Spherical Functions for the Action of a Finite Unitary Group on a Finite Heisenberg Group
نویسندگان
چکیده
The action of the unitary group on the real Heisenberg group yields a Gelfand pair. The associated spherical functions are well known and have been computed independently by many authors. In this paper we develop a discrete counterpart to this story by replacing the real numbers by a finite field of odd characteristic. This produces a finite Gelfand pair whose spherical functions are computed explicitly. Our formulae resemble classical Gauss sums.
منابع مشابه
Translation invariant surfaces in the 3-dimensional Heisenberg group
In this paper, we study translation invariant surfaces in the 3-dimensional Heisenberg group $rm Nil_3$. In particular, we completely classify translation invariant surfaces in $rm Nil_3$ whose position vector $x$ satisfies the equation $Delta x = Ax$, where $Delta$ is the Laplacian operator of the surface and $A$ is a $3 times 3$-real matrix.
متن کاملCombinatorics and Spherical Functions on the Heisenberg Group
Let V be a finite dimensional Hermitian vector space and K be a compact Lie subgroup of U(V ) for which the representation of K on C[V ] is multiplicity free. One obtains a canonical basis {pα} for the space C[VR] of K-invariant polynomials on VR and also a basis {qα} via orthogonalization of the pα’s. The polynomial pα yields the homogeneous component of highest degree in qα. The coefficients ...
متن کاملOn the Mark and Markaracter Tables of Finite Groups
Let G be a finite group and C(G) be the family of representative conjugacy classes of subgroups of G. The matrix whose H,K-entry is the number of fixed points of the set G/K under the action of H is called the table of marks of G where H,K run through all elements in C(G). Shinsaku Fujita for the first time introduced the term “markaracter” to discuss marks for permutation representati...
متن کاملA Study of Electromagnetic Radiation from Monopole Antennas on Spherical-Lossy Earth Using the Finite-Difference Time-Domain Method
Radiation from monopole antennas on spherical-lossy earth is analyzed by the finitedifference time-domain (FDTD) method in spherical coordinates. A novel generalized perfectly matched layer (PML) has been developed for the truncation of the lossy soil. For having an accurate modeling with less memory requirements, an efficient "non-uniform" mesh generation scheme is used. Also in each time step...
متن کاملGeometric Models for the Spectra of Certain Gelfand Pairs Associated with Heisenberg Groups
Let K be a compact Lie group acting on a finite dimensional Hermitian vector space V via some unitary representation. Now K acts by automorphisms on the associated Heisenberg group HV = V × R and we say that (K,HV ) is a Gelfand pair when the algebra LK(HV ) of integrable K-invariant functions on HV commutes under convolution. In this situation an application of the Orbit Method yields a inject...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008