Spherical functions and local densities of alternating forms.

*(English)*Zbl 0662.22013This paper develops the theory of the spherical Fourier transform for the space X of nondegenerate alternating 2n\(\times 2n\) matrices with entries in a p-adic number field k. Let \(G=GL(2n,k)\) and then one can identify \(X\cong GL(2n,k)/Sp(n,k)\). Let \({\mathcal O}\) be the ring of integers of k and \(K=GL(2n,{\mathcal O})\). The Hecke algebra H(G,K) acts on the space C(K\(\setminus X)\) of all K-invariant functions on X by convolution. A spherical function on X is a normalized eigenfunction of all these convolution operators. These spherical functions are given in terms of Hall-Littlewood polynomials. The spherical Fourier transform with these functions providing the analogue of the Euclidean exponential is studied. The Plancherel measure is computed. It is noted that the spherical functions on X can be viewed as generating functions of densities for solutions of certain congruences involving alternating matrices. Explicit expressions for these densities are derived as well as some properties of Hall-Littlewood polynomials. The work uses results of I. Satake [Publ. Math., Inst. Hautes Etud. Sci. 18, 229-293 (1963; Zbl 0122.285)], and I. Macdonald [Symmetric functions and Hall polynomials (1979; Zbl 0487.20007)]. There are also applications to local zeta functions associated with a prehomogeneous vector space.

Reviewer: A.Terras

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

43A90 | Harmonic analysis and spherical functions |

11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |

22E35 | Analysis on \(p\)-adic Lie groups |