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Representations of GSp(4) over a p-adic field. I, II. (English) Zbl 0662.22012
Let F be a p-adic field of odd residual characteristic; let GSp(4,F) be the group of linear transformations that take a given symplectic form on $$F^ 4$$ to a multiple of itself. In these two papers, the author gives a complete classification of the irreducible admissible representations of $$GSp_ 4(F).$$
The paper follows a strategy already used by the author for the groups U(2,1;F). Let $$\pi$$ be an irreducible admissible representation of $$G=GSp(4,F)$$. If K is a compact open subgroup of G, then $$\pi$$ $$| K$$ decomposes as a discrete direct sum of irreducibles. The author defines what it means for an irreducible representation $$\sigma$$ of K to be nondegenerate; his definition is that $$\sigma$$ is nondegenerate if it occurs in a certain list, but, as he remarks, the idea is not nearly so ad hoc as this definition may make it seem. He then shows that for any $$\pi$$, there exists K and a nondegenerate representation $$\sigma$$ of K contained in $$\pi$$, K and $$\sigma$$ are essentially unique. (To be precise, one may first need to tensor K with a quasicharacter.) He then analyzes all $$\pi$$ containing a given nondenerate $$\sigma$$ on K the basic tool here is Hecke algebra isomorphisms: roughly speaking, he shows that the Hecke algebra H(G//K,$$\sigma)$$ is isomorphic to the Hecke algebra for a smaller group. This leads to a description of the representations of G containing $$\sigma$$ in terms of the representations of this smaller group.
One interesting consequence of the analysis is that every supercuspidal representation of G is induced from a compact (mod center) subgroup. (More recently, C. Asmuth and D. Keys have obtained this result for residual characteristic 2 as well.) Implicit in the description is a list of the supercuspidals of G and of the nonsupercuspidal discrete series of G; however, these classes of representations are not described explicitly in the paper.
Reviewer: L.Corwin

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 20G05 Representation theory for linear algebraic groups 20G25 Linear algebraic groups over local fields and their integers
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##### References:
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