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Invariant Representations of $$\mathrm{GSp}(2)$$ under Tensor Product with a Quadratic Character
Ping-Shun Chan, Ohio State University, Columbus, OH
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Memoirs of the American Mathematical Society
2009; 172 pp; softcover
Volume: 204
ISBN-10: 0-8218-4822-4
ISBN-13: 978-0-8218-4822-7
List Price: US$77 Individual Members: US$46.20
Institutional Members: US\$61.60
Order Code: MEMO/204/957

Let $$F$$ be a number field or a $$p$$-adic field. The author introduces in Chapter 2 of this work two reductive rank one $$F$$-groups, $$\mathbf{H_1}$$, $$\mathbf{H_2}$$, which are twisted endoscopic groups of $$\mathrm{GSp}(2)$$ with respect to a fixed quadratic character $$\varepsilon$$ of the idèle class group of $$F$$ if $$F$$ is global, $$F^\times$$ if $$F$$ is local. When $$F$$ is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of $$\mathbf{H_1}$$, $$\mathbf{H_2}$$ to those of $$\mathrm{GSp}(2)$$. In Chapter 4, the author establishes this lifting in terms of the Satake parameters which parameterize the automorphic representations. By means of this lifting he provides a classification of the discrete spectrum automorphic representations of $$\mathrm{GSp}(2)$$ which are invariant under tensor product with $$\varepsilon$$.

• $$\varepsilon$$-endoscopy for $$\mathrm{GSp}(2)$$