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| GSp(2) | |
|---|---|
| Name | GSp(2) |
| Type | Algebraic group |
| Field | General linear group context |
GSp(2). GSp(2) is the symplectic similitude group of rank two defined over fields such as Q, R, and finite fields like F_p. It appears in the study of Adeles, Hecke operators, Siegel modular forms, and the Langlands program. GSp(2) connects to classical objects studied by figures such as André Weil, Ernst Witt, Carl Ludwig Siegel, and Harish-Chandra.
GSp(2) is the group of invertible linear transformations preserving a nondegenerate alternating bilinear form up to a scalar factor called the similitude; this definition relates to work of Émile Picard, Hermann Weyl, Richard Brauer, Willem Jacob van Stockum, and Évariste Galois. Over Q and R it is a reductive algebraic group whose center and derived subgroup interact as in descriptions by Claude Chevalley, Armand Borel, Harish-Chandra, Robert Langlands, and George Lusztig. Key invariants include the similitude character, the root system of type C_2 noted by Nicholas Bourbaki, and connections to dual groups studied by James Arthur and Robert Kottwitz.
A standard matrix model embeds GSp(2) into GL(4) via block matrices preserving the symplectic form J, a construction used by Carl Friedrich Gauss, Henri Poincaré, John von Neumann, Paul Dirac, and Claude Chevalley. The similitude character μ: GSp(2) → GL(1) generalizes determinants as in treatments by David Mumford, Alexander Grothendieck, Pierre Deligne, Jean-Pierre Serre, and Michel Raynaud. Concrete matrix representatives feature in computations by Igor Shafarevich, Jean-Louis Koszul, Hermann Minkowski, Emil Artin, and Markus Rost.
The derived subgroup Sp(2) appears alongside maximal tori and Borel subgroups analyzed by Élie Cartan, Hermann Weyl, Émile Borel, Jacques Tits, and Alain Connes. Parabolic subgroups, Levi decompositions, and unipotent radicals are central to descriptions by Serge Lang, André Weil, Nicholas Katz, Barry Mazur, and Gerd Faltings. Compact and noncompact real forms link to studies by Hermann Weyl, Wilhelm Killing, Élie Cartan, Bertram Kostant, and Harish-Chandra. Important maximal subgroups include Siegel parabolics and Klingen parabolics appearing in work of Carl Ludwig Siegel, Ilya Piatetski-Shapiro, Stephen S. Gelbart, Herb Gangl, and Henryk Iwaniec.
Smooth admissible representations of GSp(2) over local fields feature in the research of Joseph Bernstein, Pierre Deligne, Marko Tadić, David Vogan, and Flicker Joseph. The local Langlands correspondence for GSp(2) relates to parameters studied by Robert Langlands, Michael Harris, Richard Taylor, Benedict Gross, and Dipendra Prasad. Automorphic representations and cuspidal spectra for GSp(2) appear in trace formula investigations by James Arthur, Werner Müller, Laurent Clozel, Friedrich Hirzebruch, and Jean-Pierre Labesse. Techniques such as theta correspondence and Howe duality involve Roger Howe, Stephen S. Kudla, Stephen Rallis, Ilya Piatetski-Shapiro, and Satoshi Kondo.
Automorphic forms on GSp(2) include Siegel modular forms and cusp forms central to the work of Carl Ludwig Siegel, Goro Shimura, Igor Shafarevich, Haruzo Hida, and Robert Coleman. L-functions attached to automorphic representations of GSp(2) are studied by Frederick Diamond, Brian Conrad, Andrew Wiles, Christophe Breuil, and Richard Taylor. Global Arthur packets, endoscopy, and functorial lifts from GSp(2) are investigated by James Arthur, Ralph Langlands (see Robert Langlands), Colette Moeglin, Jean-Loup Waldspurger, and Weissauer. Integral representations for standard and spinor L-functions involve methods from Goro Shimura, Yoshida Hiroyuki, Fulton],[Harris (note: Fulton and Harris), Kenneth Ribet, and Ken Ono.
GSp(2) surfaces in arithmetic geometry via moduli of abelian surfaces, principally polarized abelian varieties, and Siegel modular varieties investigated by David Mumford, Pierre Deligne, Gerd Faltings, Barry Mazur, and Richard Taylor. It underpins constructions of Galois representations arising in the work of Jean-Pierre Serre, Andrew Wiles, Richard Taylor, Mark Kisin, and Christophe Breuil. Periods, algebraic cycles, and special values of L-functions connected to GSp(2) enter research by Goro Shimura, Don Zagier, Yuri Manin, Benedict Gross, and Wilfried Schmid. Relations with arithmetic compactifications, toroidal compactifications, and stratifications relate to contributions by Faltings, Chai, Ash, Mumford, and Pink.