Timeline for Formula for the Haar measure in the linear symplectic group
Current License: CC BY-SA 3.0
12 events
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| May 13, 2014 at 16:58 | comment | added | Robert Bryant | For the 'unit ball into larger ball' problem, you will most likely find a KAK decomposition more useful than a KAN decomposition. Using KAK, it should be easy. For the 'unit ball into cylinder', I'm not sure what would be the best strategy, but this will clearly be a more difficult problem. | |
| May 13, 2014 at 15:17 | history | edited | alvarezpaiva | CC BY-SA 3.0 |
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| May 13, 2014 at 15:13 | comment | added | alvarezpaiva | @JimHumphreys: thanks! That reference looks nice. | |
| May 13, 2014 at 15:10 | history | edited | alvarezpaiva | CC BY-SA 3.0 |
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| May 13, 2014 at 15:01 | comment | added | Jim Humphreys | P.S. It's also useful to keep in mind the short note by I.G. Macdonald on the volume of a compact Lie group, even though it doesn't go into details about the individual types: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002095793 | |
| May 13, 2014 at 14:26 | comment | added | Jim Humphreys | To add another reference, in the spirit of Weyl's integration formula, see $\S18$ of D. Bump's textbook Lie Groups and the classical group examples on p. 116 | |
| May 13, 2014 at 8:32 | comment | added | alvarezpaiva | @RobertBryant: thanks for the reference. | |
| May 12, 2014 at 21:03 | comment | added | Robert Bryant | Well, the general case of what you appear to want is treated in Chapter VIII ("Integration") of A. Knapp's "Lie Groups Beyond an Introduction". Particularly see Sections 3 and 4 of that Chapter. You should have no trouble specializing these formulae to the decomposition of $\mathrm{Sp}(n,\mathbb{R})$ that you want to use for your purposes. | |
| May 12, 2014 at 20:34 | answer | added | Carlo Beenakker | timeline score: 5 | |
| May 12, 2014 at 19:37 | comment | added | alvarezpaiva | @RobertBryant: I was hoping that the parametrization would be part of the answer or that the measure could be simply described in terms of the polar decomposition of symplectic matrices. | |
| May 12, 2014 at 19:03 | comment | added | Robert Bryant | Well, the obvious answer is to just wedge together a basis for the left-invariant $1$-forms, which will give you (up to a normalizing constant) the $(2n^2{+}n)$-form whose integral gives you the Haar measure. If you want something more 'explicit' than this, you need to specify which 'explicit' parametrization or factorization (such as KAN, etc.) of the group in question you want to use. | |
| May 12, 2014 at 18:38 | history | asked | alvarezpaiva | CC BY-SA 3.0 |