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Some Supercuspidal Representations of Sp4(k)

Published online by Cambridge University Press:  20 November 2018

Charles Asmuth
Charles Asmuth*
Affiliation:
Rutgers University, Newark, New Jersey
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The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.

The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 1 , 01 February 1987 , pp. 1 - 7
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Asmuth, C., Weil representations of symplectic p-adic groups, Amer. J. Math. 101 (1979), 855–908.Google Scholar
2. Asmuth, C., Tensor products of representations of a p-adic special orthogonal group, preprint.Google Scholar
3. Bernstein, I. N. and Zelivinsky, A. V., Induced representations of reductive p-adic groups, Ann. Scient. Ec. Norm. Sup. 10 (1977), 441–472.Google Scholar
4. Casselman, W., On representations of SL2(k) related to binary quadratic forms, Amer. J. Math. 94(1972), 810–834.Google Scholar
5. Corwin, L. and Howe, R., Computing characters of tamely ramified p-adic division algebras, Pacific Journal of Math. 73 (1977), 461–477.Google Scholar
6. Corwin, L., Representations of division algebras over local fields, Advances in Math. 13 (1974), 810–834.Google Scholar
7. Corwin, L., Characters of division algebras, (manuscript).Google Scholar
8. Saito, M., Representations unitaires des groupes symplectiques, J. Math. Soc. Japan 24 (1972), 232–251.Google Scholar
9. Shalika, J., Representations of the two by two unimodular group over local fields, (Princeton, 1966).Google Scholar
10. Sally, P. and Shalika, J., Characters of the discrete series of representations of SL(2) over a local field, Proc. of Nat. Acad, of Sciences (1968), 1231–1237.Google Scholar