More on D_\gamma « Tom’s Math Weblog

More on D_\gamma

By trdunlap2

Today and tomorrow I’ll post some pictures illustrating more about the $D_\gamma$ functions on my “towers”. If I remember correctly $PGL_n$ and $SL_n$ have the same Lie algebra, at least the same chamber weights (those that $\gamma$ are indexed over) so I think my diagrams below are good.

I was worried at first that the towers in the previous post are not “balanced” and so don’t represent elements of $SL_3$ . If we think of them as elements of $PGL_3$ , are $D_\gamma$ only defined modulo $3$ ?

To preview my analysis: $D_\gamma$ for $\gamma$ of level 1 describe a “pure” tower that covers the given one; $D_\gamma$ for $\gamma$ of level 2 describe when subtracted from the valuation of the determinant (which is zero for $SL$ )describes a “pure” tower that lives inside.

UPDATE:

In this picture is a $t\text{-invariant}$ subspace of $\mathscr{K}^3$ containing $\mathscr{O}^3$ and $(t^{-1},t^{-1},0)$ . Its the image of $\mathscr{O}^3$ under the matrix $\left( \begin{array}{ccc} t^{-1} & t^{-1} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$ .

The red outlines those standard coordinates contained in the subspace — this is the “inner” tower and is calculated from the $D_{w\cdot\Lambda_2}$ values. The blue outline is the “outer” tower and is calculated from the $D_{w\cdot\Lambda_1}$ values.

In this picture we have the $t\text{-invariant}$ space corresponding to the matrix $\left( \begin{array}{ccc} t^{-1} & t^{-1} & 1 \\ 0 & t & 0 \\ 0 & 0 & t \end{array}\right)$ .

The green lines indicate the generating vector $(t^{-1},t^{-1},1)$ . The black line represents $(1,1,0)$ which is also in the space. Again the red outlines the “inner” tower and the blue the “outer” tower.

This entry was posted on December 27, 2007 at 1:14 pm and is filed under Examples/exercises, Questions. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “More on D_\gamma”

trdunlap2 Says:
January 1, 2008 at 6:16 pm | Reply
Eh, I’m still trying to figure out the best way to post pictures here with labels.
trdunlap2 Says:
January 5, 2008 at 3:28 pm | Reply
To generalize then, $D_\gamma$ for $\gamma$ of level 1 describe the external tower whereas level $n-1$ describe the internal tower. Point for further investigation: what do the levels say about the tower e.g. in the case of $SL_4$ ?

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More on D_\gamma

2 Responses to “More on D_\gamma”

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