Fundamental groups and finte groups 歇口气
Oct 17

SL(2,C) 的表示

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Let a general element of SL_2\mathbb C be g= \left(\begin{array}{c}a \ b \\ c \ d \end{array}\right) .

Full principal series (indexed by k\in\mathbb Z, \ w\in\mathbb C):

(T_g^{k, w} f) (z) = |-bz+d|^{-2- w} \left(\displaystyle\frac{-bz+d}{|-bz+d|}\right)^{-k} f\left(\displaystyle\frac{az-c}{-bz+d}\right)

where f\in L^2\Big(\mathbb C,\ \frac{\mathrm{i}}{2}(1+|z|^2)^{\mathrm{Re}(w)}dz d\bar{z}\Big)

It includes two unitary series:

Unitary principal series: when w= \mathrm{i} v for some v\in\mathbb R . In this series T^{k, \mathrm{i}v} \cong T^{-k, -\mathrm{i}v} as unitary representations.

Complementary series: when k=0, \ 0<w<2, the above representatation becomes unitary w.r.t. the inner product

(f,h) = \left(\frac{\mathrm{i}}{2}\right)^2\displaystyle\int_\mathbb C \displaystyle\int_\mathbb C \displaystyle\frac{f(z)\overline{h(\zeta)}}{|z-\zeta|^{2-w}} dzd\bar{z}d\zeta d\bar{\zeta}

The trivial representation, the unitary principal series, and the complementary series are the only irreducible unitary representations of SL_2\mathbb C.

3 Responses to “SL(2,C) 的表示”

  1. xzf123 Says:
    十月 20th, 2007 at 7:00 上午

    SL(2,C)的酉表示真tnnd复杂, 得慢慢琢磨.

  2. xzf123 Says:
    十月 20th, 2007 at 8:02 上午

    这些酉表示限制到SU(2)上如何分解?

  3. xiphoid Says:
    十月 20th, 2007 at 11:15 上午

    这个好像比较复杂, 我得琢磨几天. 找本表示论的书看吧, 看看你们图书馆有没有 Michael Taylor 写的 noncommutative harmonic analysis

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