Explicit Hilbert spaces for certain unipotent representations
نویسنده
چکیده
Let G be the universal cover of the group of automorphisms of a symmetric tube domain and let P = L N be its Shilov boundary parabolic subgroup. This paper attaches an irreducible unitary representation of G to each of the (finitely many) L-orbits on n*. The Hilbert space of the representation consists of functions on the orbit which are square-integrable with respect to a certain L-equivariant measure. The representation remains irreducible when restricted to P, and descends to a quotient of G which is, at worst, the double cover of a linear group. If the L-orbit is not open (in l~*), the construction gives a unipotent representation of G.
منابع مشابه
Explicit Hilbert spaces for certain unipotent representations II
To each real semisimple Jordan algebra, the Tits-Koecher-Kantor theory associates a distinguished parabolic subgroup P = L N of a semisimple Lie group G. The groups P which arise in this manner are precisely those for which N is abelian, and P is conjugate to its opposite P. Each non-open L-orbit O on N∗ admits an L-equivariant measure dμ which is unique up to scalar multiple. By Mackey theory,...
متن کاملHilbert spaces for certain unipotent representations III
In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of ...
متن کاملOn a Class of Double Cosets in Reductive Algebraic Groups
We study a class of double coset spaces RA\G1 × G2/RC , where G1 and G2 are connected reductive algebraic groups, and RA and RC are certain spherical subgroups of G1×G2 obtained by “identifying” Levi factors of parabolic subgroups in G1 and G2. Such double cosets naturally appear in the symplectic leaf decompositions of Poisson homogeneous spaces of complex reductive groups with the Belavin–Dri...
متن کاملExplicit Methods for Hilbert Modular Forms
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.
متن کامل