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6137350aaecb531aefb08c804f74e50888e05d96 | subsection | 55 | 285 | Differential expression | Then we have(\mathcal {F}_{\tau \rho }^*f)(y_1)&=\frac{1}{\pi ^n}\int _{\mathfrak {p}^+}F_{\tau \rho }^*(z_1,z_2){\rm e}^{(y_1|z)_{\mathfrak {p}^+}}f(z){\rm e}^{-|z|_{\mathfrak {p}^+}^2}{\rm d}z\\
&=\frac{1}{\pi ^n}\left.\int _{\mathfrak {p}^+}F_{\tau \rho }^*(z_1,z_2){\rm e}^{(x|z)_{\mathfrak {p}^+}}f(z){\rm e}^{-|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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936cc3a003846140640657b9e9314cbaa4f4f195 | subsection | 56 | 285 | Differential expression | Assume G is simple. Then h(x_2,y_1)^2=h(Q(x_2)y_1,y_1)=h(x_2,Q(y_1)x_2).Lemma 3.9
Let x,y\in \mathfrak {p}^+.B(-x,y)B(x,y)=B(Q(x)y,y)=B(x,Q(y)x).
x^y=(Q(x)y)^y+x^{Q(y)x}.(1) Use and B(x,-y)=B(-x,y), Q(-x)=Q(x).(2) Both sides are computed as({\rm l.h.s.})&=B(x,y)^{-1}(x-Q(x)y),\\
({\rm r.h.s.})&=B(Q(x)y,y)^{-1}(Q(x)y-... | {
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"doi": "10.1007/978-1-4612-1366-6",
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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. 185... | 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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83c15b4f5f5dbed9d274cd04ce5aa4b7a8168864 | subsection | 57 | 285 | Differential expression | Therefore\operatorname{Proj}_2((x_2)^{y_1})=\operatorname{Proj}_2\big ((Q(x_2)y_1)^{y_1}+(x_2)^{Q(y_1)x_2}\big )=(x_2)^{Q(y_1)x_2}.(2) We extend \sigma on \fg ^\mathbb {C} holomorphically. Then since \sigma acts by +1 on \mathfrak {p}^+_1 and -1 on \mathfrak {p}^+_2,
B(-x_2,y_1)=\sigma B(x_2,y_1)\sigma holds. Therefore... | {
"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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b0d7696e07568b6a48a6c3d446ffbb213da39a66 | subsection | 58 | 285 | Differential expression | Let \mathrm {K}(x_2)\in \mathcal {P}(\mathfrak {p}^+_2)\otimes V\otimes \overline{W}\simeq \mathcal {P}(\mathfrak {p}^+_2,\operatorname{Hom}(W,V)) be an operator-valued polynomial
satisfying (REF ).Assume \mathcal {H}_\tau (D,V)_{\tilde{K}}=\mathcal {P}(\mathfrak {p}^+,V).
We define F_{\tau \rho }^*(z)\in \mathcal {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 | [
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0db68ac57b6b4c0c334a9be6e9bd32d1c52934d6 | subsection | 59 | 285 | Differential expression | Then the linear map
\mathcal {F}_{\tau \rho }\colon \mathcal {H}_\rho (D_1,W)_{\tilde{K}_1}\rightarrow \mathcal {O}_\tau (D,V)_{\tilde{K}}, \qquad (\mathcal {F}_{\tau \rho }f)(x) =F_{\tau \rho }\left(x_2;\overline{\frac{\partial }{\partial x_1}}\right)f(x_1)
intertwines the (\fg _1,\tilde{K}_1)-action.Remark 3.11 For... | {
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"raw": "Dib H., Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. 69 (1990), 403–448.",
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{
... | 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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217ee784c9d291e7587e013154290b036673f055 | subsection | 60 | 285 | Differential expression | Then for w_1\in \mathfrak {p}^+_1, \mathcal {B}_\tau (w_1) annihilates F_{\tau \rho }^*(z), because(\mathcal {B}_\tau (w_1))_zF_{\tau \rho }^*(z)
=(\mathcal {B}_\tau (w_1))_z\big \langle {\rm e}^{(x|z)_{\mathfrak {p}^+}}I_V,\mathrm {K}(\operatorname{Proj}_2(x))\big \rangle _{\hat{\tau },x} \\
=\big \langle ((Q(x)w_1|z)... | {
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"raw": "Kobayashi T., Pevzner M., Differential symmetry breaking operators: I. General theory and F-method, Selecta Math. (N.S.) 22 (2016), 801–845,... | 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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a8aaccb5d98c275f3c77824b6e37010cb2acbb1f | subsection | 61 | 285 | Differential expression | \mathcal {H}_\rho (D_1,W)_{\tilde{K}_1},
but in fact these are well-defined as maps between
\mathcal {O}_\tau (D,V) and \mathcal {O}_\rho (D_1,W) in the following sense.Theorem 3.12\mathcal {F}_{\tau \rho }^* is well-defined as the map \mathcal {F}_{\tau \rho }^*\colon \mathcal {O}_\tau (D,V)\rightarrow \mathcal {O}_\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 | [
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2ad5c3ec7572f7efffcefade3c9bd8f59d515fed | subsection | 62 | 285 | Differential expression | Especially \mathcal {F}_{\tau \rho } is well-defined as the map
\mathcal {F}_{\tau \rho }\colon \mathcal {O}_\rho (D_1,W)\rightarrow \mathcal {O}_\tau (D,V).(1) Clear since \mathcal {F}_{\tau \rho }^* is a finite-order differential operator.(2) First we decompose F_{\tau \rho }(x_2;w_1) as the sum of homogeneous polyno... | {
"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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25b193ad7ea452aa04a3e39532c4b77da6c99a75 | subsection | 63 | 285 | Differential expression | Then for v\in V we have\left|\left(F_n\left(x_2;\overline{\frac{\partial }{\partial x_1}}\right)f(x_1),v\right)_\tau \right| =\left|\left(\chi _\rho \big ({\rm e}^{t\hbar }\big )\chi _\tau \big ({\rm e}^{-t\hbar }\big )
F_n\left({\rm e}^tx_2;\overline{{\rm e}^{-t}\frac{\partial }{\partial x_1}}\right)f(x_1),v\right)_\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 | [
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78cb31f70950c33e6ac4afd35167b505779bbb26 | subsection | 64 | 285 | Differential expression | Then\left|\left(y_1\,|\,\overline{\frac{\partial }{\partial x_1}}\right)_{\mathfrak {p}^+_1}^nf(x_1)\right|_\rho &=\left|\left.\frac{{\rm d}^n}{{\rm d}t^n}\right|_{t=0}f(x_1+ty_1)\right|_\rho =\left|\frac{n!}{2\pi \sqrt{-1}}\oint _{|z|=R}\frac{f(x_1+zy_1)}{z^{n+1}}{\rm d}z\right|_\rho \\
&=\left|\frac{n!}{2\pi \sqrt{-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 | [
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d7986b05455f0f6c82c40e7ddb62c039f80b0b7a | subsection | 65 | 285 | Differential expression | Hence we have\left\Vert \left(y_1\,|\overline{\frac{\partial }{\partial x_1}}\right)_{\mathfrak {p}^+_1}^nf(x_1)\right\Vert _{\hat{\rho },y_1}^2 \\
\qquad {} =C_\rho \int _{D_1} \left(\rho \big (B(x_1)^{-1}\big )\left(y_1\,|\,\overline{\frac{\partial }{\partial x_1}}\right)_{\mathfrak {p}^+_1}^nf(x_1)
,\left(y_1\,|\,\o... | {
"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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80728942446ccae4634567c9ee2251e8f5668891 | subsection | 66 | 285 | Differential expression | By Theorem REF , \mathcal {F}_{\tau \rho } is a continuous operator from \mathcal {O}_\rho (D_1,W) to \mathcal {O}_\tau (D,V),
and therefore \mathcal {F}_{\tau \rho } f(x) must extend to whole D. | {
"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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681feec6aa41fe048dc1ecec097fe0c4326cb932 | subsection | 67 | 285 | Analytic continuation of intertwining operators | In this subsection we assume G to be simple and that (\tau ,V) is of the form (\tau ,V)=\big (\tau _0\otimes \chi ^{-\lambda },V\big ).
Then we may assume (\rho ,W) is also of the form (\rho ,W)=\big (\rho _0\otimes \chi |_{\tilde{K}_1}^{-\lambda },W\big ).
In this section we denote the representation of \tilde{G}_1
wi... | {
"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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2d3bcebf8e2bf41f60760a446f6f3d6c536f1d5a | subsection | 68 | 285 | Analytic continuation of intertwining operators | In this subsection we write \mathcal {F}_{\tau \rho }=\mathcal {F}_{\lambda ,\rho }.To do this, we consider the \tilde{K}_1-type decomposition of \mathcal {O}_\lambda (D_1,W)_{\tilde{K}_1} as\mathcal {O}_\lambda (D_1,W)_{\tilde{K}_1}&\simeq \mathcal {P}(\mathfrak {p}^+_1,W)\otimes (\chi |_{\tilde{K}_1})^{-\lambda }
\si... | {
"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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acf5e87db26b4bbf7067160c7b0f6bf6fd1a0d09 | subsection | 69 | 285 | Analytic continuation of intertwining operators | We note that \hat{\mathrm {K}}_{m,j}(x_2;y_1)^* is non-zero only if W_{m,j} appears commonly in the decomposition of both
\mathcal {P}(\mathfrak {p}_2^+,V|_{\tilde{K}_1}) and \mathcal {P}(\mathfrak {p}^+_1,W) since \hat{\mathrm {K}}_{m,j}(x_2;y_1)^* is \tilde{K}_1^\mathbb {C}-invariant.
We also note that if both \mathc... | {
"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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c8aa3a0563b67cb104f650514e2c96362771a601 | subsection | 70 | 285 | Analytic continuation of intertwining operators | In fact, this is well-defined as a map from \mathcal {O}_\lambda (D_1,W) to \mathcal {O}_\lambda (D,V) under some assumption.Theorem 3.13
Assume (REF ), (REF ) holds, and also assume that
for any \lambda \in \mathbb {C} which is not a pole of p_{m,j}(\lambda ), p_{m,j}(\lambda )q_{m,j}(\lambda ),
there exist \mu >p_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 | [
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54fddf786da19c084dfaf06b23a614afd0b991d9 | subsection | 71 | 285 | Analytic continuation of intertwining operators | As in the holomorphic discrete series case,
for f\in \mathcal {O}_\lambda (D_1,W)_{\tilde{K}_1}=\mathcal {P}(\mathfrak {p}^+_1,W), if |x|_\infty <{\rm e}^{-t} then((\mathcal {F}_{\lambda ,\rho }f)(x),v)_\tau &=\big \langle f(y_1),\hat{\mathrm {K}}_\lambda (x;y_1)^*v\big \rangle _{\lambda ,\rho _0,y_1} \\
&=\chi _\rho \... | {
"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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209a7b0bacc1f90981d9c4611b5550fc166e03af | subsection | 72 | 285 | Analytic continuation of intertwining operators | Hence for 1<s<{\rm e}^t,\chi _\rho \big ({\rm e}^{-t\hbar }\big )\chi _\tau \big ({\rm e}^{t\hbar }\big )|((\mathcal {F}_{\lambda ,\rho }f)(x),v)_\tau | \\
\qquad {}\le \sum _{m=0}^\infty \sum _{j=1}^{N_m}
|p_{m,j}(\lambda )|\big \Vert f_{m,j}\big ({\rm e}^{-t}y_1\big )\big \Vert _{F,\rho _0,y_1}\big \Vert \hat{\mathrm... | {
"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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-0.03401976078748703,
-0.018154041841626167,
-0.01920667104423046... |
5442667ddc136ee5f4553095b349ce96ba2ab43e | subsection | 73 | 285 | Analytic continuation of intertwining operators | By (REF ), as in (REF ),F_{\lambda ,\rho }(x_2;w_1)
&=\sum _{m=0}^\infty \sum _{j=1}^{N_m}p_{m,j}(\lambda )q_{m,j}(\lambda )\hat{\mathrm {K}}_{m,j}(x_2;w_1) \\
&=\chi _\rho \big ({\rm e}^{t\hbar }\big )\chi _\tau \big ({\rm e}^{-t\hbar }\big )
\sum _{m=0}^\infty \sum _{j=1}^{N_m}p_{m,j}(\lambda )q_{m,j}(\lambda )\hat{\... | {
"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 | [
-0.03945666551589966,
0.029706932604312897,
-0.013106448575854301,
-0.03545912355184555,
-0.006866008974611759,
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0.04400349035859108,
-0.023725874722003937,
0.007335186470299959,
-0.03381127864122391,
0.0... |
ba9b53ba51db0ab960a1f4172763c1eb578a103b | subsection | 74 | 285 | Analytic continuation of intertwining operators | Therefore we have|(\mathcal {F}_{\lambda ,\rho }f(x),v)_\tau |\\
\le C\chi _\rho \big ({\rm e}^{t\hbar }\big )\chi _\tau \big ({\rm e}^{-t\hbar }\big )
\sum _{m=0}^\infty \sum _{j=1}^{N_m}\big (1+m^k\big )p_{m,j}(\mu )q_{m,j}(\mu )
\left|\left(\hat{\mathrm {K}}_{m,j}\left({\rm e}^tx_2;{\rm e}^{-t}\overline{\frac{\parti... | {
"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 | [
-0.0331050381064415,
0.013890385627746582,
-0.019954556599259377,
-0.024363476783037186,
0.014050571247935295,
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0.027704492211341858,
-0.03362373262643814,
-0.019527394324541092,
-0.008459328673779964,
-... |
314fbf4914a7f5cda94d4fa6344165bec4ccb146 | subsection | 75 | 285 | Parametrization of representations of classical | Parametrization of representations of classical K^\mathbb {C}In this subsection we fix the realization of root systems and parametrization of irreducible finite-dimensional representations of K^\mathbb {C}
when it is classical.
First we set K^\mathbb {C}:={\rm GL}(r,\mathbb {C}) or \operatorname{SO}(n,\mathbb {C}). | {
"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 | [
-0.0493624173104763,
0.01547533180564642,
-0.0008656728314235806,
0.010578750632703304,
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0.03621334582567215,
0.020242253318428993,
-0.01010587252676487,
0.023903246968984604,
0.0346... |
3a40e2034274e69d1127d8cd92de88165ee78b81 | subsection | 76 | 285 | Parametrization of representations of classical | We take a Cartan subalgebra \mathfrak {h}^\mathbb {C}\subset \mathfrak {k}^\mathbb {C},
and take a basis \lbrace t_1,\ldots ,t_r\rbrace \subset \mathfrak {h}^\mathbb {C}, with the dual basis \lbrace \varepsilon _1,\ldots ,\varepsilon _r\rbrace \subset \big (\mathfrak {h}^\mathbb {C}\big )^\vee ,
where r=\big \lfloor \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 | [
-0.0359894335269928,
0.012751981616020203,
-0.037057820707559586,
-0.008921044878661633,
0.010973877273499966,
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0.033730555325746536,
0.004418550059199333,
0.003836659947410226,
-0.0203604344278574,
0.03... |
c02910b4c5469ece4da82e948d06e6415506534a | subsection | 77 | 285 | Parametrization of representations of classical | We omit the superscript (r) and [n] if there is no confusion.Next we set G:=\operatorname{Sp}(r,\mathbb {R}), U(q,s), \operatorname{SO}^*(2s), or \operatorname{SO}_0(2,n), and let K^\mathbb {C} be the complexification of their maximal compact subgroups,
that is, K^\mathbb {C}={\rm GL}(r,\mathbb {C}), {\rm GL}(q,\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 | [
-0.04963144287467003,
-0.004296202678233385,
-0.0028177136555314064,
-0.00494101457297802,
0.005482809152454138,
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0.021397072821855545,
0.051401812583208084,
0.012072100304067135,
-0.0064633809961378574,
0.016879573464393616,... |
c2d8cd5e6dfe19f33df2ce41aac4e56da4859ac1 | subsection | 78 | 285 | Parametrization of representations of classical | Also, under the suitable ordering of \Delta \big (\fg ^\mathbb {C},\mathfrak {h}^\mathbb {C}\big ), \mathcal {P}_\mathbf {m}(\mathfrak {p}^+) in Theorem REF is given by\mathcal {P}_\mathbf {m}(\mathfrak {p}^+)\simeq {\left\lbrace \begin{array}{ll}
V_{(2m_1,2m_2,\ldots ,2m_r)}^{(r)\vee }=:V_{2\mathbf {m}}^{(r)\vee }, & ... | {
"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 | [
-0.050249043852090836,
-0.0006545227952301502,
-0.04869306460022926,
-0.046435363590717316,
-0.004774726927280426,
0.013980950228869915,
0.0028450051322579384,
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0.03224847465753555,
-0.016108984127640724,
0.013584327884018421,
-0.03386547788977623,
... |
e0d2f847646371ee26d0d2a2f46236f42e4ff398 | subsection | 79 | 285 | Parametrization of representations of classical | \end{array}\right.}We have the local isomorphism \operatorname{SO}^*(6)\simeq \operatorname{SU}(1,3). Accordingly, we identify the representationV_{(m_1,m_2,m_3)}^{(3)\vee }=V_{\frac{1}{3}(2m_1-m_2-m_3,-m_1+2m_2-m_3,-m_1-m_2+2m_3)}^{(3)\vee }\otimes \chi _{\operatorname{SO}^*(6)}^{-\frac{2}{3}|\mathbf {m}|}of U(3)\subs... | {
"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 | [
-0.014346164651215076,
0.0040863677859306335,
-0.016101280227303505,
-0.02394588477909565,
-0.0032164405565708876,
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0.020893510431051254,
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0.016437042504549026,
0.019352059811353683,
0.010614635422825813,
0.005215746816247702,
0.021442938596010208,
... |
d6606cbcc4e7e7ff011467261d4f5b5a3687f58e | subsection | 80 | 285 | Explicit realization of classical groups and bounded symmetric domains | In this subsection, we review and fix the explicit realization of groupsG=\operatorname{Sp}(r,\mathbb {R}),\; U(q,s),\; \operatorname{SO}^*(2s),\; \operatorname{SO}_0(2,n).First we deal with G=\operatorname{Sp}(r,\mathbb {R}), U(q,s), and \operatorname{SO}^*(2s). For these groups we have(r,n,d,p)={\left\lbrace \begin{a... | {
"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 | [
-0.028889812529087067,
0.03287303447723389,
-0.01803896389901638,
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-0.014528843574225903,
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0.04349496215581894,
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0.02756207063794136,
0.004856937564909458,
-0.... |
6d63702e6c675266fbf0b7ee78e37de66f6cf632 | subsection | 81 | 285 | Explicit realization of classical groups and bounded symmetric domains | We embed K into G as& k \mapsto \begin{pmatrix}k&0\\0&{}^t\hspace{-1.0pt}k^{-1}\end{pmatrix}, \qquad & & G=\operatorname{Sp}(r,\mathbb {R}),\; \operatorname{SO}^*(2s),& \\
& (k_1,k_2) \mapsto \begin{pmatrix}k_1&0\\0&k_2\end{pmatrix}, \qquad & &G=U(q,s).&Clearly these extend to the embeddings of complexified Lie groups ... | {
"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 | [
-0.027102384716272354,
0.0212881900370121,
-0.01825137995183468,
0.008004054427146912,
0.005657775327563286,
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-0.009987900033593178,
0.039860039949417114,
0.0017244197661057115,
0.0019037289312109351,
-0.02177652157843113,
... |
e534683169822e25a6535633f7adff768071066e | subsection | 82 | 285 | Explicit realization of classical groups and bounded symmetric domains | Then the rational action of G on \mathfrak {p}^+ is given by\begin{pmatrix}a&b\\c&d\end{pmatrix}x=(ax+b)(cx+d)^{-1},\qquad \begin{pmatrix}a&b\\c&d\end{pmatrix}\in G,\quad x\in \mathfrak {p}^+ .The Jordan triple system structure on \mathfrak {p}^+ is given byQ(x)y=xy^*x, \qquad x,y\in \mathfrak {p}^+,the inner product (... | {
"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 | [
-0.060009025037288666,
0.010042201727628708,
-0.024250850081443787,
0.0007707161130383611,
0.001047334517352283,
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0.06617475301027298,
0.03617024049162865,
0.009775122627615929,
0.04804384708404541,
-0.026845339685678482,
-0.002604028442874551,
-0.016574211418628693,
... |
db868b899d12c5f615105ed2a1783f827cbef453 | subsection | 83 | 285 | Explicit realization of classical groups and bounded symmetric domains | Then \tilde{G} acts on \mathcal {O}(D,V) as\hat{\tau }\left(\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}\right)f(w)
=\tau \big (a^*+xb^*,(cx+d)^{-1}\big )f\big ((ax+b)(cx+d)^{-1}\big ),where we regard \big (a^*+xb^*,(cx+d)^{-1}\big ) as the lift on \tilde{K}^\mathbb {C}, and this action preserves the inner product\langle ... | {
"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 | [
-0.027553053572773933,
0.030039846897125244,
-0.004908746108412743,
0.005522815976291895,
0.03768330067396164,
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0.013898578472435474,
0.04738637059926987,
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0.045799706131219864,
-0.01553864125162363,
-0.005843200255185366,
0.00007002447819104418,
... |
e359f7a3daf314daf4a871928d443843fe8e8c7b | subsection | 84 | 285 | Explicit realization of classical groups and bounded symmetric domains | In this case, we have(r,n,d,p)=(2,n,n-2,n).We realize this group as\operatorname{SO}_0(2,n):=\left\lbrace g\in {\rm SL}(2+n,\mathbb {R})\colon g\begin{pmatrix}I_2&0\\0&-I_n\end{pmatrix}{}^t\hspace{-1.0pt}g=\begin{pmatrix}I_2&0\\0&-I_n\end{pmatrix}
\right\rbrace _0as usual, where the subscript 0 means the identity compo... | {
"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 | [
-0.017701003700494766,
0.0001362624461762607,
-0.01905909925699234,
0.017716264352202415,
-0.000167377496836707,
-0.017243219539523125,
0.04571742191910744,
0.016052979975938797,
-0.029511846601963043,
0.039552588015794754,
-0.001998992869630456,
-0.010139929130673409,
-0.0145651800557971,
... |
3575479083a60b5035823b39764baf66fcb65327 | subsection | 85 | 285 | Explicit realization of classical groups and bounded symmetric domains | For x={}^t\hspace{-1.0pt}(x_1,\ldots ,x_n),y={}^t\hspace{-1.0pt}(y_1,\ldots ,y_n)\in \mathfrak {p}^+, we writeq(x):=x_1^2+\cdots +x_n^2,\qquad q(x,y):=x_1y_1+\cdots +x_ny_n.Then the Jordan triple system structure on \mathfrak {p}^+ is given byQ(x)y=2q(x,\overline{y})x-q(x)\overline{y}, \qquad x,y\in \mathfrak {p}^+,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 | [
-0.07104186713695526,
0.007980003021657467,
-0.052060749381780624,
-0.02641182765364647,
0.02702215313911438,
0.007423081435263157,
0.03649745136499405,
0.03500215709209442,
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0.05172506719827652,
-0.030241617932915688,
0.03417821601033211,
-0.028334351256489754,
0.0116... |
b3901cd8e6854d86b12f2e77d9226a6dab519664 | subsection | 86 | 285 | Explicit realization of classical groups and bounded symmetric domains | However, for convenience, we use the same inner product as (REF ), so that\mathcal {H}_\lambda (D_{\operatorname{SO}_0(2,1)})\simeq \mathcal {H}_{2\lambda }(D_{{\rm SL}(2,\mathbb {R})}),\qquad \mathcal {H}_\lambda (D_{\operatorname{SO}_0(2,2)})\simeq \mathcal {H}_\lambda (D_{{\rm SL}(2,\mathbb {R})}) \mathbin {\hat{\bo... | {
"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 | [
-0.048188529908657074,
0.03292933851480484,
-0.0018911862280219793,
-0.00907921977341175,
0.004753238521516323,
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0.05099622160196304,
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-0.009834549389779568,
0.01501504611223936,
-0... |
2cf950b55d96270ab3b0ae5681c02537e9d62dd8 | subsection | 87 | 285 | Explicit realization of classical groups and bounded symmetric domains | Then for f=f_l\in \mathcal {P}_{\left(\left\lceil \frac{l}{2}\right\rceil ,\left\lfloor \frac{l}{2}\right\rfloor \right)}(\mathfrak {p}^+),
the ratio of two norms is given by\frac{\Vert f_l\Vert _\lambda ^2}{\Vert f_l\Vert _F^2}
=\frac{1}{(\lambda )_{\left\lceil \frac{l}{2}\right\rceil }\big (\lambda +\frac{1}{2}\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 | [
-0.08676521480083466,
0.012149875983595848,
-0.0033501775469630957,
-0.0017551727360114455,
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0.0192201416939497,
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0.028845466673374176,
-0.031133579090237617,... |
25d2f75c60fc3410af2dd1f60fd49c633cdef88e | subsection | 88 | 285 | Root systems of exceptional Lie algebras | First we consider the Lie algebra \mathfrak {e}_{7(-25)}. | {
"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 | [
-0.012874355539679527,
0.0612047016620636,
-0.0349828377366066,
0.008982285857200623,
-0.017430366948246956,
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0.042705923318862915,
0.03199128434062004,
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0.008860182017087936,
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-0.007620061747729778,
-0.07381195574998856,
0.... |
062d6d88efcaa9c3ce6b0d0107c6c26e61d0425b | subsection | 89 | 285 | Root systems of exceptional Lie algebras | We take a Cartan subalgebra
\mathfrak {h}\subset \mathfrak {so}(2)\oplus \mathfrak {e}_6\subset \mathfrak {e}_{7(-25)}, and we take three kinds of basis
\lbrace \gamma _1,\gamma _2,\gamma _3,\varepsilon _1,\varepsilon _2,\varepsilon _3,\varepsilon _4\rbrace \subset \big (\mathfrak {h}^\mathbb {C}\big )^\vee ,
\big \lbr... | {
"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 | [
-0.045698363333940506,
0.011142219416797161,
-0.06624278426170349,
-0.015370157547295094,
0.026985539123415947,
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0.06361749023199081,
0.030358731746673584,
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0.015828056260943413,
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0.017369652166962624,
-0.044568877667188644,
0... |
74ca86dcdc454d059206c8d5488627fcb8f446d9 | subsection | 90 | 285 | Root systems of exceptional Lie algebras | Here, the expression in the basis \lbrace \gamma _1,\gamma _2,\gamma _3,\varepsilon _1,\varepsilon _2,\varepsilon _3,\varepsilon _4\rbrace \subset \big (\mathfrak {h}^\mathbb {C}\big )^\vee
is a modification of the one used in . Next let\alpha _{134} =\alpha _1+\alpha _3+\alpha _4 =\delta _4^{(i)}-\delta _5^{(i)},\qqu... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 229,
"openalex_id": "https://openalex.org/W1530215243",
"raw": "Yokota I., Exceptional Lie groups, arXiv:0902.0431.",
"source_ref_id": "a627c74969e31477fdac5cfc2c6eaeb8cb8d2dfc",
"start": 0
}
]
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
0.005827564746141434,
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-0.00382... |
77b4a25c92099c23f618d46ab1176731200e60d3 | subsection | 91 | 285 | Root systems of exceptional Lie algebras | \beta _{\mathfrak {e}_{6(-14)}} =\tfrac{1}{2}(\gamma _2+\gamma _3)+\tfrac{1}{2}(-\varepsilon _1-\varepsilon _2-\varepsilon _3+\varepsilon _4),\\
\beta _{\mathfrak {su}(2,6)} =\delta _2^{(1)}-\delta _3^{(1)}
=\tfrac{1}{2}\big ({-}\delta _1^{(2)}+\delta _2^{(2)}-\delta _3^{(2)}+\delta _4^{(2)}+\delta _5^{(2)}+\delta _6^{... | {
"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 | [
-0.05910670757293701,
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0.006583516951650381,
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0.03976047411561012,
-0.04345273599028587,
0.... |
017edf792ab604fbc4c1e888e1b13a30ab81bb24 | subsection | 92 | 285 | Root systems of exceptional Lie algebras | Next let\beta _{\mathfrak {so}(2,8)} =\tfrac{1}{2} (\gamma _1+\gamma _2)+\tfrac{1}{2}(-\varepsilon _1-\varepsilon _2-\varepsilon _3-\varepsilon _4),\\
\beta _{\mathfrak {so}(2,8)^{\prime }} =\tfrac{1}{2} (\gamma _1+\gamma _3)+\tfrac{1}{2}(-\varepsilon _1-\varepsilon _2-\varepsilon _3+\varepsilon _4),\\
\beta _{\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 | [
-0.049029309302568436,
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... |
c5096e9085c0cb4269be67e3a6c685ccdbe234cd | subsection | 93 | 285 | Root systems of exceptional Lie algebras | Next we take another simple system of positive roots of \mathfrak {e}_{7(-25)} as\alpha _1^{\prime } =\delta _1^{(i)}-\delta _2^{(i)}=\alpha _{23445},\qquad \alpha _2^{\prime } =\delta _3^{(i)}-\delta _4^{(i)}=\alpha _5,\\
\alpha _3^{\prime } =\tfrac{1}{2}\big ({-}\delta _1^{(i)}+\delta _2^{(i)}-\delta _3^{(i)}-\delta ... | {
"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 | [
-0.05195387825369835,
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... |
7afc76c2a5f4899a11516d0a86262699e6a796aa | subsection | 94 | 285 | Root systems of exceptional Lie algebras | The Vogan diagrams for each Lie algebra are as in Fig. REF , and the roots in \Delta _{\mathfrak {p}^+}(\mathfrak {e}_{7(-25)}) are described in Fig. REF (quoted from ), where each arrow with label j means that adding the simple root \alpha _j to the root at the source of the arrow we get the root at the target of the ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0022-1236(83)90076-9",
"end": 327,
"openalex_id": "https://openalex.org/W2017206176",
"raw": "Jakobsen H.P., Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385–412.",
"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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... |
f7c0999c1fc1c513a89513fa0ab78bf5c1292594 | subsection | 95 | 285 | Root systems of exceptional Lie algebras | First, for \mathfrak {e}_{7(-25)} we have\gamma _1(\mathfrak {e}_{7(-25)}) =\gamma _1
=\tfrac{1}{2}\big (\delta _1^{(1)}+\delta _2^{(1)}+\delta _3^{(1)}+\delta _4^{(1)}-\delta _5^{(1)}-\delta _6^{(1)}-\delta _7^{(1)}-\delta _8^{(1)}\big )
=\delta _3^{(2)}+\delta _4^{(2)},\\
\gamma _2(\mathfrak {e}_{7(-25)}) =\gamma _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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-0.06708550453186035,
-0.00... |
47dd297a02f0040d36c210f59211f36bc96c0b2d | subsection | 96 | 285 | Root systems of exceptional Lie algebras | Then we have\gamma _1(\mathfrak {sl}(2,\mathbb {R})) =2{\rm d}\chi _{\mathfrak {sl}(2,\mathbb {R})}={\rm d}\chi _{\mathfrak {e}_{6(-14)}}-2{\rm d}\chi _{\mathfrak {u}(1)},\\
\gamma _1(\mathfrak {so}(2,10)) ={\rm d}\chi _{\mathfrak {so}(2,10)}+\varepsilon _0=\tfrac{1}{2}({\rm d}\chi _{\mathfrak {e}_{6(-14)}}+4{\rm d}\ch... | {
"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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... |
e0269dfe7488817e7c51172359f63765d34b375c | subsection | 97 | 285 | Root systems of exceptional Lie algebras | Then as \mathfrak {u}(1)\oplus \mathfrak {u}(1)\oplus \mathfrak {so}(10)-modules we have\mathcal {P}_m(\mathfrak {p}^+(\mathfrak {sl}(2,\mathbb {R}))) \simeq -2m{\rm d}\chi _{\mathfrak {sl}(2,\mathbb {R})}\simeq -m({\rm d}\chi _{\mathfrak {e}_{6(-14)}}-2{\rm d}\chi _{\mathfrak {u}(1)}),\\
\mathcal {P}_{(m_1,m_2)}(\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 | [
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... |
37adbd27eb6af08e7184814ebcca6169582a9ae6 | subsection | 98 | 285 | Root systems of exceptional Lie algebras | We have\gamma _1(\mathfrak {su}(2,6))=\delta _1^{(1)}-\delta _8^{(1)}
=\tfrac{1}{2}\big (\delta _1^{(2)}-\delta _2^{(2)}+\delta _3^{(2)}+\delta _4^{(2)}+\delta _5^{(2)}+\delta _6^{(2)}+\delta _7^{(2)}-\delta _8^{(2)}\big ),\\
\gamma _2(\mathfrak {su}(2,6))=\delta _2^{(1)}-\delta _7^{(1)}
=\tfrac{1}{2}\big ({-}\delta _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 | [
-0.04649156332015991,
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-0.023856306448578835,
-0.04649156332015991,
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0.0019164795521646738,
-0... |
9a09391de8f294d986eab28a807a67e0010734cc | subsection | 99 | 285 | Root systems of exceptional Lie algebras | We also write V_{(a_1,a_2;a_3,\ldots ,a_8)}^{(2,6)\vee }\simeq V_{(a_1+c,a_2+c;a_3+c\ldots ,a_8+c)}^{(2,6)\vee },
V_{(b_1,b_2)}^{(2)\vee }\boxtimes V_{(b_3,\ldots ,b_8)}^{(6)\vee }\simeq V_{(b_1+d,b_2+d)}^{(2)\vee }\boxtimes V_{(b_3,\ldots ,b_8)}^{(6)\vee }
for any c,d\in \mathbb {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 | [
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-0.0004862263158429414... |
becccb471e1c36d507040ba5f310550f2118b264 | subsection | 100 | 285 | Root systems of exceptional Lie algebras | Then as \mathfrak {s}(\mathfrak {u}(2)\oplus \mathfrak {u}(6))\simeq \mathfrak {su}(2)\oplus \mathfrak {u}(6)-modules we have\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+(\mathfrak {su}(2,6))) \simeq V_{(m_1,m_2;0,0,0,0,-m_2,-m_1)}^{(2,6)\vee }\\
\hphantom{\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+(\mathfrak {su}(2,6)))}{} \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 | [
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0.0... |
a7f49d7f5fa217b61b188d44acd0f457c19d2c13 | subsection | 101 | 285 | Root systems of exceptional Lie algebras | We have\gamma _1(\mathfrak {so}(2,8))=\tfrac{1}{2} (\gamma _1+\gamma _2)+\tfrac{1}{2}(\varepsilon _1+\varepsilon _2+\varepsilon _3+\varepsilon _4),\\
\gamma _2(\mathfrak {so}(2,8))=\tfrac{1}{2} (\gamma _1+\gamma _2) -\tfrac{1}{2}(\varepsilon _1+\varepsilon _2+\varepsilon _3+\varepsilon _4),\\
\gamma _1(\mathfrak {so}(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 | [
-0.024347666651010513,
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0.006925965193659067,
-0.014729117974638939,
-0... |
a08a7d2365078789a291a4641bb2f26f9e3dc796 | subsection | 102 | 285 | Root systems of exceptional Lie algebras | Then as \mathfrak {u}(1)\oplus \mathfrak {u}(1)\oplus \mathfrak {so}(8)-modules we have\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+(\mathfrak {so}(2,8))) \simeq -(m_1+m_2){\rm d}\chi _{\mathfrak {so}(2,8)}
\boxtimes V_{\left(\frac{m_1-m_2}{2},\frac{m_1-m_2}{2},\frac{m_1-m_2}{2},\frac{m_1-m_2}{2}\right)}^{[10]\vee },\\
\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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... |
c1a64b64f478716d056603a5af8a323ee20d4d01 | subsection | 103 | 285 | Root systems of exceptional Lie algebras | We have\gamma _1(\mathfrak {su}(2,4))=\delta _1^{(1)}-\delta _8^{(1)},\qquad \gamma _2(\mathfrak {su}(2,4))=\delta _2^{(1)}-\delta _7^{(1)},\\
\gamma _1(\mathfrak {su}(4,2))=\tfrac{1}{2}\big (\delta _1^{(1)}+\delta _2^{(1)}+\delta _5^{(1)}-\delta _6^{(1)}-\delta _7^{(1)}-\delta _8^{(1)}+\delta _3^{(1)}-\delta _4^{(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 | [
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-0.... |
5e0b7134d6542d96c99eaae2116da894c9e289bb | subsection | 104 | 285 | Root systems of exceptional Lie algebras | Then as \mathfrak {s}(\mathfrak {u}(2)\oplus \mathfrak {u}(4))\oplus \mathfrak {su}(2)-modules we have\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+(\mathfrak {su}(2,4))) \simeq V_{(m_1,m_2;0,0,-m_2,-m_1)}^{(2,4)\vee }\boxtimes V_{(0,0)}^{(2)\vee },\\
\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+(\mathfrak {su}(4,2))) \simeq V_{{... | {
"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.... |
810a0b3ec2d71556c8af623ddab5c85dd4ccf8a3 | subsection | 105 | 285 | Root systems of exceptional Lie algebras | We have\gamma _1(\mathfrak {sl}(2,\mathbb {R})) =\delta _2^{(1)}-\delta _8^{(1)}
=\tfrac{1}{2}\big ({-}\delta _1^{(2)}+\delta _2^{(2)}+\delta _3^{(2)}+\delta _4^{(2)}+\delta _5^{(2)}+\delta _6^{(2)}+\delta _7^{(2)}-\delta _8^{(2)}\big ),\\
\gamma _1(\mathfrak {su}(1,5)) =\delta _1^{(1)}-\delta _7^{(1)}
=\tfrac{1}{2}\bi... | {
"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 | [
-0.07270925492048264,
0.03104337304830551,
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0.03269169479608536,
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-0.... |
844bbf1c56bc76a97e39ab1f5094e80dd4e9accc | subsection | 106 | 285 | Root systems of exceptional Lie algebras | Especially we have\gamma _1(\mathfrak {sl}(2,\mathbb {R})) =2{\rm d}\chi _{\mathfrak {sl}(2,\mathbb {R})}
=-3{\rm d}\chi _{\mathfrak {u}(1)}+\tfrac{1}{2}\big (\delta _3^{(2)}+\delta _4^{(2)}+\delta _5^{(2)}+\delta _6^{(2)}+\delta _7^{(2)}\big ),\\
\gamma _1(\mathfrak {su}(1,5)) =\delta _1^{(1)}-\delta _7^{(1)}
=3{\rm d... | {
"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 | [
-0.04303908720612526,
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0.008676497265696526,
-0.012751474045217037,
0.013... |
27599ef04bf2c0e4166ba5e011cff170afae89da | subsection | 107 | 285 | Root systems of exceptional Lie algebras | Then as \mathfrak {u}(1)\oplus \mathfrak {s}(\mathfrak {u}(1)\oplus \mathfrak {u}(5))\simeq \mathfrak {u}(1)\oplus \mathfrak {u}(5)-modules we have\mathcal {P}_m(\mathfrak {p}^+(\mathfrak {sl}(2,\mathbb {R}))) \simeq -2m{\rm d}\chi _{\mathfrak {sl}(2,\mathbb {R})}
\simeq 3m{\rm d}\chi _{\mathfrak {u}(1)}\boxtimes V_{\l... | {
"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 | [
-0.015835439786314964,
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0.025110919028520584,
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-0.0010135520715266466,
-0.025141431018710136,... |
fa88c2b12b7d26b5d797d39b6f464564160b8500 | subsection | 108 | 285 | Exceptional Jordan triple systems | When \fg =\mathfrak {e}_{7(-25)}, we have \mathfrak {p}^+=\operatorname{Herm}(3,\mathbb {O})^\mathbb {C}. In this subsection we consider the Jordan triple system structure
of \operatorname{Herm}(3,\mathbb {O})^\mathbb {C} and its subsystems. For x\in \operatorname{Herm}(3,\mathbb {O})^\mathbb {C}, the adjoint element x... | {
"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 | [
-0.021161044016480446,
-0.0024429792538285255,
-0.04561753571033478,
-0.017377382144331932,
0.015424524433910847,
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0.020321926102042198,
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0.02140515111386776,
-0.08732935786247253,
... |
efc35ace0ab1bb32baf379cf51076fb6662e2c61 | subsection | 109 | 285 | Exceptional Jordan triple systems | Then the Jordan triple system structure of \operatorname{Herm}(3,\mathbb {O})^\mathbb {C} is given byQ(x)y:=(x|y)x-x^\sharp \times \overline{y},and the generic norm h(x,y) is given byh(x,y):=1-(x|y)+(x^\sharp |y^\sharp )-(\det x)\overline{(\det y)}.We have the linear isomorphism\mathbb {C}\oplus M(1,2;\mathbb {O})^\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 | [
-0.051594626158475876,
0.015637047588825226,
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0.022852972149848938,
-0.022364791482686996,
... |
15220350235d1df9232452afa8201ad6b1879dbd | subsection | 110 | 285 | Exceptional Jordan triple systems | Then we have the linear isomorphismM(1,3;\mathbb {K}^{\prime })\oplus \operatorname{Herm}(3,\mathbb {K}^{\prime })\simeq \operatorname{Skew}(3,\mathbb {K}^{\prime })\oplus \operatorname{Herm}(3,\mathbb {K}^{\prime })\xrightarrow{}\operatorname{Herm}(3,\mathbb {K}), \\
\left((a_1,a_2,a_3),\begin{pmatrix}\xi _1&x_3&\hat{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1431,
"openalex_id": "",
"raw": "Yokota I., Realizations of involutive automorphisms \\sigma and G^\\sigma of exceptional linear Lie groups G. I. G=G_2, F_4 and E_6, Tsukuba J. Math. 14 (1990), 185–223.",
"source_ref_id": "b... | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
-0.03277193009853363,
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... |
9e701133d0c6e1bbcc3f1d4f82c5885af4934b61 | subsection | 111 | 285 | Exceptional Jordan triple systems | Then sinceQ(x)y=(x|y)x-x^\sharp \times y=xyx, \qquad x,y\in \operatorname{Herm}(3,\mathbb {K}^{\prime }),\quad \mathbb {K}^{\prime }=\mathbb {R},\mathbb {C},\mathbb {H}holds, we haveQ((a,x))(b,y)=\bigl (a{}^t\hspace{-1.0pt}ba-axy+bx^\sharp +\operatorname{Re}_{\mathbb {K}^{\prime }}\operatorname{Tr}(xy)a,\\
\hphantom{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 | [
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-0... |
1a74112a0c3aec09d9348a889372735a9bcc3f46 | subsection | 112 | 285 | Exceptional Jordan triple systems | Since we have the isomorphism \mathbb {H}\simeq \lbrace a\in M(2,\mathbb {C})\colon aJ_2=J_2\overline{a}\rbrace
where J_2:=\left(\begin{}0&1\\-1&0\end{}\right),
M(1,3;\mathbb {H}) and \operatorname{Herm}(3,\mathbb {H}) are naturally identified with\begin{split}&
M(1,3;\mathbb {H}) \simeq \lbrace a\in M(2,6;\mathbb {C}... | {
"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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... |
8dc7f31afe1493bd5c6a1f9e8cd41459202f9834 | subsection | 113 | 285 | Exceptional Jordan triple systems | Then \sharp in \operatorname{Herm}(3,\mathbb {H})^{\prime } and \# in \operatorname{Skew}(6,\mathbb {C}) are related as(x^\sharp )J_6^{-1}=(J_6x)^\#,\qquad J_6^{-1}(x^\sharp )=(xJ_6)^\#, \qquad x\in \operatorname{Herm}(3,\mathbb {H})^{\prime }.For x,y\in \operatorname{Skew}(6,\mathbb {C}) we definex\mathbin {\dot{\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 | [
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-0.013300608843564987... |
5b62736db7ad08abc16140cb12655d243150f47c | subsection | 114 | 285 | Exceptional Jordan triple systems | Then the Jordan triple system structure
is induced on M(2,6;\mathbb {C})\oplus \operatorname{Skew}(6,\mathbb {C}) asQ((a,x))(b,y)=\big (ab^*a-axy^*+J_2\overline{b}x^\#+\tfrac{1}{2}\operatorname{Tr}(xy^*)a,\\
\hphantom{Q((a,x))(b,y)=\big (}{} xy^*x+({}^t\hspace{-1.0pt}aJ_2a)\mathbin {\dot{\times }}y^*+\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 | [
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fddaf8903ddae9e718afff086eca93c7a2e51650 | subsection | 115 | 285 | Exceptional Jordan triple systems | Then we have the isomorphism M(1,2;\mathbb {O})^\mathbb {C}\simeq \mathbb {O}^\mathbb {C}\oplus \mathbb {O}^\mathbb {C}\simeq \mathbb {C}^8\oplus \mathbb {C}^8.
Similarly, we have the isomorphismM(1,2;\mathbb {O})^\mathbb {C}\simeq M(2,4;\mathbb {C})\oplus M(4,2;\mathbb {C})\subset M(2,6;\mathbb {C})\oplus \operatornam... | {
"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.... |
ad565059c0b971a0ece6fcd71b2bf7997138f1ca | subsection | 116 | 285 | Exceptional Jordan triple systems | Then the Jordan triple system structure of M(2,4;\mathbb {C})\oplus M(4,2;\mathbb {C}) is given byQ((a,x))(b,y)=\big (ab^*a-J_2\overline{b}\big (xJ_2{}^t\hspace{-1.0pt}x\big )^\#+\operatorname{Tr}(xy^*)a-axy^*,\\
\hphantom{Q((a,x))(b,y)=\big (}{} xy^*x-\big ({}^t\hspace{-1.0pt}aJ_2a\big )^\#\overline{y}J_2+\operatornam... | {
"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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... |
9160448d5f03ae3f4c78c5891dbcb61c70e34d45 | subsection | 117 | 285 | Exceptional Jordan triple systems | Then we have the isomorphismM(1,2;\mathbb {O})^{\mathbb {C}\prime }\simeq \mathbb {C}\oplus \operatorname{Skew}(5,\mathbb {C})\oplus M(1,5;\mathbb {C})\subset M(2,6;\mathbb {C})\oplus \operatorname{Skew}(6,\mathbb {C})\simeq \operatorname{Herm}(3,\mathbb {O})^\mathbb {C},where the inclusion is given by
(\alpha ,x,a)\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 | [
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... |
5ea5ce2e6fcd8fbf6a6864f1ec613b93c162c115 | subsection | 118 | 285 | Exceptional Jordan triple systems | Then the Jordan triple system structure is given byQ((\alpha ,x,a))(\beta ,y,b)=\bigg (\alpha \overline{\beta }\alpha +\tfrac{1}{2}\operatorname{Tr}(xy^*)\alpha +\overline{b}\,{}^t\mathbf {Pf}(x),\\
\hphantom{Q((\alpha ,x,a))(\beta ,y,b)=\bigg (}{} xy^*x+\alpha \operatorname{Proj}\left(\begin{pmatrix}y^*&-{}^t\hspace{-... | {
"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.... |
8d990f1db0c24bc7c0f5c0982d15ebe63dfd7e8e | subsection | 119 | 285 | Normal derivative case | In this subsection, we find a sufficient condition for \mathcal {F}_{\tau \rho }^* to become a normal derivative,
that is, a differential operator in the direction of \mathfrak {p}^+_2,
and a sufficient condition for \mathcal {F}_{\tau \rho } to become a multiplication operator.
Let G\supset G_1 be two real reductive g... | {
"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... |
60b7f10077a50ad830c7356a9ad124c2aba1eac4 | subsection | 120 | 285 | Normal derivative case | Then the linear map
\mathcal {F}_{\tau \rho }\colon \ \mathcal {O}_\rho (D_1,W)\rightarrow \mathcal {O}_\tau (D,V), \qquad (\mathcal {F}_{\tau \rho }f)(x_1,x_2)=\mathrm {K}(x_2)f(x_1)
intertwines the \tilde{G}_1-action.(1) Since {\rm e}^{(x|z)_{\mathfrak {p}^+}}I_V is the reproducing kernel of \mathcal {P}(\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 | [
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-0.... |
5e67072c9775b25c0b6edfefdd497fc99f4f4554 | subsection | 121 | 285 | Normal derivative case | Since this is a finite-order differential operator, this extends to the operator between the spaces of all holomorphic functions,
and the claim follows.(2) By the assumption, we haveF_{\tau \rho }(x_2;w_1)
&=\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak {p}^+}}I_W,\bigl (\tau (B(x_2,y_1))\mathrm {K}\big ((x_2)^{Q(y_1)x_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 | [
-0.020257875323295593,
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-0.0... |
a7d2974278d9955ed735799c71a67ab1fde39c09 | subsection | 122 | 285 | Normal derivative case | Therefore \mathcal {F}_{\tau \rho }^* and \mathcal {F}_{\tau \rho } intertwine \tilde{G}_1-action for any \lambda .The condition in Theorem REF (1) is the same as when (G,G_1) is of split rank 1
(i.e., (G,G_1)=(U(q,s),U(q,s-1)\times U(1)), (\operatorname{SO}^*(2s),\operatorname{SO}^*(2(s-1))\times \operatorname{SO}(2))... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00029-015-0208-8",
"end": 394,
"openalex_id": "https://openalex.org/W2220694899",
"raw": "Kobayashi T., Pevzner M., Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22 (... | 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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0.0055327401496469975,
-0.00... |
72e0ea80534768f89e43a65d56ac8cb40a9646e3 | subsection | 123 | 285 | Normal derivative case | In the first case we have \mathfrak {p}^+=M(q,s;\mathbb {C}), \mathfrak {p}^+_1=M(q,s^{\prime };\mathbb {C}), \mathfrak {p}^+_2=M(q,s^{\prime \prime };\mathbb {C}), and\mathcal {P}(\mathfrak {p}^+)=\bigoplus _{\mathbf {m}\in \mathbb {Z}_{++}^{\min \lbrace q,s\rbrace }}\mathcal {P}_\mathbf {m}(\mathfrak {p}^+)
=\bigoplu... | {
"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 | [
-0.010651156306266785,
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... |
eecaa072c8c753b2f1286025e5461cdf92a089b1 | subsection | 124 | 285 | Normal derivative case | In the second case we have \mathfrak {p}^+=\operatorname{Skew}(s,\mathbb {C}), \mathfrak {p}^+_1=\mathbb {C}^{s-1},
\mathfrak {p}^+_2=\operatorname{Skew}(s-1,\mathbb {C}), and\mathcal {P}(\mathfrak {p}^+) =\bigoplus _{\mathbf {m}\in \mathbb {Z}_{++}^{\lfloor s/2\rfloor }}\mathcal {P}_\mathbf {m}(\mathfrak {p}^+)
=\bigo... | {
"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.... |
1c66297bee41100c670e99767b42265cf7ab815c | subsection | 125 | 285 | Normal derivative case | In the third case \mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+) is isomorphic to\mathcal {P}_{(m_1,m_2)}(\mathfrak {p}^+)\simeq \chi _{\mathfrak {e}_{6(-14)}}^{-\frac{3}{4}(m_1+m_2)}\boxtimes V_{\left(\frac{m_1+m_2}{2},\frac{m_1-m_2}{2},\frac{m_1-m_2}{2},\frac{m_1-m_2}{2},\frac{m_1-m_2}{2}\right)}^{[10]\vee },and by we can... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 332,
"openalex_id": "",
"raw": "Tsukamoto C., Spectra of Laplace–Beltrami operators on {\\rm SO}(n+2)/{\\rm SO}(2)\\times {\\rm SO}(n) and {\\rm Sp}(n+1)/{\\rm Sp}(1)\\times {\\rm Sp}(n), Osaka J. Math. 18 (1981), 407–426.",
... | 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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... |
48216251280e1c00a831adbeaec139c54b38bcc3 | subsection | 126 | 285 | Normal derivative case | Therefore
(W=)\mathcal {P}_\mathbf {m}(\mathfrak {p}^+_2)\otimes \chi ^{-\lambda }\subset \mathcal {P}_\mathbf {m}(\mathfrak {p}^+)\otimes \chi ^{-\lambda }(=V^{\prime }) holds as a concrete submodule,
and the condition in Theorem REF (1) is also satisfied.Next we consider \mathcal {F}_{\tau \rho }. We again consider(G... | {
"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... |
7aa9ee5d5b6e54d2b66e225599b8e38c6caea8f8 | subsection | 127 | 285 | Normal derivative case | We realize G_1\subset G such that\mathfrak {p}^+_1=\fg _1\cap \mathfrak {p}^+&={\left\lbrace \begin{array}{ll}
\left\lbrace y_1=\begin{pmatrix}y&0\end{pmatrix}\colon y\in M(q,s^{\prime };\mathbb {C})\right\rbrace & (\text{Case }1),\vspace{2.84526pt}\\
\left\lbrace y_1=\begin{pmatrix}y&0\\0&0\end{pmatrix}\colon y\in \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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0.0... |
cac6dbdc3d15ed0f82df16420e08af1f2688af25 | subsection | 128 | 285 | Normal derivative case | \end{array}\right.}Then for (y_1,x_2)\in \mathfrak {p}^+_1\times \mathfrak {p}^+_2, (x_2)^{Q(y_1)x_2}=x_2 holds sinceQ(y_1)x_2={\left\lbrace \begin{array}{ll}
\begin{pmatrix}y&0\end{pmatrix}\begin{pmatrix}0\\x^*\end{pmatrix}\begin{pmatrix}y&0\end{pmatrix}=0& (\text{Case }1),\\
\begin{pmatrix}y&0\\0&0\end{pmatrix}\begin... | {
"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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... |
98f8607e38ffb5b1344c02098453adbe55ac8f31 | subsection | 129 | 285 | Normal derivative case | \begin{pmatrix}I_s& 0\\ -(xy^*-\overline{y}\hspace{1.0pt}{}^t\hspace{-1.0pt}x)& I_s\end{pmatrix}\!
\begin{pmatrix}1&-\sqrt{-1}\\1&\sqrt{-1}\end{pmatrix}\in \operatorname{End}(\mathfrak {p}^+)\qquad \!\! (\text{Case }4),for Cases 1–4, andB(x_2,y_1)z =\begin{pmatrix}z_1& z_2\end{pmatrix}
-\begin{pmatrix}0& x\end{pmatrix}... | {
"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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932a95f98ba66cf7016ce4eb0fa76f1807c815f5 | subsection | 130 | 285 | Normal derivative case | Therefore, for the representationV={\left\lbrace \begin{array}{ll} \chi ^{-\lambda _1-\lambda _2}_{U(q,s)}\otimes \big (V_\mathbf {k}^{(q)\vee }\boxtimes V_\mathbf {m}^{(s)}\big ) & (\text{Case }1),\\
\chi ^{-\lambda }_{\operatorname{SO}^*(2s)}\otimes V_\mathbf {m}^{(s)\vee } & (\text{Cases }2,3),\\
\chi ^{-\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 | [
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5627d38dbaec611ac82ca06de1658c314f3f5982 | subsection | 131 | 285 | Normal derivative case | Thus we have proved the following.Corollary 5.3Let (G,G_1)=(U(q,s),U(q,s^{\prime })\times U(s^{\prime \prime })),
and (\tau ,V)=\big (\chi ^{-\lambda _1-\lambda _2}\otimes \big (\tau _\mathbf {k}^{(q)\vee }\boxtimes \tau _\mathbf {m}^{(s)}\big ),V_\mathbf {k}^{(q)\vee }\otimes V_\mathbf {m}^{(s)}\big ).
Then for any 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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19e8954f3dd5f242f71d9aa0715c50dac460571f | subsection | 132 | 285 | Normal derivative case | Then for any subrepresentation W\subset \mathcal {P}\big (\mathfrak {p}^+_2,V_{(m_2,\ldots ,m_s)}^{(s-1)\vee }\boxtimes \mathbb {C}_{-m_1}\big ) of \tilde{K}^\mathbb {C}_1,
the multiplication operator
\mathcal {F}_{\tau \rho }\colon \mathcal {O}_\lambda (D_1,W)\rightarrow \mathcal {O}_{\frac{\lambda }{2}+\frac{\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 | [
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cd0819e58d143ed4db067f642dbccd2dc189a2b5 | subsection | 133 | 285 | Body | \mathcal {F}_{\tau \rho }^* for (G,G_1)=(G_0\times G_0, \Delta G_0)In this subsection we find the operator \mathcal {F}_{\tau \rho }^* for (G,G_1)=(G_0\times G_0, \Delta G_0),
where G_0 is a simple Lie group of Hermitian type, although it is already done by Peng–Zhang (see also, e.g., , , ).
We denote the complexified ... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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c88ef230c403adabae0e71ef7a526b0fb79dfefa | subsection | 134 | 285 | Body | Then the function F_{\tau \rho }^*(z_L,z_R)\in \mathcal {P}(\overline{\mathfrak {p}^+},\operatorname{Hom}(V,W)) in Theorem REF (1) is given byF_{\tau \rho }^*(z_L,z_R)=\big \langle {\rm e}^{(x_L|z_L)_{\mathfrak {p}^+_0}+(x_R|z_R)_{\mathfrak {p}^+_0}}I_V,\mathrm {K}\big (\tfrac{1}{2}(x_L-x_R)\big )
\big \rangle _{\hat{\... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
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7554eb86297c0448fb8b1e652712613388eb06b6 | subsection | 135 | 285 | Body | Then if \mathrm {K}(x_L-x_R,y_2)\in \mathcal {P}(\mathfrak {p}^+_0)\otimes \mathcal {P}(\mathfrak {p}^+_0)\otimes \overline{\mathcal {P}_\mathbf {k}(\mathfrak {p}^+_0)} is expanded as\mathrm {K}(x_L-x_R,y_2)={} &\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{r_0}}\sum _{\mathbf {n}\in \mathbb {Z}_{++}^{r_0}}\mathcal {K}_{\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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61a338f4a0002a6b01a89eeb6c334366c176c249 | subsection | 136 | 285 | Body | Then by Theorem REF , the linear map\mathcal {F}_{\tau \rho }^*\colon \mathcal {H}_\lambda (D_0)_{\tilde{K}_0}\boxtimes \mathcal {H}_\mu (D_0)_{\tilde{K}_0}\rightarrow \mathcal {H}_{\lambda +\mu }(D_0,\mathcal {P}_\mathbf {k}(\mathfrak {p}^+_0))_{\tilde{K}_0}, \\
\mathcal {F}_{\lambda ,\mu ,k}^*f(y_1,y_2)
:=\sum _{\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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6962ae72b01fb5a8fa14e2fcd75cfff011e78ddc | subsection | 137 | 285 | Body | Then since this is \tilde{K}^\mathbb {C}_0-invariant, its orthogonal projection
\mathcal {K}_{\mathbf {m},\mathbf {n}}(x_L,x_R;y_2)\in \mathcal {P}_\mathbf {m}(\mathfrak {p}^+_0)\otimes \mathcal {P}_\mathbf {n}(\mathfrak {p}^+_0)\otimes \overline{\mathcal {P}_\mathbf {k}(\mathfrak {p}^+_0)} is also \tilde{K}^\mathbb {C... | {
"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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53ec06899e5fe0d6dc05d629d96f7db54c7d552b | subsection | 138 | 285 | Body | Since \tilde{K}^\mathbb {C}_0 acts transitively on an open dense subset of \mathfrak {p}^+_0, by \tilde{K}^\mathbb {C}_0-invariance of \mathcal {K}_j
it suffices to show \mathcal {K}_1=\mathcal {K}_2 on \mathfrak {p}^+_0\oplus \mathfrak {p}^+_0\oplus \overline{\mathfrak {p}^+_{\mathrm {T},0}}.
We consider B(te,te)\in \... | {
"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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fe868c984fc91c60aa179c21a203107c279ec740 | subsection | 139 | 285 | Body | Especially, by taking a limit |t|\rightarrow 1, we have\mathcal {K}_j(x_L,x_R;y_2)=\mathcal {K}_j (x_{L\mathrm {T}},x_{R\mathrm {T}};y_2 ).Therefore, \mathcal {K}_1=\mathcal {K}_2 on \mathfrak {p}^+_{\mathrm {T},0}\oplus \mathfrak {p}^+_{\mathrm {T},0}\oplus \overline{\mathfrak {p}^+_{\mathrm {T},0}} implies
\mathcal {... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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54a0876d2ba3d4b1a9c674957190dde9a0032969 | subsection | 140 | 285 | Body | This lies in \mathcal {P}_\mathbf {m}(\mathfrak {p}^+_{\mathrm {T},0}) as a polynomial in x_R, and lies in \mathcal {P}_{k-\mathbf {m}^*}(\mathfrak {p}^+_{\mathrm {T},0}) as a polynomial in x_L,
where k-\mathbf {m}^*:=(k-m_{r_0},k-m_{r_0-1},\ldots ,k-m_1).
Now let \Psi _{k-\mathbf {m}^*,\mathbf {m}}^{(d_0,r_0)}(x_L,x_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 | [
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1a6ebbd30061c9f9758a74743381a1f3ff572dc5 | subsection | 141 | 285 | Body | Then we have\mathrm {K}(x_L-x_R,y_2)=\sum _{\mathbf {m}\in \mathbb {Z}_{++}^{r_0}}(-k)_{\mathbf {m},d_0}\frac{d_\mathbf {m}^{(d_0,r_0,b_0)}}{\big (\frac{n_0}{r_0}\big )_{\mathbf {m},d_0}}
\Psi _{k-\mathbf {m}^*,\mathbf {m}}^{(d_0,r_0)}(x_L,x_R;y_2).We write\overline{\Psi _{k-\mathbf {m}^*,\mathbf {m}}^{(d_0,r_0)}(x_L,x... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
] | 2,018 | en | Mathematics | [
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b9dfcc86d7f1f5083d6820af01a0be134d85b35a | subsection | 142 | 285 | Body | Therefore we have proved the following.Theorem 5.5
Let k\in \mathbb {Z}_{\ge 0}. Then the linear map\mathcal {F}_{\lambda ,\mu ,k}^*\colon \ \mathcal {O}_\lambda (D_0)\mathbin {\hat{\boxtimes }}\mathcal {O}_\mu (D_0)\rightarrow \mathcal {O}_{\lambda +\mu }(D_0,\mathcal {P}_{(k,\ldots ,k)}(\mathfrak {p}^+_0)), \\
\math... | {
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Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
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08c1960e1ca8e6e679af75606025734fe032c0f3 | subsection | 143 | 285 | Body | If G_0 is of tube type, i.e., G_0=G_{0,\mathrm {T}}, then \mathcal {P}_{(k,\ldots ,k)}(\mathfrak {p}^+_0) is 1-dimensional, and we have
\mathcal {O}_{\lambda +\mu }(D_0,\mathcal {P}_{(k,\ldots ,k)}(\mathfrak {p}^+_0))\simeq \mathcal {O}_{\lambda +\mu +2k}(D_0) via f\Delta (y)^k\mapsto f,
and thus it gives the intertwin... | {
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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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668ea321f5f4038019b3e58e2016396a8371d1b1 | subsection | 144 | 285 | Body | Then the maximal compact subgroups (K,K_1)=(K,K_{11}\times K_{22})\subset (G,G_{11}\times G_{22}) 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(q)\times U(s), (U(q^{\prime })\times U(s^{\prime }))\times (U(... | {
"cite_spans": []
} | 10.3842/SIGMA.2019.036 | 1804.07100 | Construction of Intertwining Operators between Holomorphic Discrete
Series Representations | [
"Ryosuke Nakahama"
] | [
"math.RT"
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3fc96d40ba557ad36d92906512cedb6f735d42cf | subsection | 145 | 285 | Body | Also we have\mathfrak {p}^+={\left\lbrace \begin{array}{ll}\operatorname{Sym}(s,\mathbb {C}) & (\text{Case }d=1),\\ M(q,s;\mathbb {C}) & (\text{Case }d=2),\\ \operatorname{Skew}(s,\mathbb {C}) & (\text{Case }d=4),\\
M(1,2;\mathbb {O})^\mathbb {C}& (\text{Case }d=6),\\ \operatorname{Herm}(3,\mathbb {O})^\mathbb {C}& (\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 | [
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449edafd14fae24230cbb89417bf3f082b021896 | subsection | 146 | 285 | Body | Then we have \chi |_{K_{jj}}=\chi _{jj} (j=1,2).
Similarly, for d=2, let \chi ^{-\lambda _1-\lambda _2}, \chi _{11}^{-\lambda _1-\lambda _2} and \chi _{22}^{-\lambda _1-\lambda _2} be
the characters of K^\mathbb {C}, K_{11}^\mathbb {C} and K_{22}^\mathbb {C} respectively, as (REF ).
Then similarly we have \chi ^{-\lamb... | {
"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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... |
eb145186fb2aacdb3cdf7fe53a85058190c16492 | subsection | 147 | 285 | Body | For x_2=x_{12}\in \mathfrak {p}^+_2=\mathfrak {p}^+_{12}, w_1=w_{11}+w_{22}\in \mathfrak {p}^+_1=\mathfrak {p}^+_{11}\oplus \mathfrak {p}^+_{22},
we want to computeF_{\tau \rho }(x_2;w_1)&=F_{\tau \rho }(x_{12};w_{11},w_{22})\\
& =\big \langle {\rm e}^{(y_1|w_1)_{\mathfrak {p}^+_1}}I_W,
\big (h(Q(x_2)y_1,y_1)^{-\lambda... | {
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{
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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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774d4712d90452ca657314c7b5afd55678ef6f41 | subsection | 148 | 285 | Body | Moreover we have the following.Lemma 5.6(x_{12})^{Q(y_{11}+y_{22})x_{12}}=B(Q(x_{12})y_{11},y_{22})^{-1}x_{12}.By the definition of the quasi-inverse, we have&(x_{12})^{Q(y_{11}+y_{22})x_{12}}=B(x_{12},Q(y_{11}+y_{22})x_{12})^{-1}(x_{12}-Q(x_{12})Q(y_{11}+y_{22})x_{12})\\
&=B(Q(x_{12})(y_{11}+y_{22}),y_{11}+y_{22})^{-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 | [
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0.... |
72e812874b0c5a1102db1e7de16bf8c6f7035dcb | subsection | 149 | 285 | Body | Thus it suffices to showB(Q(x_{12})y_{22},y_{11})^{-1}(x_{12}-Q(x_{12})Q(y_{11},y_{22})x_{12})=x_{12}.This follows fromB(Q(x_{12})y_{22},y_{11})x_{12}&=x_{12}-D(Q(x_{12})y_{22},y_{11})x_{12}+Q(Q(x_{12})y_{22})Q(y_{11})x_{12}\\
&=x_{12}-Q(Q(x_{12})y_{22},x_{12})y_{11}+Q(Q(x_{12})y_{22})0\\
&=x_{12}-Q(x_{12})D(y_{22},x_{... | {
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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. 185... | 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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333fa2b5145a0a40a43968e7e69c0c72a7b2c70e | subsection | 150 | 285 | Body | Then we haveF_{\tau \rho }(x_{12};w_{11},w_{22})
=\bigl \langle {\rm e}^{(y_{11}|w_{11})_{\mathfrak {p}^+_{11}}}{\rm e}^{(y_{22}|w_{22})_{\mathfrak {p}^+_{22}}}I_{W_{11}\boxtimes W_{22}},\\
\quad \big (h_{22}(Q(x_{12})y_{11},y_{22})^{-\lambda }
\mathrm {K}\big (B(Q(x_{12})y_{11},y_{22})^{-1}x_{12}\big )\big )^*\bigr \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 | [
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aeb048dcb06f5c8ce9c413f8b2b9bd943c60d7e3 | subsection | 151 | 285 | Body | In the following we omit \boxtimes I_{W_{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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8cdb9113c321cdc39bf1e1c249190f73dbc42f0e | subsection | 152 | 285 | Body | Now we assume s^{\prime }\le s^{\prime \prime } when d=1, q^{\prime }\le s^{\prime \prime } when d=2, 2\le s^{\prime }\le s^{\prime \prime } when d=4, and set W=W_{11}\boxtimes W_{22} asW&=\mathcal {P}_{(\underbrace{\scriptstyle k+1,\ldots ,k+1}_l,k,\ldots ,k)}(M(s^{\prime },s^{\prime \prime };\mathbb {C}))\otimes \chi... | {
"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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01a3d1a5aa30d42ccffa26ca82746b611b9debd5 | subsection | 153 | 285 | Body | We note that when d=4, s^{\prime }=3, we identify \operatorname{SO}^*(6)\simeq \operatorname{SU}(1,3) up to covering. We write\mathrm {K}(x_{12})={\left\lbrace \begin{array}{ll}
\mathrm {K}_{k\langle s^{\prime }\rangle +\langle l\rangle }^{(2)}(x_{12}) & (d=1),\\
\mathrm {K}_{(k,\ldots ,k)}^{(2)}(x_{12})\mathrm {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 | [
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-0.... |
a6a557b7838edc479ffb0fa8649b9e0313ae1df7 | subsection | 154 | 285 | Body | Then \mathcal {P}(\mathfrak {p}^+_{11})\otimes W_{11} is decomposed under \tilde{K}_{11}^\mathbb {C} as\mathcal {P}(\mathfrak {p}^+_{11})\otimes W_{11}
&\simeq \bigoplus _{\mathbf {m}\in \mathbb {Z}_{++}^{s^{\prime }}} V_{2\mathbf {m}}^{(s^{\prime })\vee }\otimes V_{\langle l\rangle }^{(s^{\prime })\vee }\otimes \chi _... | {
"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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... |
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