c1c4bd815b08bd2a6a76244d857 | subsection | 170 | 285 | Body | Therefore, when (G,G_1)=(\operatorname{Sp}(s,\mathbb {R}), \operatorname{Sp}(s^{\prime },\mathbb {R})\times \operatorname{Sp}(s^{\prime \prime },\mathbb {R})), the intertwining operator is reduced to\mathcal {F}_{\lambda ,k}\colon \ \mathcal {O}_{\lambda +k}(D_{11}) \mathbin {\hat{\boxtimes }}\mathcal {O}_{\lambda +k}(... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05393713340163231,
0.016773318871855736,
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-0.021962517872452736,
0.018314816057682037,
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0.047832194715738297,
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-0.0070588355883955956,
... |
1092ac7da3ebe5f6f862eaa900cd16ceb6c902a7 | subsection | 171 | 285 | Body | Then the maximal compact subgroups (K,K_1)=(K,K_{11}\times K_{22})\subset (G,G_1) are given by(K,K_1) =(K,K_{11}\times K_{22})
={\left\lbrace \begin{array}{ll} (U(s), U(s^{\prime })\times U(s^{\prime \prime }))& (\text{Cases }d=1,4),\\
(U(1)\times \operatorname{Spin}(10), U(1)\times U(5))& (\text{Case }d=6),\\
(U(1)\ti... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03070421889424324,
0.018404219299554825,
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-0.010896029882133007,
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0.05023771896958351,
0.00916395802050829,
-0.005268703680485487,
-0.03128411993384361,
0... |
0896efd6219a3219cf7bb3a1fb27cf432b2e6f6b | subsection | 172 | 285 | Body | \end{array}\right.}Now let (\tau ,V)=\big (\chi ^{-\lambda },\mathbb {C}\big ) with \lambda sufficiently large,
W=W_{11}\boxtimes W_{22}\subset (\mathcal {P}(\mathfrak {p}^+_{11})\boxtimes \mathcal {P}(\mathfrak {p}^+_{22}))\otimes \chi ^{-\lambda }
be an irreducible \tilde{K}^\mathbb {C}_1=\tilde{K}^\mathbb {C}_{11}\t... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.010146193206310272,
0.030758986249566078,
-0.013388398103415966,
-0.0028683976270258427,
-0.01283913105726242,
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0.050898801535367966,
0.002696751384064555,
0.018720870837569237,
0.02921798638999462,
-0.014776824973523617,
-0.012411922216415405,
-0.00457723019644618,
... |
caf70b2921892d4bb0d4a8b2f3b607af4ecc436c | subsection | 173 | 285 | Body | For x_2=x_{11}+x_{22}\in \mathfrak {p}^+_2=\mathfrak {p}^+_{11}\oplus \mathfrak {p}^+_{22}, w_1=w_{12}\in \mathfrak {p}^+_1=\mathfrak {p}^+_{12},
we want to computeF_{\tau \rho }(x_2;w_1)=F_{\tau \rho }(x_{11},x_{22};w_{12})
=\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak {p}^+_1}}I_W,
\big (h(x_2,Q(y_1)x_2)^{-\lambda /2}\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.... |
194ce98dd173210417642c635107b9fd9b600a9c | subsection | 174 | 285 | Body | Now we assume that W=W_{11}\boxtimes W_{22}\subset \mathcal {P}(\mathfrak {p}^+_2)\otimes \chi ^{-\lambda } is of the formW=W_{11}\boxtimes W_{22}=\big (\mathcal {P}_{(k,\ldots ,k)}(\mathfrak {p}^+_{11})\otimes \chi _{11}^{-\lambda }\big )
\boxtimes \big (\mathcal {P}_\mathbf {l}(\mathfrak {p}^+_{22})\otimes \chi _{22}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
50f6076c8ff318d97bd49f54ac8fba619f889b3e | subsection | 175 | 285 | Body | Then we haveF_{\tau \rho }(x_{11},x_{22};w_{12})\\
{}= \big \langle {\rm e}^{(y_{12}|w_{12})_{\mathfrak {p}^+_{12}}}I_{W_{22}},\big (h_{22}(x_{22},Q(y_{12})x_{11})^{-\lambda }
\Delta \big ((x_{11})^{Q(y_{12})x_{22}}\big )^k
\mathrm {K}_\mathbf {l}^{(d)}\big ((x_{22})^{Q(y_{12})x_{11}}\big )\big )^*\big \rangle _{\hat{\... | {
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"raw": "Faraut J., Kaneyuki S., Korányi A., Lu Q.k., Roos G., Analysis and geometry on complex homogeneous domains, Progress in Mathematics, Vol. 18... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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ed0d3818ae23c95cb62bcd30600fa06d064ae6f0 | subsection | 176 | 285 | Body | For a while we omit the subscript 22. We realize \overline{W_{22}}=\overline{\mathcal {P}_\mathbf {l}(\mathfrak {p}^+_{22})\otimes \chi _{22}^{-\lambda }} as a space of polynomials in y, and write \mathrm {K}_\mathbf {l}^{(d)}(x)=\mathbf {K}_\mathbf {l}^{(d)}(x,y)\in \mathcal {P}(\mathfrak {p}^+_{22}\times \overline{\m... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
6d0a90e8ff757546ee4ae101740a58b4bf71317b | subsection | 177 | 285 | Body | Clearly \mathcal {K}_{\mathbf {n},\mathbf {l}}^{(d)} is non-zero only if \mathcal {P}_\mathbf {n} appears abstractly in the decomposition of \mathcal {P}_\mathbf {l}\otimes \mathcal {P}.
Then the following holds.Proposition 5.8h(x,z)^{-\mu }\mathbf {K}_\mathbf {l}^{(d)}(x^z,y)
&=\sum _{\mathbf {n}\in \mathbb {Z}_{++}^{... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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44ec84c949a833ac36493278f0a36ed116448dc6 | subsection | 178 | 285 | Body | Then by projecting both sides to \mathcal {P}_\mathbf {l} with respect to the variable \bar{y} and dividing by (\mu )_{\mathbf {l},d}, we get the desired formula.Corollary 5.9
When we define \mathcal {K}_{\mathbf {n},\mathbf {l}}^{(d)}(x;z)\in \mathcal {P}(\mathfrak {p}^+_{22}\times \overline{\mathfrak {p}^+_{22}},\op... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.... |
6ba7a800036f574ad7c8e510861e63b2060ae821 | subsection | 179 | 285 | Body | In these cases we haveW=\big (\mathcal {P}_{(k,\ldots ,k)}(\mathfrak {p}^+_{11})\otimes \chi _{11}^{-\lambda }\big )\boxtimes \big (\mathcal {P}_\mathbf {l}(\mathfrak {p}^+_{22})\otimes \chi _{22}^{-\lambda }\big )
\simeq {\left\lbrace \begin{array}{ll} \big (\mathbb {C}\boxtimes V_{2\mathbf {l}}^{(s^{\prime \prime })}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.036... |
bdaa844a45bc9e638ef3735ba8b60309c6428aa3 | subsection | 180 | 285 | Body | Therefore by the result of and (REF ), we getF_{\tau \rho }(x_{11},x_{22};w_{12})\\
={\left\lbrace \begin{array}{ll}\displaystyle \det (x_{11})^k\sum _{\mathbf {n}\in \mathbb {Z}_{++}^{s^{\prime \prime }}}
\frac{(\lambda +k+\mathbf {l})_{\mathbf {n}-\mathbf {l},1}}{(2(\lambda +k+\mathbf {l}))_{2(\mathbf {n}-\mathbf {l... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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56ebfef4fde16a69c1c475ed8eba36076ead809e | subsection | 181 | 285 | Body | Next we consider d=6 case. | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0... |
e1194f965fa4c7a79eac96dfed998e65dbf38f21 | subsection | 182 | 285 | Body | In this case we haveW=(\mathcal {P}_k(\mathbb {C})\boxtimes \mathcal {P}_l(M(1,5;\mathbb {C})))\otimes \chi ^{-\lambda }
&\simeq \Big (V_{\left(\frac{l}{2},\frac{l}{2},\frac{l}{2},\frac{l}{2},-\frac{l}{2}\right)}^{(5)\vee }
\otimes \chi _{\operatorname{SO}^*(10)}^{-\lambda -k}\Big )\boxtimes \chi _{U(1)}^{-\lambda +3k-... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
155db5f5a7f0a864e95711f6d436e0b0f3e2493d | subsection | 183 | 285 | Body | Therefore we haveF_{\tau \rho }(x_{11},x_{22};w_{12})\\
=x_{11}^k\sum _{n=l}^\infty (\lambda +k+l)_{n-l}\left\langle {\rm e}^{(y_{12}|w_{12})_{\mathfrak {p}^+_{12}}}I_{W_{22}},
\left(\frac{1}{(n-l)!}(x_{22}|Q(y_{12})x_{11})_{\mathfrak {p}_{22}}^{n-l}\mathrm {K}_l(x_{22})\right)^*\right\rangle _{\hat{\rho },y_{12}} \\
=... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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953cfd5a2040144491f88d2445f056211f8a4cf6 | subsection | 184 | 285 | Body | In this case we haveW=\big (\mathcal {P}_k(\mathbb {C})\boxtimes \mathcal {P}_\mathbf {l}\big (\operatorname{Herm}(2,\mathbb {O})^\mathbb {C}\big )\big )\otimes \chi ^{-\lambda }
\simeq V_{(l_1-l_2,0,0,0,0)}^{[10]\vee }
\boxtimes \chi _{E_{6(-14)}}^{-\lambda -k-\frac{|\mathbf {l}|}{2}}\boxtimes \chi _{U(1)}^{-\lambda +... | {
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"source_re... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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... |
acbeb80809e72cc43da9f6666e18a146cdb3af32 | subsection | 185 | 285 | Body | Therefore, if we assume on the norms of holomorphic discrete series representations of E_{6(-14)} is true,
then we haveF_{\tau \rho }(x_{11},x_{22};w_{12}) \\
=x_{11}^k\sum _{\mathbf {n}\in \mathbb {Z}_{++}^{2}}(\lambda +k+\mathbf {l})_{\mathbf {n}-\mathbf {l},8}\Bigl \langle {\rm e}^{(y_{12}|w_{12})_{\mathfrak {p}^+}}... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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1aa9638e1ec65b77ad79c70aa938d72473ed3ef5 | subsection | 186 | 285 | Body | Also, by Theorem REF , this continues meromorphically for all \lambda \in \mathbb {C}.
Therefore we get the following.Theorem 5.10Let (G,G_1)=(\operatorname{Sp}(s,\mathbb {R}), U(s^{\prime },s^{\prime \prime })) with s=s^{\prime }+s^{\prime \prime }. Let k\in \mathbb {Z}_{\ge 0}, \mathbf {l}\in \mathbb {Z}_{++}^{s^{\pr... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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51aa1a9bf001110cff348b77d52e394a54d28fd8 | subsection | 187 | 285 | Body | Let k\in \mathbb {Z}_{\ge 0} if s^{\prime } is even, k=0 if s^{\prime } is odd, and \mathbf {l}\in \mathbb {Z}_{++}^{\lfloor s^{\prime \prime }/2\rfloor }. Then the linear map
\mathcal {F}_{\lambda ,k,\mathbf {l}}\colon \ \mathcal {O}_{\left(\frac{\lambda }{2}+k\right)+\frac{\lambda }{2}}\big (D_1,\mathbb {C}\boxtimes... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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b8ade0a115709ce55187af8415f5d5b3b49e2710 | subsection | 188 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{\lambda +k}\big (D_1,\mathbb {C}\boxtimes V_{\left(\frac{l}{2},\frac{l}{2},\frac{l}{2},\frac{l}{2},-\frac{l}{2}\right)}^{(5)\vee }\big )
\boxtimes \chi _{U(1)}^{-\lambda +3k-3l}\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k,l}f)(... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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efea50d5cf89c59a3e766fede55ac6e85b5f3f58 | subsection | 189 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,\mathbf {l}}\colon \ \mathcal {O}_{\lambda +k+\frac{|\mathbf {l}|}{2}}\big (D_1,V_{(l_1-l_2,0,0,0,0)}^{[10]\vee }\big )
\boxtimes \chi _{U(1)}^{-\lambda +2k-2|\mathbf {l}|}\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k,\mathbf {l}}f)\begin{pmatrix}x_{11}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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e0d5e278f4ad29e9997acc3c21ed0e825b7bd32e | subsection | 190 | 285 | Body | Therefore by replacing \mathbf {n}-\mathbf {l}=\mathbf {m}, the intertwining operators are rewritten as, when (G,G_1)=(\operatorname{Sp}(s,\mathbb {R}), U(s^{\prime },s^{\prime \prime })),\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{(\lambda +2k)+(\lambda +2l)}(D_1)\rightarrow \mathcal {O}_\lambda (D),\\
(\mathcal... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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f9a5dbe03cb9082d05b4279a7639c68b3b5e3c79 | subsection | 191 | 285 | Body | We can also compute for (G,G_1)=(\operatorname{SO}^*(12), \operatorname{SO}^*(6)\times \operatorname{SO}^*(6)) in a similar way,
but we omit this case since this is contained in Theorem REF (5).
Then the maximal compact subgroups are(K,K_1)={\left\lbrace \begin{array}{ll} (S(U(3)\times U(3)), U(3))\simeq (S(U(3)\times ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02605319209396839,
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0.0... |
2cddeaed549fca905ca5ccfa2b3de9b328611245 | subsection | 192 | 285 | Body | M(2,6;\mathbb {C})\oplus \operatorname{Skew}(6,\mathbb {C}) by(k_1,k_2).(x_1,x_2)=\big (k_1x_1k_2^{-1},\det (k_2)^{-2/\varepsilon }k_2x_2{}^t\hspace{-1.0pt}k_2\big ),where \varepsilon =1 if d_2=1, \varepsilon =2 if d_2=4.
Let \chi , \chi _1 be the characters of K^\mathbb {C}, K_1^\mathbb {C} respectively, normalized as... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.010772769339382648,
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0.... |
06bae2ed8894887577d431bca66849056ce24ef4 | subsection | 193 | 285 | Body | (x_2)^{Q(y_1)x_2}=\det (x_2)^{-1}\big ((I-x_2^\sharp y_1^*\overline{y_1})^{-1}x_2^\sharp \big )^\sharp .For x_2\in \operatorname{Skew}(6,\mathbb {C}), y_1\in M(2,6;\mathbb {C}),h(Q(x_2)y_1,y_1)=\det \big (I-x_2^\#y_1^*J_2\overline{y_1}\big ).
(x_2)^{Q(y_1)x_2}=\operatorname{Pf}(x_2)^{-1}\big ((I-x_2^\#y_1^*J_2\overlin... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.020130842924118042,
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... |
ec7c2309675a339b967ed7d437ce45ee611fb742 | subsection | 194 | 285 | Body | Then we have Q(y_1)x_2=\big ({}^t\hspace{-1.0pt}\hat{y}_1y_1\big )\times x_2\in \operatorname{Herm}(3,\mathbb {K}^{\prime }),
where x\times y is as (REF ), and hence(x_2)^{Q(y_1)x_2}=x_2\big (I-\big (\big ({}^t\hspace{-1.0pt}\hat{y}_1y_1\big )\times x_2\big )x_2\big )^{-1}.By (REF ), it holds that\big ({}^t\hspace{-1.0... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.026574328541755676,
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... |
86d28b7220ad5b7ec63b48c8e8311daf09b2b938 | subsection | 195 | 285 | Body | Then by the previous lemma,h(Q(x_2)y_1,y_1)^{-\lambda /2}\mathrm {K}_{(k_1,k_2,k_3)}^{(d_2)}\big ((x_2)^{Q(y_1)x_2}\big ) \\
={\left\lbrace \begin{array}{ll} \det \big (I-y_1^*\overline{y_1}x_2^\sharp \big )^{-\lambda }\mathrm {K}_{(k_1,k_2,k_3)}^{(1)}
\big (\det (x_2)^{-1}\big (\big (I-x_2^\sharp y_1^*\overline{y_1}\b... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.035764437168836594,
0.02879159152507782,
-0.036527328193187714,
-0.03793105110526085,
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0.01116875745356083,
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0.01846201717853546,
0.005229633301496506,
0.... |
5e415553f1275af9189aad01eec16c0bfd986033 | subsection | 196 | 285 | Body | \end{array}\right.}Then since the map f(x_2)\mapsto f(x_2^{\sharp (\#)}) yields
\mathcal {P}_{(k_1,k_2,k_3)}(\mathfrak {p}^+_2)\rightarrow \mathcal {P}_{(k_1+k_2,k_1+k_3,k_2+k_3)}(\mathfrak {p}^+_2),
we write \mathrm {K}_{(k_1,k_2,k_3)}^{(d_2)}(x_2^{\sharp (\#)})=\mathrm {K}_{(k_1+k_2,k_1+k_3,k_2+k_3)}^{(d_2)}(x_2). Th... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03654923290014267,
0.020227162167429924,
-0.04286449775099754,
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0.03456617519259453,
-0.010227988474071026,... |
94f122cf875643b5be376c9d4e106d2c7c0c1886 | subsection | 197 | 285 | Body | \operatorname{Skew}(6,\mathbb {C}) let\mathcal {K}_{\mathbf {n},(k_1+k_2,k_1+k_3,k_2+k_3)}^{(d_2)}(x_2;z_2)
:=\operatorname{Proj}_{\mathbf {n},x}\big ({\rm e}^{\frac{1}{\varepsilon }\operatorname{tr}(x_2z_2^*)}\mathrm {K}_{(k_1+k_2,k_1+k_3,k_2+k_3)}^{(d_2)}(x_2)\big ) \\
\hphantom{\mathcal {K}_{\mathbf {n},(k_1+k_2,k_1... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.0227205827832222,
0.013969572260975838,
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-0.0012416973477229476,
... |
eb8e577e0f756d1c6299b91853c7993b8475e60a | subsection | 198 | 285 | Body | \end{array}\right.}We define \mathcal {K}_{\mathbf {m},\mathbf {k}}^{(d_2)\prime }(x_2;y_1)\in \mathcal {P}\big (\mathfrak {p}^+_2\times \overline{\mathfrak {p}^+_1},\operatorname{Hom}\big (W,\chi ^{-\lambda }\big )\big ) by\mathcal {K}_{\mathbf {m},\mathbf {k}}^{(1)\prime }(x_2;y_1) :=\det (x_2)^{-|\mathbf {k}|}
\math... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.010484933853149414,
0.020389914512634277,
-0.040413543581962585,
-0.0639168843626976,
-0.015719769522547722,
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0.029669156298041344,
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0.0165896974503994,
0.013957018963992596,
0.0... |
fe5ad59b4368a088f11f70ab1cd11e71bf1e0fe3 | subsection | 199 | 285 | Body | \end{array}\right.}Then we geth(Q(x_2)y_1,y_1)^{-\lambda /2}\mathrm {K}_{(k_1,k_2,k_3)}^{(d_2)}\big ((x_2)^{Q(y_1)x_2}\big ) \\
\qquad {} =\sum _{\mathbf {m}\in (\mathbb {Z}_{\ge 0})^3}(\lambda +(k_1+k_2,k_1+k_3,k_2+k_3))_{(m_3,m_2,m_1),d_2}\mathcal {K}_{\mathbf {m},\mathbf {k}}^{(d_2)\prime }(x_2;y_1)Now as a function... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.008452767506241798,
0.030500149354338646,
-0.045406926423311234,
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-0.003659941488876939,
... |
266845b61debe6d91a337aa218aa94a78c25a804 | subsection | 200 | 285 | Body | \mathcal {K}_{\mathbf {m},\mathbf {k}}^{(d_2)\prime }(x_2;y_1) is non-zero only if these inclusions hold, that is,
0\le m_1\le k_1-k_2, 0\le m_2\le k_2-k_3, 0\le m_3 hold.
Therefore by the result of and (REF ) we getF_{\tau \rho }(x_2;w_1)\\
= \sum _{{\mathbf {m}\in (\mathbb {Z}_{\ge 0})^3\\ 0\le m_j\le k_j-k_{j+1}}} ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1506.05919",
"end": 328,
"openalex_id": "https://openalex.org/W1193240370",
"raw": "Nakahama R., Norm computation and analytic continuation of vector valued holomorphic discrete series representations, J. Lie Theory 26 (2016)... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02324158325791359,
0.005692127626389265,
-0.02872006595134735,
-0.03424432873725891,
-0.0115139689296484,
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0.03998223692178726,
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-0.012048082426190376,
0.018983932211995125,
0.004... |
dab5754b85224d09233531d42e5ffda5e6883b9d | subsection | 201 | 285 | Body | \frac{1}{\left(\lambda +(k_1+k_2,k_1+k_3,k_2+k_3)+\frac{1}{2}\right)_{(m_3,m_2,m_1),2}}
\mathcal {K}_{\mathbf {m},\mathbf {k}}^{(1)\prime }\left(\!x_2;\frac{1}{2}w_1\!\right) \hspace{-18.0pt}& (d_2=1), \\
\displaystyle \sum _{{\mathbf {m}\in (\mathbb {Z}_{\ge 0})^3\\ 0\le m_j\le k_j-k_{j+1}}}
\frac{1}{(\lambda +(k_1+k_... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02866186574101448,
0.03623177111148834,
-0.02344229258596897,
-0.04798344150185585,
0.007928561419248581,
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0.009241086430847645,
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-0.014445397071540356,
0.009470014832913876,
0.0... |
30ed79761250f96b9697e94771c01715018ccc0a | subsection | 202 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,\mathbf {k}}\colon \ \mathcal {O}_\lambda (D_1,V_{2(k_1,k_2,k_3)}^{(3)\vee })\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,\mathbf {k}}f)(x_1,x_2)=\sum _{{\mathbf {m}\in (\mathbb {Z}_{\ge 0})^3\\ 0\le m_j\le k_j-k_{j+1}}}
\frac{1}{\left(\lambda +(k_1+k_2,k_... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04079163819551468,
0.02925892546772957,
-0.00641469145193696,
-0.04680207371711731,
0.011715773493051529,
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0.03832034394145012,
0.00863428134471178,
-0.012165794149041176,
0.03728301078081131,
-0.01668887585401535,
-0.006445201113820076,
0.01463708933442831,
-0.022... |
806405d506a9343fbfdd9b074d24f446582a24e5 | subsection | 203 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,\mathbf {k}}\colon \ \mathcal {O}_\lambda (D_1,V_{(0,0;-k_2-k_3,-k_2-k_3,-k_1-k_3,-k_1-k_3,-k_1-k_2,-k_1-k_2)}^{(2,6)\vee })
\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,\mathbf {k}}f)(x_1,x_2)=\sum _{{\mathbf {m}\in (\mathbb {Z}_{\ge 0})^3\\ 0\le m_j\le k... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03527028486132622,
0.012150857597589493,
-0.019740376621484756,
-0.05644465982913971,
0.010297336615622044,
0.011960165575146675,
0.03627713769674301,
0.012227133847773075,
0.008993007242679596,
0.040151987224817276,
-0.029519032686948776,
-0.02308128960430622,
0.008115827105939388,
-0.... |
94805c10b670e61b4a35b88ab7e949fcaf9f0fcd | subsection | 204 | 285 | Body | \end{array}\right.}Therefore, for (G,G_1)=(\operatorname{SU}(3,3),\operatorname{SO}^*(6)) we get\mathcal {F}_{\lambda ,k}\colon \ \mathcal {O}_{\lambda +2k}(D_1)\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k}f)(x_1,x_2)=\det (x_2)^k\sum _{m=0}^\infty \frac{1}{\left(\lambda +2k+\frac{1}{2}\right)_m}\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05270359292626381,
0.020919695496559143,
-0.03237898647785187,
-0.06683316081762314,
0.01960744522511959,
0.00370214506983757,
0.006275145802646875,
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0.011062567122280598,
0.03869609534740448,
-0.016479410231113434,
-0.005428287200629711,
-0.0060691530816257,
-0.013... |
931ab3d70c77d32effc04725600b79cc5e857c5f | subsection | 205 | 285 | Body | Then the maximal compact subgroups are (K,K_1)=(S(U(s)\times U(s)), U(s)), and \mathfrak {p}^+, \mathfrak {p}^+_1:=\fg _1^\mathbb {C}\cap \mathfrak {p}^+, \mathfrak {p}^+_2:=(\mathfrak {p}^+_1)^\bot are realized as\mathfrak {p}^+=M(s,\mathbb {C}),\qquad (\mathfrak {p}^+_1,\mathfrak {p}^+_2)={\left\lbrace \begin{array}{... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.005841887090355158,
0.01031011063605547,
-0.05076451227068901,
0.011699037626385689,
0.002008983399719,
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0.03623420000076294,
-0.020269174128770828,
0.014469259418547153,
0.05195502191781998,
0.016560280695557594,
0.00274351192638278,
-0.005101635120809078,
0.024619... |
130fc39fd6ac8cd33d1f5c4907f5cda1f38b93f7 | subsection | 206 | 285 | Body | For x_2\in \mathfrak {p}^+_2, w_1\in \mathfrak {p}^+_1, we want to computeF_{\tau \rho }(x_2;w_1)
&=\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak {p}^+_1}}I_W,
\big (h(Q(x_2)y_1,y_1)^{-\lambda /2}\mathrm {K}\big ((x_2)^{Q(y_1)x_2}\big )\big )^*\big \rangle _{\hat{\rho },y_1} \\
&=\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02417438104748726,
0.03534587472677231,
-0.007642248645424843,
0.006562490016222,
-0.0017503153067082167,
0.012071928940713406,
0.03494907543063164,
0.03476593643426895,
0.0013544673565775156,
0.04123685508966446,
-0.0026078266091644764,
-0.02731827273964882,
-0.0025887496303766966,
0.0... |
0a753f70a9d87c1e2659b2b19fb723e3b59c45a4 | subsection | 207 | 285 | Body | Then \mathcal {H}_{\varepsilon \lambda }(D_1,W)_{\tilde{K}_1} becomes multiplicity-free under \tilde{K}_1.
However, when (G,G_1)=(\operatorname{SU}(3,3),\operatorname{SO}^*(6)) this list does not exhaust all \tilde{K}_1-multiplicity-free submodules of \mathcal {H}_\lambda (D).
For this pair see Theorem REF (1). | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.032100893557071686,
0.0194527767598629,
-0.0024411326739937067,
-0.02250419184565544,
0.018308496102690697,
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0.022107508033514023,
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0.032070379704236984,
0.016096219420433044,
-0.010893554426729679,
-0.048731110990047455,
0.059777237474918365,
... |
fb961d713f980ea524e1fa1850431ddd388a2921 | subsection | 208 | 285 | Body | We write the polynomial \mathrm {K}(x_2) as&\mathrm {K}(x_2)=\operatorname{Pf}(x_2)^k \mathrm {K}_{\langle l\rangle }^{(4)}(x_2)\qquad & &(\text{Case }1),&\\
&\mathrm {K}(x_2)=\det (x_2)^k \mathrm {K}_{(l,0,\ldots ,0)}^{(1)}(x_2)\qquad & &(\text{Case }2),&\\
&\mathrm {K}(x_2)=\det (x_2)^k \mathrm {K}_{(l,\ldots ,l,0)}^... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.019785962998867035,
0.04530787095427513,
-0.008642056956887245,
-0.0230200607329607,
-0.017543451860547066,
0.012791465036571026,
0.04753512889146805,
0.015186527743935585,
0.00032202721922658384,
0.017116306349635124,
-0.022547150030732155,
0.011403243988752365,
-0.046375732868909836,
... |
86fea01397e6e16e9270a7fe9be6d572b4fd2575 | subsection | 209 | 285 | Body | Since \mathcal {P}(\mathfrak {p}^+_2) and \mathcal {P}(\mathfrak {p}^+_1)\otimes W_{(l)}^{\prime } are decomposed under K_1 as& \mathcal {P}(\mathfrak {p}^+_2) \simeq \bigoplus _{\mathbf {m}\in \mathbb {Z}_{++}^{\lfloor s/2\rfloor }}\mathcal {P}_\mathbf {m}(\operatorname{Skew}(s,\mathbb {C}))
\simeq \bigoplus _{\mathbf... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.021413691341876984,
0.033608656376600266,
-0.038614850491285324,
-0.014446227811276913,
0.012706270441412926,
0.00249546580016613,
0.056350212544202805,
-0.00005085608427179977,
0.0057960436679422855,
0.05048929899930954,
-0.029197711497545242,
-0.023413116112351418,
-0.020085828378796577... |
f7cc90c04de169a07dd7352ba6717213ebc37f1a | subsection | 210 | 285 | Body | Then \det (I-x_2y_1^*x_2y_1^*)^{-\mu /2}\mathrm {K}_{(l)}^{\prime }\big (x_2(I-y_1^*x_2y_1^*x_2)^{-1}\big ) is expanded as follows.Proposition 5.13For Case 1,
\det (I-x_2y_1^*x_2y_1^*)^{-\mu /2}\mathrm {K}_{\langle l\rangle }^{(4)}\big (x_2(I-y_1^*x_2y_1^*x_2)^{-1}\big )\\
=\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{\lfl... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.026197826489806175,
0.029569823294878006,
-0.034055646508932114,
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0.052944935858249664,
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0.009292065165936947,
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0.007278021425008774,
-0.032468825578689575,... |
6cf1e22bbe6730b0cda23f3841ec174fb0064fb6 | subsection | 211 | 285 | Body | For Case 3 with s even,
\det (I-x_2y_1^*x_2y_1^*)^{-\mu /2}\mathrm {K}_{(\underbrace{\scriptstyle l,\ldots ,l}_{s-1},0)}^{(1)}\big (x_2(I-y_1^*x_2y_1^*x_2)^{-1}\big )\\
\qquad {} =\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{s/2}}\sum _\mathbf {l}\frac{(\mu +l)_{\mathbf {m}-\mathbf {l}+(\smash{\overbrace{\scriptstyle l,\ld... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.03774697706103325,
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... |
d950f64bf44b5ab7a188d8b87545f7f4aa176b02 | subsection | 212 | 285 | Body | For Case 3 with s odd, there exist monic polynomials \varphi _{\mathbf {m},-\mathbf {l}}(\mu )\in \mathbb {C}[\mu ] of degree l-l_{\lceil s/2\rceil }
such that
\det (I-x_2y_1^*x_2y_1^*)^{-\mu /2}\mathrm {K}_{(\underbrace{\scriptstyle l,\ldots ,l}_{s-1},0)}^{(1)}\big (x_2(I-y_1^*x_2y_1^*x_2)^{-1}\big )\\
\qquad {} =\su... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.011566394940018654,
-0.054108757525682... |
72a3f63314be4a54f578371f1a1288e40a2c5663 | subsection | 213 | 285 | Body | We realize\overline{W_{(l)}^{\prime }}={\left\lbrace \begin{array}{ll}
\overline{V_{\langle 2l\rangle }^{(2r)\vee }}\simeq \overline{\mathcal {P}_{\langle l\rangle }(\operatorname{Skew}(2r,\mathbb {C}))}&(\text{Case }1),\\
\overline{V_{(2l,0,\ldots ,0)}^{(2r)\vee }}\simeq \overline{\mathcal {P}_{(l,0,\ldots ,0)}(\opera... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.0... |
4117e25e92e303a519ccfa9607b5e9c0858498ab | subsection | 214 | 285 | Body | For Case 3, \operatorname{Rest}(\mathcal {P}_{(\smash{\overbrace{\scriptstyle l,\ldots ,l}^{2r-1}},0)}(\operatorname{Sym}(2r,\mathbb {C})))
=\mathcal {P}_{(\smash{\overbrace{\scriptstyle 2l,\ldots ,2l}^{r-1}},l)}(M(r,\mathbb {C})).Since W_{(l)}^{\prime } and \mathcal {P}(M(r,\mathbb {C})) are decomposed under U(r)\time... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.010248630307614803,
0.0517086535692215,
-0.0030848912429064512,
-0.012454031966626644,
0.01781109720468521,
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0.022313473746180534,
0.0073030078783631325,
... |
8dee2be5934b9f6cf2013e77e8790102501f70c2 | subsection | 215 | 285 | Body | (W_{(l)}^{\prime \prime }\boxtimes \overline{W_{(l)}^{\prime \prime }})^{U(r)\times U(r)}. | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.02570357732474804,
0.02062387950718403,
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0.006570810452103615,
0.0237205121666193,
0.0047... |
8f2936785812f5ec346f8438f6875043d4358165 | subsection | 216 | 285 | Body | Next we consider \mathcal {P}(M(r,\mathbb {C})\oplus M(r,\mathbb {C})), on which U(r)\times U(r)\times U(r)=:K_{xL}\times K_{zL}\times K_R acts byf(x,z)\mapsto f\big (k_{xL}^{-1}xk_R,k_{zL}^{-1}zk_R\big ) \qquad ((k_{xL},k_{zL},k_R)\in K_{xL}\times K_{zL}\times K_R).Under this action we expand {\rm e}^{\operatorname{tr... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.028629692271351814,
0.06342484056949615,
-0.023990340530872345,
-0.008782722987234592,
0.003790458431467414,
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0.04001442342996597,
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0.007256620097905397,
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-0.030048968270421028,
-0.01697026565670967,
... |
2053bdda57f012aaba914c1975afa5e4c8ecc364 | subsection | 217 | 285 | Body | For \mathbf {m}\in \mathbb {Z}_{++}^r, as (REF ) let\tilde{\Phi }_\mathbf {m}^{(2)}(t_1,\ldots ,t_r)=\frac{\prod \limits _{i<j}(m_i-m_j-i+j)}{\prod \limits _{i=1}^r (m_i+r-i)!}
\frac{\det \big (\big (t_i^{m_j+r-j}\big )_{i,j}\big )}{\det \big (\big (t_i^{r-j}\big )_{i,j}\big )}be the renormalized Schur polynomial, so t... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.0677541196346283,
0.007015755865722895,
-0.03607448935508728,
0.005867445841431618,
0.013451633974909782,
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0.04935063421726227,
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0.00442919647321105,
-0.03500629588961601,
0.0202... |
be96bf2cc2203d0f47549b90935fa2aaf73ed8e2 | subsection | 218 | 285 | Body | Then the following holds.Lemma 5.16For x\in \operatorname{Sym}(2r,\mathbb {C}),
\tilde{\Phi }_\mathbf {m}^{(2)\prime }\left(\left(x\begin{pmatrix}0&I\\-I&0\end{pmatrix}\right)^2\right)
\in \mathcal {P}_{\mathbf {m}^2}(\operatorname{Sym}(2r,\mathbb {C}))^{\operatorname{Sp}(r,\mathbb {C})}\simeq \big (V_{(2\mathbf {m})^... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1006/jfan.2002.3957",
"end": 819,
"openalex_id": "https://openalex.org/W2088015995",
"raw": "Zhang G., Branching coefficients of holomorphic representations and Segal–Bargmann transform, J. Funct. Anal. 195 (2002), 306–349, arXiv:math.RT... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04080134630203247,
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0.010246112011373043,
-0.01791352964937687,
-0... |
06c3b6f9ca64552bfe2b11db407490ba15e8dd08 | subsection | 219 | 285 | Body | (2) Follows from with (G,K,H,L)=(\operatorname{SO}^*(4r),U(2r),\operatorname{SO}(2r,\mathbb {C}),\operatorname{SO}(2r)).[Proof of Lemma REF ]
We define two linear maps \alpha , \beta by&\alpha \colon \ \mathcal {P}(M(r,\mathbb {C}))\boxtimes \overline{\mathcal {P}(M(r,\mathbb {C}))\otimes \mathcal {P}(M(r,\mathbb {C}))... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1006/jfan.2002.3957",
"end": 121,
"openalex_id": "https://openalex.org/W2088015995",
"raw": "Zhang G., Branching coefficients of holomorphic representations and Segal–Bargmann transform, J. Funct. Anal. 195 (2002), 306–349, arXiv:math.RT... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04627123847603798,
0.013589886948466301,
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-0.000894197029992938,
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0.00539856543764472,
0.00953810103237629,
0... |
3c5e61e55ba0f28257c95d0ba38f972c4aae266c | subsection | 220 | 285 | Body | Therefore it holds that\alpha \big (\mathbf {K}_{\mathbf {m},\mathbf {l}}^{(2)}(x,x;yx^*y,w)\big )=\mathbf {K}_{\mathbf {m},\mathbf {l}}^{(2)}\big (I,I;y^2,I\big )\in \overline{\mathbb {C}\tilde{\Phi }_\mathbf {m}^{(2)}\big (y^2\big )}.Next, as a function of x, under the action of K_L=\Delta U(r)\subset K_{xL}\times K_... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.002904856577515602,
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0.011344771832227707,
0.002807582961395383,
-0.016982825472950935,
... |
2e9d9c523f4dbeaf3955aea8758a360a3bd27a8a | subsection | 221 | 285 | Body | Hence it holds that\mathbf {K}_{\mathbf {m},\mathbf {l}}^{(2)}(x,x;y,y)\in {\left\lbrace \begin{array}{ll}
\big (\mathcal {P}_{\mathbf {m}+\mathbf {l}}(M(r,\mathbb {C}))_x\boxtimes \overline{\mathcal {P}_{\mathbf {m}+\mathbf {l}}(M(r,\mathbb {C}))_y}\big )^{\Delta U(r)}\\
\hspace{105.0pt} {}=\mathbb {C}\mathbf {K}_{\ma... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.041717320680618286,
0.030929410830140114,
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0.0011892259353771806,
0.02473437413573265,
0.00966639444231987,
0.03... |
fe697e1bbfd829b7742df8426b4cbc839c0cbd29 | subsection | 222 | 285 | Body | We have\mathcal {K}_{\mathbf {m},\mathbf {l}}\left(\begin{pmatrix}0&I\\-I&0\end{pmatrix};y_1,\begin{pmatrix}0&I\\-I&0\end{pmatrix}\right)
\in \overline{\mathcal {P}_{\mathbf {m}^2}(\operatorname{Sym}(2r,\mathbb {C}))^{\operatorname{Sp}(r,\mathbb {C})}}\\
\qquad {}=\overline{\mathbb {C}\tilde{\Phi }_\mathbf {m}^{(2)\pri... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.0438666008412838,
0.01828538253903389,
-0.034281276166439056,
-0.04649187996983528,
-0.005036874674260616,
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0.023184888064861298,
0.005555825307965279,
0.02590174786746502,
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0.012096131220459938,
-0.01854485645890236,
0.... |
087af6b426917f66972bb0d2c277d3ed1d99ba3e | subsection | 223 | 285 | Body | Therefore,&\mathcal {K}_{\mathbf {m},\mathbf {l}}^{(4,1)}\left(x_2;\begin{pmatrix}0&I\\I&0\end{pmatrix},
\begin{pmatrix}0&I\\I&0\end{pmatrix}x_2^*\begin{pmatrix}0&I\\I&0\end{pmatrix}\right)&&&\\
&\qquad \quad {} \in \left(V_{(2\mathbf {m}+\mathbf {l})^2}^{(2r)\vee }\otimes V_{\langle 2l\rangle }^{(2r)\vee }\right)^{O(2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.027077550068497658,
0.05388050898909569,
-0.030723484233021736,
-0.03258458897471428,
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0.03058619052171707,
0.011685297824442387,
0.011692924425005913,
0.007627478800714016,
0.024240126833319664,
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0.018443243578076363,
-0.024514716118574142,
... |
47e5f48e5e4525a8811e11be146497070a5deb93 | subsection | 224 | 285 | Body | \end{array}\right.}Therefore both \mathcal {K}_{\mathbf {m},\mathbf {l}}\left(\Big (\begin{}0&x\\ \mp {}^t\hspace{-1.0pt}x&0\end{}\Big );
\Big (\begin{}0&y\\ \pm {}^t\hspace{-1.0pt}y&0\end{}\Big ),
\Big (\begin{}0&w\\ \mp {}^t\hspace{-1.0pt}w&0\end{}\Big )\right)
and \mathbf {K}_{\mathbf {m},\mathbf {l}}^{(2)}(x,x;yx^*... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.057326626032590866,
0.01749103143811226,
-0.0398966446518898,
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-0.023061908781528473,
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0.0007340395241044462,
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0.02423713542521,
-0.007493976037949324,
0.02... |
1732076063f1910a01f8a46ceafb32703c3b4275 | subsection | 225 | 285 | Body | Finally, we have{\rm e}^{\frac{1}{2}\operatorname{tr}\left(\Big (\begin{}0&x\\ \mp {}^t\hspace{-1.0pt}x&0\end{}\Big )
\Big (\begin{}0&y\\ \pm {}^t\hspace{-1.0pt}y&0\end{}\Big )^*
\Big (\begin{}0&x\\ \mp {}^t\hspace{-1.0pt}x&0\end{}\Big )
\Big (\begin{}0&y\\ \pm {}^t\hspace{-1.0pt}y&0\end{}\right)^*\Big )}
\mathrm {K}_{... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.054394837468862534,
0.04148293286561966,
-0.006719227880239487,
-0.033088669180870056,
-0.01069124136120081,
0.020771991461515427,
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0.003937673754990101,
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0.01683431677520275,
0.0038861632347106934,
0.009302372112870216,
-0.016681695356965065,
... |
3f15ed1e036d13bd6e565cf5949909b302fbcf73 | subsection | 226 | 285 | Body | \begin{pmatrix}0&x\\ \mp {}^t\hspace{-1.0pt}x&0\end{pmatrix}
\begin{pmatrix}0&y\\ \pm {}^t\hspace{-1.0pt}y&0\end{pmatrix}\!\!{\vphantom{\biggr )}}^*\right)^{-\mu /2}\\
\quad {} \times \mathrm {K}_{(l)}^{\prime }\left(\begin{pmatrix}0&x\\ \mp {}^t\hspace{-1.0pt}x&0\end{pmatrix}\left(I
-\begin{pmatrix}0&y\\ \pm {}^t\hspa... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.028421960771083832,
0.02561485394835472,
-0.012067509815096855,
-0.032220710068941116,
-0.009359566494822502,
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0.007894989103078842,
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0.024882564321160316,
-0.027781207114458084,
... |
408406f668b211191985c3587daa4b12a541271d | subsection | 227 | 285 | Body | Thus by it holds that& \det (I-xy^*)^{-\mu }\mathbf {K}_{\langle l\rangle }^{(2)}\big (z(I-y^*x)^{-1},w\big )&&&\\
& \qquad {} =\sum _{\mathbf {m}\in \mathbb {Z}_{++}^r}\sum _\mathbf {l}\frac{(\mu )_{\mathbf {m}+\mathbf {l},2}}{(\mu )_{\langle l\rangle ,2}}\mathbf {K}_{\mathbf {m},\mathbf {l}}^{(2)}(x,z;y,w)\quad && (\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1506.05919",
"end": 1248,
"openalex_id": "https://openalex.org/W1193240370",
"raw": "Nakahama R., Norm computation and analytic continuation of vector valued holomorphic discrete series representations, J. Lie Theory 26 (2016... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.004035033285617828,
-0.039938438683748245,... |
521c19b16433a40f2ff4cd6e02fef21424488a6f | subsection | 228 | 285 | Body | We redefine the restriction map\operatorname{Rest}\colon \ \mathcal {P}(\operatorname{Sym}(2r+2,\mathbb {C}))\longrightarrow \mathcal {P}(\operatorname{Sym}(2r+1,\mathbb {C})),\\
\operatorname{Rest}\colon \ \mathcal {P}(\operatorname{Skew}(2r+2,\mathbb {C}))\longrightarrow \mathcal {P}(\operatorname{Skew}(2r+1,\mathbb ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.018535302951931953,
0.02611723355948925,
-0.008253168314695358,
0.018779389560222626,
0.0004891260759904981,
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0.005507197231054306,
0.028573352843523026,
0.001426379312761128,
0.039786066859960556,
-0.022577982395887375,
0.023340752348303795,
-0.027703795582056046,
... |
406f63f0d7866cf5da020bf4f57ec203a8ddb292 | subsection | 229 | 285 | Body | Since\operatorname{Rest}(\mathcal {P}_{2\mathbf {m}+\mathbf {l}}(\operatorname{Skew}(2r+2,\mathbb {C})))={\left\lbrace \begin{array}{ll}
\mathcal {P}_{2\mathbf {m}+\mathbf {l}}(\operatorname{Skew}(2r+1,\mathbb {C})) & (2m_{r+1}+l_{r+1}=0), \\ \lbrace 0\rbrace & (2m_{r+1}+l_{r+1}>0), \end{array}\right.}by projecting {\r... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.029654385522007942,
0.043535809963941574,
-0.00075318175368011,
-0.012516164220869541,
-0.0017675936687737703,
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-0.007261053193360567,
-0.0064182523638010025,
0.014705158770084381,
0.03606119751930237,
-0.04008833691477775... |
5bd7efa6774ce22375fe4db1e08b3982957f2239 | subsection | 230 | 285 | Body | In this case we have\operatorname{Rest}\big (\mathcal {P}_{{(m_1+l_1,m_1,m_2+l_2,m_2,\ldots ,\\\hspace{25.0pt} m_{r+1}+l_{r+1},m_{r+1})}}(\operatorname{Sym}(2r+2,\mathbb {C}))\big ) \\
\qquad {} \subset \bigoplus _{{\mathbf {n}\in \mathbb {Z}_{++}^{2r+1}\\ m_j+l_j\ge n_{2j-1}\ge m_j\\ m_j\ge n_{2j}\ge m_{j+1}+l_{j+1}}}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.0019146334379911423,
0.01842929981648922,
-0.018902236595749855,
-0.0025706433225423098,
-0.0014197655254974961,
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-0.007204666268080473,
0.012769307941198349,
-0.015042458660900593,
-0.005690503865480423,
-0.029276346787810... |
45db319955c5ef50c460920cd6525dac116468b7 | subsection | 231 | 285 | Body | Therefore when m_{r+1}=0 we have\operatorname{Rest}\otimes \overline{\operatorname{Rest}\otimes \operatorname{Proj}}\Big (\Big (
\mathcal {P}_{{(m_1+l_1,m_1,m_2+l_2,m_2,\ldots ,\;\\\hspace{30.0pt} m_r+l_r,m_r,l_{r+1},0)}}(\operatorname{Sym}(2r+2,\mathbb {C}))_{x_2}\\
\qquad \quad {} \otimes \overline{\mathcal {P}_{2\ma... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.014452182687819004,
0.03253648802638054,
-0.005963242147117853,
-0.017031295225024223,
0.017626473680138588,
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-0.0059136440977454185,
-0.003971679601818323,
-0.019183097407221794... |
8de1f0c79ebbf820015723111ee26bb1fe5da441 | subsection | 232 | 285 | Body | In this case we have\operatorname{Rest}\big (\mathcal {P}_{{(m_1+l,m_1+l-k_1,m_2+l,m_2+l-k_2,\ldots ,\;\\\hspace{45.0pt} m_{r+1}+l,m_{r+1}+l-k_{r+1})}}(\operatorname{Sym}(2r+2,\mathbb {C}))\big ) \\
\qquad {} \subset \bigoplus _{{\mathbf {n}\in \mathbb {Z}_{++}^{2r+1}\\ m_j+l\ge n_{2j-1}\ge m_j+l-k_j\\ m_j+l-k_j\ge n_{... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.008591176010668278,
0.026475470513105392,
-0.020783627405762672,
-0.017594361677765846,
0.010521519929170609,
0.0195170771330595,
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-0.00489452900364995,
-0.014664512127637863,
... |
e9a0fb784b81749553202d15262e300c0c21c22b | subsection | 233 | 285 | Body | Comparing two expansion of {\rm e}^{\frac{1}{2}\operatorname{tr}(x_2y_1^*x_2y_1^*)}\mathrm {K}_{(l,\ldots ,l,0)}^{(1)}(x_2),{\rm e}^{\frac{1}{2}\operatorname{tr}(x_2y_1^*x_2y_1^*)}\mathrm {K}_{(l,\ldots ,l,0)}^{(1)(2r+1)}(x_2)
=\operatorname{Rest}\otimes \overline{\operatorname{Rest}\otimes \operatorname{Proj}}\big ({\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.06720990687608719,
0.03763388469815254,
-0.025791266933083534,
-0.0135747529566288,
0.008645416237413883,
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0.029148710891604424,
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0.02982020005583763,
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-0.022372780367732048,
-0.04709576815366745,
0.0036... |
578765deea88c73d440c9d780639dcbc0b180e61 | subsection | 234 | 285 | Body | Now we put\varphi _{\mathbf {m},-\mathbf {l}}(\mu ):=\sum _{{\mathbf {k}\in (\mathbb {Z}_{\ge 0})^{r+1},\; |\mathbf {k}|=l\\ 0\le k_j\le l_j\\ l_{r+1}\le k_{r+1}}}
c_{\mathbf {m},\mathbf {k},\mathbf {l}}(\mu +2l+\mathbf {m}-\mathbf {l}^{\prime })_{\mathbf {l}^{\prime }-\mathbf {k}^{\prime },2}(\mu +l-r)_{l-k_{r+1}}.The... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
0.00649925647303462,
-0.01244165189564228,
-0.017621222883462906,
-0.05660150572657585,
-0.025371510535478592,
0.03570013865828514,
0.020291104912757874,
0.02743113413453102,
0.0054007903672754765,
0.008940291590988636,
-0.014623326249420643,
0.007742658723145723,
-0.03332013264298439,
0.0... |
ed13bf51504637a69a0943db75f8c59710bdb35f | subsection | 235 | 285 | Body | \sum _\mathbf {m}\!\sum _\mathbf {l}(\lambda \!+\!k\!+\!\langle l\rangle )_{\mathbf {m}{+}\mathbf {l}{-}\langle l\rangle ,2}
\big \langle {\rm e}^{\operatorname{tr}(y_1w_1^*)}I_W,\overline{\mathcal {K}_{\mathbf {m},\mathbf {l}}^{(4,1)}(x_2;y_1)} \big \rangle _{\mathcal {H}_{\lambda {+}k}(D_1,W_{(l)}^{\prime }),y_1} \\
... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04250939562916756,
0.030775826424360275,
0.0004021970962639898,
-0.046232398599386215,
0.02397065982222557,
0.04763615503907204,
0.04015962779521942,
-0.0028361212462186813,
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0.004894073586910963,
-0.032713618129491806,
0.009139672853052616,
-0.0192863866686821,
0... |
47e04809a9ae1e5bd06a5c947a127980e7f7214f | subsection | 236 | 285 | Body | Also, by Theorem REF , this continues meromorphically for all \lambda \in \mathbb {C}. Therefore we have the following.Theorem 5.17Let (G,G_1)=(\operatorname{SU}(s,s), \operatorname{Sp}(s,\mathbb {R})) with s\ge 2. Let k\in \mathbb {Z}_{\ge 0} if s is even, k=0 if s is odd,
and l\in \left\lbrace 0,\ldots ,\left\lceil \... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05032634735107422,
0.05572826787829399,
-0.0069889225997030735,
-0.04095691815018654,
0.022721629589796066,
0.006225939840078354,
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0.009018457494676113,
0.020219044759869576,
0.029527435079216957,
-0.00629460858181119,
-0.01785379834473133,
-0.015030762180685997,
0... |
45c29ed3b92848604fc518f145f41a37a5b99fd0 | subsection | 237 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{2\lambda +4k}\big (D_1,V_{(2l,0,\ldots ,0)}^{(s)\vee }\big )\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k,l}f)(x_1+x_2)=\det (x_2)^k\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{\lfloor s/2\rfloor }}
\sum _{{\mathbf {l}\in (\mathbb {Z... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.027947651222348213,
0.03429384157061577,
-0.024591492488980293,
-0.028298523277044296,
0.006136431824415922,
0.02475930005311966,
0.012051662430167198,
0.0067428285256028175,
0.011006675660610199,
0.03429384157061577,
-0.017070645466446877,
-0.02748999372124672,
0.003583081066608429,
-0... |
b44a3f9e697f7f23c3efe3d3b910838b09f7f815 | subsection | 238 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{2\lambda +4k}\big (D_1,V_{(2l,\dots ,2l,0)}^{(s)\vee }\big )\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k,l}f)(x_1+x_2)=\det (x_2)^k\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{s/2}}
\sum _{{\mathbf {l}\in (\mathbb {Z}_{\ge 0})^{s/2}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03234542906284332,
0.03197925537824631,
-0.020200636237859726,
-0.04177442565560341,
0.010291033424437046,
0.0305450689047575,
0.018507078289985657,
0.013449289835989475,
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0.025967886671423912,
-0.007121334318071604,
-0.031246904283761978,
0.009451883845031261,
-0... |
c0d26ed35dac56e7bb9307cd17ae03cc0a99f1d2 | subsection | 239 | 285 | Body | Then there exist monic polynomials \varphi _{\mathbf {m},-\mathbf {l}}(\mu )\in \mathbb {C}[\mu ] of degree l-l_{\lceil s/2\rceil } such that the linear map\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{2\lambda +4k}\big (D_1,V_{(2l,\dots ,2l,0)}^{(s)\vee }\big )\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.011935383081436157,
0.029533205553889275,
-0.032112590968608856,
-0.025778595358133316,
-0.02651120163500309,
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0.0008671086397953331,
0.003500867635011673,
-0.0003393552324268967,
0.004586422815918922,
-0.004731417633593082,
-0.018422003835439682,
-0.0336693786084651... |
2b995fa338bed9898f700894d368cca2aa54770a | subsection | 240 | 285 | Body | Here for \mathbf {l}=(l_1,\ldots ,l_{\lfloor s/2\rfloor },l_{\lceil s/2\rceil })\in (\mathbb {Z}_{\ge 0})^{\lceil s/2\rceil }, we put
\mathbf {l}^{\prime }=(l_1,\ldots ,l_{\lfloor s/2\rfloor })\in (\mathbb {Z}_{\ge 0})^{\lfloor s/2\rfloor }.Later we prove \varphi _{\mathbf {m},-\mathbf {l}}(\mu )=\left(\mu +l-\left\lfl... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.04482365772128105,
0.036432571709156036,
-0.015355688519775867,
-0.014806454069912434,
-0.008352945558726788,
0.009863341227173805,
0.024196842685341835,
0.012670541182160378,
0.022610165178775787,
0.03451025113463402,
-0.018704494461417198,
0.008993718773126602,
-0.002959765028208494,
... |
3508971918197d9ca8b5194bfbe809620b761d34 | subsection | 241 | 285 | Body | We note that the difference between \frac{\partial }{\partial x_1} in G_1=\operatorname{Sp}(s,\mathbb {R}) case and \frac{1}{2}\frac{\partial }{\partial x_1} in G_1=\operatorname{SO}^*(2s) case is caused by the difference of the normalization of the inner product on \operatorname{Sym}(s,\mathbb {C}) and \operatorname{S... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05521208792924881,
0.011735239066183567,
-0.021303502842783928,
-0.024386102333664894,
0.018327727913856506,
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0.028567448258399963,
-0.0050702644512057304,
0.06372738629579544,
0.005238128360360861,
-0.006031668744981289,
-0.028369063511490822,... |
99af99d1963c9d6a7b92c25ea5b7aaef77629068 | subsection | 242 | 285 | Body | Also we have\mathfrak {p}^+={\left\lbrace \begin{array}{ll}\mathbb {C}^n & (\text{Case }1),\\ M(1,2;\mathbb {O})^\mathbb {C}& (\text{Case }2),\\ \operatorname{Herm}(3,\mathbb {O})^\mathbb {C}& (\text{Case }3), \end{array}\right.}and \mathfrak {p}^+_1:=\fg _1^\mathbb {C}\cap \mathfrak {p}^+, \mathfrak {p}^+_2:=(\mathfra... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.009832811541855335,
0.027827847748994827,
-0.03579173982143402,
-0.015454829670488834,
0.010168454609811306,
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0.05837135389447212,
0.02788887359201908,
0.038263291120529175,
0.023769620805978775,
-0.017636509612202644,
0.013250266201794147,
-0.01161019317805767,
0.0... |
79c6d3bb6c1a83dce72a6fee18feee79ea360cd7 | subsection | 243 | 285 | Body | For x_2\in \mathfrak {p}^+_2, w_1\in \mathfrak {p}^+_1, we want to computeF_{\tau \rho }(x_2;w_1)
=\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak {p}^+_1}}I_W,
\big (h(x_2,Q(y_1)x_2)^{-\lambda /2}\mathrm {K}\big ((x_2)^{Q(y_1)x_2}\big )\big )^*\big \rangle _{\hat{\rho },y_1}.Now for y_1\in \mathfrak {p}^+_1 and x_2\in \mat... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.024120839312672615,
0.01574491150677204,
-0.023449543863534927,
0.00386757287196815,
0.008055536076426506,
0.010527120903134346,
0.040887948125600815,
0.03743993490934372,
0.0040544671937823296,
0.045525986701250076,
-0.000014481943253485952,
-0.0265161395072937,
-0.015447406098246574,
... |
20845228ec2f01c1c98b4c5f10c1f53d84a2e677 | subsection | 244 | 285 | Body | Similarly, h(x_2,Q(y_1)x_2)^{-\lambda /2}=h_2(x_2,Q(y_1)x_2)^{-\lambda /2} is given byh_2(x_2,Q(y_1)x_2)^{-\lambda /2}&={\left\lbrace \begin{array}{ll}
\big (1-2q(x_2,-q(\overline{y_1})x_2)+q(x_2)q(-q(\overline{y_1})x_2)\big )^{-\lambda /2} & (\text{Case }1),\\
\det \big (I_2+J_2{}^t\hspace{-1.0pt}x_2(y_1^*J_2\overline... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.07444137334823608,
0.028037549927830696,
-0.026451094076037407,
-0.039417315274477005,
-0.0024082851596176624,
-0.004057664889842272,
0.017908642068505287,
-0.006700485944747925,
0.013332327827811241,
0.028785014525055885,
-0.021249350160360336,
-0.013263682834804058,
-0.01827474683523178... |
ccbab47d0057e32490dbb241dbb3acecd5a36a42 | subsection | 245 | 285 | Body | Also, if n^{\prime \prime }=2 we do not assume k_1\ge k_2 (see Section REF ). Then \mathrm {K}(x_2)=\mathrm {K}_\mathbf {k}^{(d_2)}(x_2) is homogeneous of degree |\mathbf {k}|, where \mathbf {k}=(k_1,k_2) for Cases 1, 2, and \mathbf {k}=(k,0) for Case 3. Also, we have\big (\tfrac{1}{2}(x_2|Q(y_1)x_2)_{\mathfrak {p}^+}\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05283712223172188,
0.01304139755666256,
-0.01965748518705368,
-0.048655327409505844,
-0.04071907326579094,
0.00525776669383049,
0.023457728326320648,
-0.008233861066401005,
-0.0010139705846086144,
0.027639521285891533,
-0.0019354152027517557,
-0.018314426764845848,
-0.016559293493628502,
... |
c9e85cb0a00aecaf55453ffadfe7fb6a003276a0 | subsection | 246 | 285 | Body | This space is computed as\Big (\bigoplus _{{\mathbf {m}\in \mathbb {Z}_{++}^2\\ |\mathbf {m}|=2m+|\mathbf {k}|}}\chi _1^{-|\mathbf {m}|}\boxtimes \mathbb {C}^{[n^{\prime }]}\boxtimes V_{(m_1-m_2,0,\ldots ,0)}^{[n^{\prime \prime }]\vee } \\
\qquad \quad {} \otimes \overline{\bigoplus _{{\mathbf {n}\in \mathbb {Z}_{++}^2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.010995502583682537,
0.021350044757127762,
-0.032139524817466736,
-0.014040058478713036,
-0.014154515229165554,
0.018221553415060043,
0.026065673679113388,
0.024768494069576263,
0.018999861553311348,
0.0011359857162460685,
-0.015161736868321896,
0.005554979667067528,
-0.037145111709833145,... |
60494bca5702e573d8a59b3427eb65ec0c49f5c2 | subsection | 247 | 285 | Body | Therefore by using the results of and (REF ) we haveF_{\tau \rho }(x_2;w_1) ={\left\lbrace \begin{array}{ll}
\displaystyle \sum _{m=0}^{\infty }\frac{(\lambda +k_1+k_2)_m}{(\lambda +k_1+k_2)_{(m,m),n^{\prime }-2}}\frac{1}{m!}& \\
\quad {}\times \big (\tfrac{1}{2}(x_2|Q(w_1)x_2)_{\mathfrak {p}^+}\big )^m
\mathrm {K}_{(k... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.029382694512605667,
0.026957020163536072,
-0.017803533002734184,
-0.03496631979942322,
-0.010625673457980156,
0.026407809928059578,
0.027399437502026558,
0.04912371560931206,
0.00837544072419405,
0.056385479867458344,
-0.007376185152679682,
-0.0009620696655474603,
-0.010450230911374092,
... |
5aee81d04654d0ee1baabd9b022452687390811c | subsection | 248 | 285 | Body | \end{array}\right.}By Theorem REF , by substituting w_1 with \overline{\frac{\partial }{\partial x_1}}, we get the intertwining operator from (\mathcal {H}_1)_{\tilde{K}_1} to \mathcal {H}_{\tilde{K}}, and by Theorem REF , this extends to the intertwining operator between the spaces of all holomorphic functions if \mat... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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0.040791936218738556,
0.007543609477579594,
-0.027214964851737022,
0.023813093081116676,
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0.03600185737013817,
0.0030624461360275745,
-0.057450421154499054,
-0.009496252983808517,
... |
b080fabed7007f2fa32968b0c40e5ed7f578454d | subsection | 249 | 285 | Body | Then the linear map
\mathcal {F}_{\lambda ,k_1,k_2}\colon \ \mathcal {O}_{\lambda }\big (D_1,V_{(0,0;-k_2,-k_1,-k_1-k_2,-k_1-k_2)}^{(2,4)\vee }\big )
\boxtimes V_{(k_1-k_2,0)}^{(2)\vee }\rightarrow \mathcal {O}_\lambda (D), \\
(\mathcal {F}_{\lambda ,k_1,k_2}f)(x_1,x_2)\\
=\sum _{m=0}^\infty \frac{1}{(\lambda +k_1+k_2... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05758458375930786,
0.02538970485329628,
-0.0167993176728487,
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-0.04958926886320114,
0.010421374812722206,
-0.011... |
141a169ed0bb3e04f277bdc7a849e0e117b86fa3 | subsection | 250 | 285 | Behavior of | Behavior of \mathcal {F}_{\tau \rho } when \lambda is a poleIn this section, we look at the behavior of \mathcal {F}_\lambda when \lambda is a pole.
For simplicity we only treat the case that both G and G_1 are classical and both \mathcal {H} and \mathcal {H}_1 are of scalar type.
In this case, the underlying (\fg ,\ti... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05676979199051857,
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-0.010735900141298771,
-0... |
4e6d3bb5acb7b9e6af791da51574dec1df8aee38 | subsection | 251 | 285 | Behavior of | \end{array}\right.}For these \lambda and for i=1,2,\ldots ,r, letM_i(\lambda )=M_i^\fg (\lambda ):=\bigoplus _{{\mathbf {m}\in \mathbb {Z}_{++}^{r}\\ m_i\le \frac{d}{2}(i-1)-\lambda }}\mathcal {P}_\mathbf {m}(\mathfrak {p}^+).Also, since \mathfrak {so}(2,1)\simeq \mathfrak {sl}(2,\mathbb {R}) and \mathcal {O}_{\lambda ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05043958127498627,
0.027477823197841644,
-0.020734237506985664,
-0.017621813341975212,
0.034053582698106766,
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0.0021683983504772186,
-0.04784589260816574,... |
7d007f50e7cc62c8caf359c6c1b7a8a11f5a44fa | subsection | 252 | 285 | Behavior of | \end{array}\right.}Then the composition series are given by, when G=\operatorname{Sp}(r,\mathbb {R}) with r\ge 2,&\mathcal {O}_\lambda (D)_{\tilde{K}}\supset M_{2\left\lceil \frac{r}{2}\right\rceil -1}(\lambda )\supset M_{2\left\lceil \frac{r}{2}\right\rceil -3}(\lambda )
\supset \cdots \supset M_{\max \lbrace 2\lambda... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.05892350152134895,
0.022168785333633423,
-0.01070296112447977,
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0.03640379756689072,
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-0.006202072370797396,
-0.030895931646227837,
0... |
6c9b1b6b69bbeeb0c05eb3999a3642309545fd49 | subsection | 253 | 285 | Behavior of | For G=U(q,s) case, we use the same symbol M_i(\lambda _1+\lambda _2)\subset \mathcal {O}_{\lambda _1+\lambda _2}(D)_{\tilde{K}} as in the G=\operatorname{SU}(q,s) case. We also write M_0(\lambda )=M_{-1}(\lambda )=\lbrace 0\rbrace , M_{r+1}(\lambda )=M_{r+2}(\lambda )=\mathcal {O}_\lambda (D)_{\tilde{K}}. We note that ... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03597429022192955,
0.039879895746707916,
-0.04049014672636986,
-0.02114519290626049,
0.009169019758701324,
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0.045158565044403076,
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-0.028513971716165543,
0.007403105963021517,
... |
7646035791af3ff06aabbe96199007e3c60a40b0 | subsection | 254 | 285 | Behavior of | Then the intertwining operator\mathcal {F}_{\lambda ,k,l}\colon \ \mathcal {O}_{(\lambda _1+k)+(\lambda _2+l)}(D_{11}) \mathbin {\hat{\boxtimes }}\mathcal {O}_{(\lambda _1+l)+(\lambda _2+k)}(D_{22})
\rightarrow \mathcal {O}_{\lambda _1+\lambda _2}(D)is given by (REF ),(\mathcal {F}_{\lambda ,k,l}f)\begin{pmatrix}x_{11}... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.013064541853964329,
-0.016544349491596222,
-0.034767553210258484,
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0.019886797294020653,
-0.03363814204931259,
-0.006833701394498348,
0.002653734991326928,
... |
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