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Invariant representations of GSp(2)
Chan, Ping Shun

2005, Doctor of Philosophy, Ohio State University, Mathematics.
Let F be a number field or a p-adic field. We introduce in Chapter 2 of this work two reductive rank one F-groups, H1, H2, which are twisted endoscopic groups of GSp(2) with respect to a fixed quadratic character varepsilon of the idele class group of F if F is global, F* if F is local. If F is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of H1, H2 to those of GSp(2). In Chapter 4, we establish this lifting in terms of the Satake parameters which parametrize the automorphic representations. By means of this lifting we provide a classification of the discrete spectrum automorphic representations of GSp(2) which are invariant under tensor product with varepsilon. The techniques through which we arrive at our results are inspired by those of Kazhdan’s in [K]. In particular, they involve comparing the spectral sides of the trace formulas for the groups under consideration. We make use of the twisted extension of Arthur’s trace formula, and Kottwitz-Shelstad’s stabilization of the elliptic component of the geometric side of the twisted trace formula. If F is local, in Chapter 5 we provide a classification of the irreducible admissible representations of GSp(2, F) which are invariant under tensor product with the quadratic character varepsilon of F*. Here, our techniques are also directly inspired by [K]. More precisely, we use the global results from Chapter 4 to express the twisted characters of these invariant representations in terms of the characters of the admissible representations of Hi(F) (i = 1, 2). These (twisted) character identities provide candidates for the liftings predicted by the local component of the conjectural Langlands functoriality. The proofs rely on Sally-Tadic’s classification of the irreducible admissible representations of GSp(2, F), and Flicker’s results on the lifting from PGSp(2) to PGL(4).
Yuval Flicker (Advisor)
266 p.

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Chan, Ping Shun "Invariant representations of GSp(2)." Electronic Thesis or Dissertation. Ohio State University, 2005. https://etd.ohiolink.edu/

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