Representations of GSp(4) over a p-adic field. I, II.

*(English)*Zbl 0662.22012Let 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.

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 |

##### Keywords:

p-adic field; GSp(4,F); group of linear transformations; symplectic form; irreducible admissible representations; nondegenerate representation; Hecke algebra; supercuspidal representation; discrete series##### References:

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