First, let us formulate the theorem of Harish-Chandra in a little more precise manner: it is a priori obvious that the character of $V$ is well-defined as a distribution. Now the theorem says that this distribution is given by a locally $L^1$-function locally constant which is well defined and is locally constant on an open dense subset of $G$ (but there is no good way to define on the whole of $G$). For example for principal series the function will be well defined on the open subset of regular semi-simple elements. What you can prove is that if such an element $g$ is not split, then the value of the character of an unramified principal series representation at $g$ is just equal to $0$. If $g$ is split, then up to conjugacy it lies in the standard split torus and the character is equal to the Weyl group average of the original character $\chi:T\to {\mathbb C}^*$ from which the principal series representation was induced (up to the standard "$\rho$-shift").
|
2 | added 7 characters in body | ||
|
|
||||
|
|
1 |
|
||
|
First, let us formulate the theorem of Harish-Chandra in a little more precise manner: it is a priori obvious that the character of $V$ is well-defined as a distribution. Now the theorem says that this distribution is given by a locally $L^1$-function locally constant which is well defined on an open dense subset of $G$ (but there is no good way to define on the whole of $G$). For example for principal series the function will be well defined on the open subset of regular semi-simple elements. What you can prove is that if such an element $g$ is not split, then the value of the character of an unramified principal series representation at $g$ is just equal to $0$. If $g$ is split, then up to conjugacy it lies in the standard split torus and the character is equal to the Weyl group average of the original character $\chi:T\to {\mathbb C}^*$ from which the principal series representation was induced (up to the standard "$\rho$-shift"). |
||||
