Kirillov的character公式、全纯Peter-Weyl定理和Blattner-Kostant-Sternberg配对
Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing
摘要：
设K是紧李群，具有双不变黎曼度量. K的复化G继承了一个Kaehler结构，其势能是度量的动能的两倍，左平移和右平移将G上平方可积全纯函数的希尔伯特空间相对于适当的测度转化为酉（KxK）表示。我们证实了彼得的说法-这个希尔伯特空间的Weyl定理，其效果是这个希尔伯特空间包含G上的代表函数的向量空间作为稠密子空间，并且其傅里叶系数的全纯函数的赋值产生Hilbert代数从G上的卷积代数到由K的不可约表示的自同态代数产生的代数的同构，希尔伯特空间和的适当运算。其结果是（i）全纯Plancherel定理和存在唯一确定的酉同构之间的空间的平方可积函数的K和希尔伯特空间的全纯函数在G上，和（ii）证明这个同构符合相应的Blattner-Kostant-Sternberg配对映射，乘以一个适当的常数。然后我们证明了G上的全纯函数的Hilbert空间上的能量算子的谱分解与K上的度量相关联，以通常的方式精化这个希尔伯特空间的Peter-Weyl分解，从而得到这个希尔伯特空间分解为不可约的同构（KxK）-表示。我们的重要工具之一是基里洛夫的性格公式。我们的方法是几何的和独立的热核，这是使用B。C. Hall获得了许多这样的结果[Journal of泛函分析122（1994），103- 151]，[Comm.in Math. Physics 226（2002），233- 268]。
英文摘要：
Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitable measure into a unitary (KxK)-representation. We establish the statement of the Peter-Weyl theorem for this Hilbert space to the effect that this Hilbert space contains the vector space of representative functions on G as a dense subspace and that the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra on G onto the algebra which arises from the endomorphism algebras of the irreducible representations of K by the appropriate operation of Hilbert space sum. Consequences are (i) a holomorphic Plancherel theorem and the existence of a uniquely determined unitary isomorphism between the space of square-integrable functions on K and the Hilbert space of holomorphic functions on G, and (ii) a proof that this isomorphism coincides with the corresponding Blattner-Kostant-Sternberg pairing map, multiplied by a suitable constant. We then show that the spectral decomposition of the energy operator on the Hilbert space of holomorphic functions on G associated with the metric on K refines to the Peter-Weyl decomposition of this Hilbert space in the usual manner and thus yields the decomposition of this Hilbert space into irreducible isotypical (KxK)-representations. Among our crucial tools is Kirillov's character formula. Our methods are geometric and independent of heat kernels, which are used by B. C. Hall to obtain many of these results [Journal of Functional Analysis 122 (1994), 103--151], [Comm. in Math. Physics 226 (2002), 233--268].
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