Yan Zhi-da and Zhang Da-gan classified finite dimensional real representations of real semisimple Lie group in Ref. [1]. In this note, we shall discuss real orthogonal representations (infinite dimension), mainly usin...Yan Zhi-da and Zhang Da-gan classified finite dimensional real representations of real semisimple Lie group in Ref. [1]. In this note, we shall discuss real orthogonal representations (infinite dimension), mainly using lowest K-type theory that D. Vogan described in Ref. [2]. Let G be a connected real semisimple Lie group, K the maximal compact subgroup of G and , the Lie algebras of G, K respectively. Let V be a real Hilbert space展开更多
We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β...We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.展开更多
We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.
基金Project supported by the National Natural Science Foundation of China.
文摘Yan Zhi-da and Zhang Da-gan classified finite dimensional real representations of real semisimple Lie group in Ref. [1]. In this note, we shall discuss real orthogonal representations (infinite dimension), mainly using lowest K-type theory that D. Vogan described in Ref. [2]. Let G be a connected real semisimple Lie group, K the maximal compact subgroup of G and , the Lie algebras of G, K respectively. Let V be a real Hilbert space
文摘We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.
文摘We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.
