Timeline for How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?
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| May 3, 2016 at 12:33 | comment | added | Jianrong Li | I have another question. How to write the map $\mathbb{C}[G/U_-] \hookrightarrow \mathbb{C}[B]$ explicitly? For example, let $G = GL_2$. Then we have $\mathbb{C}[G/U_-] = \mathbb{C}[g_{11}, g_{21}, g_{11}g_{22} - g_{12} g_{21}]$ and $\mathbb{C}[B] = \mathbb{C}[b_{11}, b_{12}, b_{22}]$. What are the images of $g_{11}, g_{21}, g_{11}g_{22} - g_{12} g_{21}$? | |
| May 3, 2016 at 10:59 | comment | added | Friedrich Knop | First: $t'=t$. Second: $f$ is homogeneous of degree $\lambda$ if the coaction sends $f$ to $f\otimes\lambda$. | |
| May 3, 2016 at 8:40 | comment | added | Jianrong Li | thank you very much. Is the action of $T$ on $G/U^-$ given by the following formula $G/U^- \times T \to G/U^-$, $(gU^-, t) \mapsto gt'U^-$ ($t'$ satisfies $t'U^- = U^- t$)? Since we have an action $G/U^- \times T \to G/U^-$, there is a coaction $\mathbb{C}[G/U^-] \to \mathbb{C}[G/U^-] \otimes \mathbb{C}[T]$. What is the corresponding multigrading on $\mathbb{C}[G/U^-]$? | |
| May 3, 2016 at 8:06 | vote | accept | Jianrong Li | ||
| May 3, 2016 at 6:54 | history | answered | Friedrich Knop | CC BY-SA 3.0 |