I assume you want to get the to the page URL params within Java code and you are overriding one of the executeImpl() methods in a sub-class?
You probably shouldn't cast the webscript request to LocalWebScriptRequest as that assumes a lot...
You have a couple of options though, the "clean" way is to "bind" in the url params into the component webscript url, then the params will be automatically made available to the "args" map in the model for your webscript. So in your webscript descriptor, you do something like this:
<shortname>Search Results List</shortname>
<description>Search: Results List Component</description>
If you *don't* know the name of the param and so can't bind it onto the WebScript component URL or you don't want to do it this way, then you can access the current RequestContext via the ThreadLocal accessor:
You can then call:
RequestContext context = ThreadLocalRequestContext.getRequestContext();
to retrieve any page URL params.
Hope this helps,