Geodesic
Transfer of Characters in the Theta Correspondence with One Compact Member
Journal of Lie theory, Tome 30 (2020) no. 4, pp. 997-1026
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\newcommand{\Sp}{\textrm{Sp}\,} For an irreducible dual pair $(G, G') \subseteq \Sp(W)$ with one member compact and two representations $\Pi \leftrightarrow \Pi'$ appearing in the Howe duality, we give an expression of the character $\Theta_{\Pi'}$ of $\Pi'$ via the character of $\Pi$. We compute the value of $\Theta_{\Pi'}$ on the maximal compact torus $T'$ of $G'$ for the dual pair $(G = U(n, \mathbb{C}),\, G' = U(p, q, \mathbb{C}))$, which are explicit in low dimensions. For $(G = U(1, \mathbb{C}),\, G' = U(1, 1, \mathbb{C}))$, we determine the value of the character on both Cartan subgroups of $G'$.
Classification : 22E45, 22E46, 22E30
Mots-clés : Howe correspondence, characters, oscillator semigroup, reductive dual pairs 
@article{JLT_2020_30_4_JLT_2020_30_4_a5,
     author = {A. Merino},
     title = {Transfer of {Characters} in the {Theta} {Correspondence} with {One} {Compact} {Member}},
     journal = {Journal of Lie theory},
     pages = {997--1026},
     year = {2020},
     volume = {30},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JLT_2020_30_4_JLT_2020_30_4_a5/}
}
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A. Merino. Transfer of Characters in the Theta Correspondence with One Compact Member. Journal of Lie theory, Tome 30 (2020) no. 4, pp. 997-1026. http://geodesic.mathdoc.fr/item/JLT_2020_30_4_JLT_2020_30_4_a5/